The Division of Applied Mathematics at Brown University is one of the most prominent departments at Brown, and is also one of the oldest and strongest of its type in the country. The Division of Applied Mathematics is a world renowned center of research activity in a wide spectrum of traditional and modern mathematics. It explores the connections between mathematics and its applications at both the research and educational levels. The principal areas of research activities are ordinary, functional, and partial differential equations: stochastic control theory; applied probability, statistics and stochastic systems theory; neuroscience and computational molecular biology; numerical analysis and scientific computation; and the mechanics of solids, materials science and fluids. The effort in virtually all research ranges from applied and algorithmic problems to the study of fundamental mathematical questions. The Division emphasizes applied mathematics as a unifying theme. To facilitate cooperation among faculty and students, some research programs are partly organized around interdepartmental research centers. These centers facilitate funding and cooperative research in order to maintain the highest level of research and education in the Division. It is this breadth and the discovery from mutual collaboration which marks the great strength and uniqueness of the Division of Applied Mathematics at Brown.

For additional information, please visit the department's website: https://www.brown.edu/academics/applied-mathematics/

**APMA 0070. Introduction to Applied Complex Variables**.

Applications of complex analysis that do not require calculus as a prerequisite. Topics include algebra of complex numbers, plane geometry by means of complex coordinates, complex exponentials, and logarithms and their relation to trigonometry, polynomials, and roots of polynomials, conformal mappings, rational functions and their applications, finite Fourier series and the FFT, iterations and fractals. Uses MATLAB, which has easy and comprehensive complex variable capabilities.

**APMA 0090. Introduction to Mathematical Modeling**.

We will explore issues of mathematical modeling and analysis. Five to six self-contained topics will be discussed and developed. The course will include seminars in which modeling issues are discussed, lectures to provide mathematical background, and computational experiments. Required mathematical background is knowledge of one-variable calculus, and no prior computing experience will be assumed.

**APMA 0100. Elementary Probability for Applications**.

This course serves as an introduction to probability and stochastic processes with applications to practical problems. It will cover basic probability and stochastic processes such as basic concepts of probability and conditional probability, simple random walk, Markov chains, continuous distributions, Brownian motion and option pricing. Enrollment limited to 19 first year students.

**APMA 0110. What’s the big deal with Data Science?**.

This seminar serves as a practical introduction to the interdisciplinary field of data science. Over the course of the semester, students will be exposed to the diversity of questions that data science can address by reading current scholarly works from leading researchers. Through hands-on labs and experiences, students will gain facility with computational and visualization techniques for uncovering meaning from large numerical and text-based data sets. Ultimately, students will gain fluency with data science vocabulary and ideas. There are no prerequisites for this course.

**APMA 0120. Mathematics of Finance**.

The current volatility in international financial markets makes it imparative for us to become competent in financial calculations early in our liberal arts and scientific career paths. This course is designed to prepare the student with those elements of mathematics of finance appropriate for the calculations necessary in financial transactions.

**APMA 0160. Introduction to Scientific Computing**.

For students in any discipline that may involve numerical computations. Includes instruction for programming in MATLAB. Applications discussed include solution of linear equations (with vectors and matrices) and nonlinear equations (by bisection, iteration, and Newton's method), interpolation, and curve-fitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. Prerequisite: MATH 0100 or its equivalent.

**APMA 0180. Modeling the World with Mathematics: An Introduction for Non-Mathematicians**.

Mathematics is the foundation of our technological society and most of its powerful ideas are quite accessible. This course will explain some of these using historical texts and Excel. Topics include the predictive power of 'differential equations' from the planets to epidemics, oscillations and music, chaotic systems, randomness and the atomic bomb. Prerequisite: some knowledge of calculus.

**APMA 0200. Introduction to Modelling**.

This course provides an introduction to the mathematical modeling of selected biological, chemical, engineering, and physical processes. The goal is to illustrate the typical way in which applied mathematicians approach practical applications, from understanding the underlying problem, creating a model, analyzing the model using mathematical techniques, and interpreting the findings in terms of the original problem. Single-variable calculus is the only requirement; all other techniques from differential equations, linear algebra, and numerical methods, to probability and statistics will be introduced in class. Prerequisites: Math 0100 or equivalent.

**APMA 0330. Methods of Applied Mathematics I, II**.

This course will cover mathematical techniques involving ordinary differential equations used in the analysis of physical, biological, and economic phenomena. The course emphasizes established methods and their applications rather than rigorous foundation. Topics include: first and second order differential equations, an introduction to numerical methods, series solutions, and Laplace transformations.

Fall | APMA0330 | S01 | 16978 | MWF | 9:00-9:50(01) | (S. Akopian) |

Fall | APMA0330 | C01 | 17856 | M | 8:00-8:50 | (S. Akopian) |

Fall | APMA0330 | C02 | 17857 | W | 8:00-8:50 | (S. Akopian) |

Fall | APMA0330 | C03 | 17858 | F | 8:00-8:50 | (S. Akopian) |

Fall | APMA0330 | C04 | 17859 | T | 2:30-3:50 | (S. Akopian) |

Fall | APMA0330 | C05 | 17860 | Th | 2:30-3:50 | (S. Akopian) |

Fall | APMA0330 | C06 | 17861 | T | 3:00-4:20 | (S. Akopian) |

Fall | APMA0330 | C07 | 17862 | Th | 3:00-4:20 | (S. Akopian) |

**APMA 0340. Methods of Applied Mathematics I, II**.

Mathematical techniques involving differential equations used in the analysis of physical, biological and economic phenomena. Emphasis on the use of established methods, rather than rigorous foundations. I: First and second order differential equations. II: Applications of linear algebra to systems of equations; numerical methods; nonlinear problems and stability; introduction to partial differential equations; introduction to statistics. Prerequisite: MATH 0100, 0170, 0180, 0190, 0200, or 0350, or advanced placement.

**APMA 0350. Applied Ordinary Differential Equations**.

This course provides a comprehensive introduction to ordinary differential equations and their applications. During the course, we will see how applied mathematicians use ordinary differential equations to solve practical applications, from understanding the underlying problem, creating a differential-equations model, solving the model using analytical, numerical, or qualitative methods, and interpreting the findings in terms of the original problem. We will also learn about the underlying rigorous theoretical foundations of differential equations. Format: lectures and problem-solving workshops.

Prerequisites: MATH 0100, MATH 0170, MATH 0180, MATH 0190, MATH 0200, MATH 0350 or advanced placement. MATH 0520 (can be taken concurrently).

Fall | APMA0350 | S01 | 16980 | TTh | 1:00-2:20(08) | (J. Mallet-Paret) |

**APMA 0360. Applied Partial Differential Equations I**.

This course builds on APMA 0350 which covers ordinary differential equations and systems involving a single independent variable. We will look at processes with two or more independent variables formulated as partial differential equations (PDE) using concepts from multivariable calculus. We will see how problems are described quantitatively as PDEs, how seemingly unrelated contexts can result in similar equations; and develop methods for solution using analytical, numerical or qualitative methods. Contexts include first order equations; the second order wave equation and problems involving diffusion processes; steady state balances for systems in two or three dimensions; together with insights from theory.

Fall | APMA0360 | S01 | 17023 | MWF | 1:00-1:50(06) | (B. Sandstede) |

Fall | APMA0360 | C03 | 17874 | M | 10:00-10:50 | (B. Sandstede) |

Fall | APMA0360 | C04 | 17875 | W | 10:00-10:50 | (B. Sandstede) |

Fall | APMA0360 | C05 | 17877 | F | 10:00-10:50 | (B. Sandstede) |

**APMA 0410. Mathematical Methods in the Brain Sciences**.

Basic mathematical methods commonly used in the neural and cognitive sciences. Topics include: introduction to probability and statistics, emphasizing hypothesis testing and modern nonparametric methods; introduction to differential equations and systems of differential equations, emphasizing qualitative behavior and simple phase-plane analysis. Examples from neuroscience, cognitive science, and other sciences. Prerequisite: MATH 0100 or equivalent.

**APMA 0650. Essential Statistics**.

A first course in probability and statistics emphasizing statistical reasoning and basic concepts. Topics include visual and numerical summaries of data, representative and non-representative samples, elementary discrete probability theory, the normal distribution, sampling variability, elementary statistical inference, measures of association. Examples and applications from the popular press and the life, social and physical sciences. No prerequisites.

**APMA 1070. Quantitative Models of Biological Systems**.

Quantitative dynamic models help understand problems in biology and there has been rapid progress in recent years. The course provides an introduction to the concepts and techniques, with applications to population dynamics, infectious diseases, enzyme kinetics, aspects of cellular biology. Additional topics covered will vary. Mathematical techniques will be discussed as they arise in the context of biological problems. Prerequisites: APMA 0330, 0340 or 0350, 0360, or written permission.

**APMA 1080. Inference in Genomics and Molecular Biology**.

Massive quantities of fundamental biological and geological sequence data have emerged. Goal of APMA1080 is to enable students to construct and apply probabilistic models to draw inferences from sequence data on problems novel to them. Statistical topics: Bayesian inferences; estimation; hypothesis testing and false discovery rates; statistical decision theory; change point algorithm; hidden Markov models; Kalman filters; and significances in high dimensions. Prerequisites: One of following APMA1650, APMA1655, MATH1610, CSCI1450; and one of the following AMPA0160, CSCI0040, CSCI0150, CSCI0170, CSCI0190, CLPS0950, waver for students with substantial computing experience and their acceptance of responsibility for their own computing.

Fall | APMA1080 | S01 | 16981 | TTh | 10:30-11:50(13) | (C. Lawrence) |

Fall | APMA1080 | C01 | 17879 | M | 2:00-2:50 | (C. Lawrence) |

Fall | APMA1080 | C02 | 17880 | Th | 9:00-10:20 | (C. Lawrence) |

**APMA 1150. Machine Learning for Scientific Modeling: Data-Driven Discovery of Differential Equations(MATH 1150)**.

Interested students must register for MATH 1150.

**APMA 1160. An Introduction to Numerical Optimization**.

This course provides a thorough introduction to numerical methods and algorithms for solving non-linear continuous optimization problems. A particular attention will be given to the mathematical underpinnings to understand the theoretical properties of the optimization problems and the algorithms designed to solve them. Topics will include: line search methods, trust-region methods, nonlinear conjugate gradient methods, an introduction to constrained optimization (Karush-Kuhn-Tucker conditions, mini-maximization, saddle-points of Lagrangians). Some applications in signal and image processing will be explored. Basic programming skills at the level of APMA 16 or CSCI 40 are assumed.

**APMA 1170. Introduction to Computational Linear Algebra**.

Focuses on fundamental algorithms in computational linear algebra with relevance to all science concentrators. Basic linear algebra and matrix decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical analysis of errors and convergence. Iterative methods and conjugate gradient techniques. Computation of eigenvalues and eigenvectors, and an introduction to least squares methods.

Fall | APMA1170 | S01 | 16982 | MWF | 10:00-10:50(14) | (J. Darbon) |

Fall | APMA1170 | C01 | 17881 | T | 9:00-10:20 | (J. Darbon) |

Fall | APMA1170 | C02 | 17882 | Th | 9:00-10:20 | (J. Darbon) |

**APMA 1180. Introduction to Numerical Solution of Differential Equations**.

Fundamental numerical techniques for solving ordinary and partial differential equations. Overview of techniques for approximation and integration of functions. Development of mult-istep and multi-stage methods, error analysis, step-size control for ordinary differential equations. Solution of two-point boundary value problems, introduction to methods for solving linear partial differential equations. Students will be required to use Matlab (or other computer languages) to implement the mathematical algorithms under consideration: experience with a programming language is therefore strongly recommended. Prerequisites: APMA 0330, 0340 or 0350, 0360.

**APMA 1190. Finite Volume Method for CFD: A Survey**.

This course will provide students with an overview of the subjects necessary to perform robust simulations of computational fluid dynamics (CFD) problems. After an initial overview of the finite volume method and fluid mechanics, students will use the finite volume library OpenFOAM to explore the different components that make up a modern CFD code (discretization, linear algebra, timestepping, boundary conditions, splitting schemes, and multiphysics) and learn how to navigate a production scale software library.

**APMA 1200. Operations Research: Probabilistic Models**.

Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birth-death processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Applications. Prerequisite: APMA 1650, 1655 or MATH 1610, or equivalent.

**APMA 1210. Operations Research: Deterministic Models**.

An introduction to the basic mathematical ideas and computational methods of optimizing allocation of effort or resources, with or without constraints. Linear programming, network models, dynamic programming, and integer programming.

Fall | APMA1210 | S01 | 16993 | TTh | 9:00-10:20(02) | (C. Klivans) |

**APMA 1250. Advanced Engineering Mechanics (ENGN 1370)**.

Interested students must register for ENGN 1370.

**APMA 1260. Introduction to the Mechanics of Solids and Fluids**.

An introduction to the dynamics of fluid flow and deforming elastic solids for students in the physical or mathematical sciences. Topics in fluid mechanics include statics, simple viscous flows, inviscid flows, potential flow, linear water waves, and acoustics. Topics in solid mechanics include elastic/plastic deformation, strain and stress, simple elastostatics, and elastic waves with reference to seismology. Offered in alternate years.

**APMA 1330. Methods of Applied Mathematics**.

Review of vector calculus and curvilinear coordinates. Partial differential equations. Heat conduction and diffusion equations, the wave equation, Laplace and Poisson equations. Separation of variables, special functions, Fourier series and power series solution of differential equations. Sturm-Liouville problem and eigenfunction expansions.

Fall | APMA1330 | S01 | 16983 | MWF | 1:00-1:50(06) | (H. Dong) |

**APMA 1340. Methods of Applied Mathematics III, IV**.

See Methods Of Applied Mathematics III, IV (APMA 1330) for course description.

**APMA 1360. Applied Dynamical Systems**.

This course gives an overview of the theory and applications of dynamical systems modeled by differential equations and maps. We will discuss changes of the dynamics when parameters are varied, investigate periodic and homoclinic solutions that arise in applications, and study the impact of additional structures such as time reversibility and conserved quantities on the dynamics. We will also study systems with complicated "chaotic" dynamics that possess attracting sets which do not have an integer dimension. Applications to chemical reactions, climate, epidemiology, and phase transitions will be discussed.

**APMA 1650. Statistical Inference I**.

APMA 1650 is an integrated first course in mathematical statistics. The first half of APMA 1650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing.

Prerequisite: One year of university-level calculus. At Brown, this corresponds to MATH 0100, MATH 0170, MATH 0180, MATH 0190, MATH 0200, or MATH 0350. A score of 4 or 5 on the AP Calculus BC exam is also sufficient.

Fall | APMA1650 | S01 | 17022 | TTh | 1:00-2:20(08) | (N. Kaihnsa) |

Fall | APMA1650 | C01 | 17896 | M | 1:00-1:50 | (N. Kaihnsa) |

Fall | APMA1650 | C02 | 17897 | W | 1:00-1:50 | (N. Kaihnsa) |

Fall | APMA1650 | C03 | 17898 | F | 1:00-1:50 | (N. Kaihnsa) |

Fall | APMA1650 | C04 | 17899 | M | 2:00-2:50 | (N. Kaihnsa) |

Fall | APMA1650 | C05 | 17900 | W | 2:00-2:50 | (N. Kaihnsa) |

Fall | APMA1650 | C06 | 17901 | F | 2:00-2:50 | (N. Kaihnsa) |

Fall | APMA1650 | C07 | 17902 | T | 10:30-11:50 | (N. Kaihnsa) |

Fall | APMA1650 | C08 | 17903 | Th | 10:30-11:50 | (N. Kaihnsa) |

Fall | APMA1650 | C09 | 17904 | T | 9:00-10:20 | (N. Kaihnsa) |

Fall | APMA1650 | C10 | 17905 | Th | 9:00-10:20 | (N. Kaihnsa) |

Fall | APMA1650 | C11 | 17906 | W | 3:00-5:30 | (N. Kaihnsa) |

Fall | APMA1650 | C12 | 17907 | Th | 4:00-6:30 | (N. Kaihnsa) |

Fall | APMA1650 | C13 | 17908 | M | 3:00-5:30 | (N. Kaihnsa) |

**APMA 1655. Statistical Inference I**.

Students may opt to enroll in 1655 for more in depth coverage of APMA 1650. Enrollment in 1655 will include an optional recitation section and required additional individual work. Applied Math concentrators are encouraged to take 1655.

Prerequisite (for either version): MATH 0100, 0170, 0180, 0190, 0200, or 0350.

Fall | APMA1655 | S01 | 16984 | MWF | 11:00-11:50(16) | (Y. Shin) |

Fall | APMA1655 | C01 | 17892 | M | 12:00-12:50 | (Y. Shin) |

Fall | APMA1655 | C02 | 17893 | T | 4:00-6:30 | (Y. Shin) |

Fall | APMA1655 | C03 | 17894 | W | 12:00-12:50 | (Y. Shin) |

Fall | APMA1655 | C04 | 17895 | F | 12:00-12:50 | (Y. Shin) |

**APMA 1660. Statistical Inference II**.

APMA 1660 is designed as a sequel to APMA 1650 to form one of the alternative tracks for an integrated year's course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihood-ratio tests, nonparametric tests, introduction to statistical computing, matrix approach to simple-linear and multiple regression, analysis of variance, and design of experiments. Prerequisite: APMA 1650, 1655 or equivalent, basic linear algebra.

**APMA 1670. Statistical Analysis of Time Series**.

Time series analysis is an important branch of mathematical statistics with many applications to signal processing, econometrics, geology, etc. The course emphasizes methods for analysis in the frequency domain, in particular, estimation of the spectrum of time-series, but time domain methods are also covered. Prerequisites: elementary probability and statistics on the level of APMA 1650-1660.

**APMA 1680. Nonparametric Statistics**.

A systematic treatment of distribution-free alternatives to classical statistical tests. These nonparametric tests make minimum assumptions about distributions governing the generation of observations, yet are of nearly equal power to the classical alternatives. Prerequisite: APMA 1650 or equivalent. Offered in alternate years.

**APMA 1681. Computational Neuroscience (NEUR 1680)**.

Interested students must register for NEUR 1680.

**APMA 1690. Computational Probability and Statistics**.

Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, dimensionality reduction. Prerequistes: A calculus-based course in probability or statistics (e.g. APMA1650 or MATH1610) is required, and some programming experience is strongly recommended. Prerequisite: MATH 0100, 0170, 0180, 0190, 0200, or 0350, or equivalent placement.

Fall | APMA1690 | S01 | 16994 | MWF | 2:00-2:50(10) | (S. Geman) |

Fall | APMA1690 | C01 | 17885 | M | 10:00-10:50 | (S. Geman) |

Fall | APMA1690 | C02 | 17886 | W | 10:00-10:50 | (S. Geman) |

Fall | APMA1690 | C03 | 17887 | F | 10:00-10:50 | (S. Geman) |

Fall | APMA1690 | C04 | 17890 | T | 4:00-6:30 | (S. Geman) |

Fall | APMA1690 | C05 | 17891 | Th | 4:00-6:30 | (S. Geman) |

**APMA 1700. The Mathematics of Insurance**.

The course consists of two parts: the first treats life contingencies, i.e. the construction of models for individual life insurance contracts. The second treats the Collective Theory of Risk, which constructs mathematical models for the insurance company and its portfolio of policies as a whole. Suitable also for students proceeding to the Institute of Actuaries examinations. Prerequisites: Probability Theory to the level of APMA 1650 or MATH 1610.

**APMA 1710. Information Theory**.

Information theory is the study of the fundamental limits of information transmission and storage. This course, intended primarily for advanced undergraduates and beginning graduate students, offers a broad introduction to information theory and its applications: Entropy and information, lossless data compression, communication in the presence of noise, channel capacity, channel coding, source-channel separation, lossy data compression. Prerequisite: one course in probability.

**APMA 1720. Monte Carlo Simulation with Applications to Finance**.

The course will cover the basics of Monte Carlo and its applications to financial engineering: generating random variables and simulating stochastic processes; analysis of simulated data; variance reduction techniques; binomial trees and option pricing; Black-Scholes formula; portfolio optimization; interest rate models. The course will use MATLAB as the standard simulation tool. Prerequisites: APMA 1650 or MATH 1610

**APMA 1740. Recent Applications of Probability and Statistics**.

This course develops the mathematical foundations of modern applications of statistics to the computational, cognitive, engineering, and neural sciences. The course is rigorous, but the emphasis is on application. Topics include: Gibbs ensembles and their relation to maximum entropy, large deviations, exponential models, and information theory; statistical estimation and the generative, discriminative and algorithmic approaches to classification; graphical models, dynamic programming, MCMC computing, parameter estimation, and the EM algorithm. For 2,000-level credit enroll in 2610; for 1,000-level credit enroll in 1740. Rigorous calculus-based statistics, programming experience, and strong mathematical background are essential. For 2610, some graduate level analysis is strongly suggested.

**APMA 1850. Introdution to High Performance Parallel Computing**.

No description available.

**APMA 1860. Graphs and Networks**.

Selected topics about the mathematics of graphs and networks with an emphasis on random graph models and the dynamics of processes operating on these graphs. Topics include: empirical properties of biological, social, and technological networks (small-world effects, scale-free properties, transitivity, community structure); mathematical and statistical models of random graphs and their properties (Bernoulli random graphs, preferential attachment models, stochastic block models, phase transitions); dynamical processes on graphs and networks (percolation, cascades, epidemics, queuing, synchronization). Prereq: APMA 0360 and MATH 0520 and either APMA 1650 or MATH 1610, or equivalents of these, and programming experience. APMA 1200 or APMA 1690 or similar courses recommended.

**APMA 1880. Advanced Matrix Theory**.

Canonical forms of orthogonal, Hermitian and normal matrices: Rayleigh quotients. Norms, eigenvalues, matrix equations, generalized inverses. Banded, sparse, non-negative and circulant matrices. Prerequisite: APMA 0340 or 0360, or MATH 0520 or 0540, or permission of the instructor.

**APMA 1910. Race and Gender in the Scientific Community**.

This course examines the (1) disparities in representation in the scientific community, (2) issues facing different groups in the sciences, and (3) paths towards a more inclusive scientific environment. We will delve into the current statistics on racial and gender demographics in the sciences and explore their background through texts dealing with the history, philosophy, and sociology of science. We will also explore the specific problems faced by underrepresented and well-represented racial minorities, women, and LGBTQ community members. The course is reading intensive and discussion based. To be added to the waitlist for this course, please go to https://goo.gl/forms/foK0fyGxm5Eu2irA2

**APMA 1911. Race and Gender in the Scientific Community (MATH 1910)**.

Interested students must register for MATH 1910.

**APMA 1930A. Actuarial Mathematics**.

A seminar considering selected topics from two fields: (1) life contingencies-the study of the valuation of life insurance contracts; and (2) collective risk theory, which is concerned with the random process that generates claims for a portfolio of policies. Topics are chosen from *Actuarial Mathematics*, 2nd ed., by Bowers, Gerber, Hickman, Jones, and Nesbitt. Prerequisite: knowledge of probability theory to the level of APMA 1650 or MATH 1610. Particularly appropriate for students planning to take the examinations of the Society of Actuaries.

**APMA 1930B. Computational Probability and Statistics**.

Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from: random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, probabilistic grammars.

**APMA 1930C. Information Theory**.

Information theory is the mathematical study of the fundamental limits of information transmission (or coding) and storage (or compression). This course offers a broad introduction to information theory and its real-world applications. A subset of the following is covered: entropy and information; the asymptotic equipartition property; theoretical limits of lossless data compression and practical algorithms; communication in the presence of noise-channel coding, channel capacity; source-channel separation; Gaussian channels; Lossy data compression.

**APMA 1930D. Mixing and Transport in Dynamical Systems**.

Mixing and transport are important in several areas of applied science, including fluid mechanics, atmospheric science, chemistry, and particle dynamics. In many cases, mixing seems highly complicated and unpredictable. We use the modern theory of dynamical systems to understand and predict mixing and transport from the differential equations describing the physical process in question. Prerequisites: APMA 0330, 0340; or APMA 0350, 0360.

**APMA 1930E. Ocean Dynamics**.

Works through the popular book by Henry Stommel entitled *A View of the Sea.* Introduces the appropriate mathematics to match the physical concepts introduced in the book.

**APMA 1930G. The Mathematics of Sports**.

Topics to be discussed will range from the determination of who won the match, through biomechanics, free-fall of flexible bodies and aerodynamics, to the flight of ski jumpers and similar unnatural phenomena. Prerequisite: APMA 0340 or equivalent, or permission of the instructor.

**APMA 1930H. Scaling and Self-Similarity**.

The themes of scaling and self-similarity provide the simplest, and yet the most fruitful description of complicated forms in nature such as the branching of trees, the structure of human lungs, rugged natural landscapes, and turbulent fluid flows. This seminar is an investigation of some of these phenomena in a self-contained setting requiring little more mathematical background than high school algebra.

Topics to be covered: Dimensional analysis; empirical laws in biology, geosciences, and physics and the interplay between scaling and function; an introduction to fractals; social networks and the "small world" phenomenon.

**APMA 1930I. Random Matrix Theory**.

In the past few years, random matrices have become extremely important in a variety of fields such as computer science, physics and statistics. They are also of basic importance in various areas of mathematics. This class will serve as an introduction to this area. The focus is on the basic matrix ensembles and their limiting distributions, but several applications will be considered. Prerequisites: MATH 0200 or 0350; and MATH 0520 or 0540; and APMA 0350, 0360, 1650, and 1660. APMA 1170 and MATH 1010 are recommended, but not required.

**APMA 1930J. Mathematics of Random Networks**.

An intro to the emerging field of random networks and a glimpse of some of the latest developments. Random networks arise in a variety of applications including statistics, communications, physics, biology and social networks. They are studied using methods from a variety of disciplines ranging from probability, graph theory and statistical physics to nonlinear dynamical systems. Describes elements of these theories and shows how they can be used to gain practical insight into various aspects of these networks including their structure, design, distributed control and self-organizing properties. Prerequisites: Advanced calculus, basic knowledge of probability. Enrollment limited to 40.

**APMA 1930M. Applied Asymptotic Analysis**.

Many problems in applied mathematics and physics are nonlinear and are intractable to solve using elementary methods. In this course we will systematically develop techniques for obtaining quantitative information from nonlinear systems by exploiting small scale parameters. Topics will include: regular and singular perturbations, boundary layer theory, multiscale and averaging methods and asymptotic expansions of integrals. Along the way, we will discuss many applications including nonlinear waves, coupled oscillators, nonlinear optics, fluid dynamics and pattern formation.

**APMA 1930P. Mathematics and Climate**.

The study of Earth’s climate involves many scientific components; mathematical tools play an important role in relating these through quantitative models, computational experiments and data analysis. The course aims to introduce students in applied mathematics to several of the conceptual models, the underlying physical principles and some of the ways data is analyzed and incorporated. Students will develop individual projects later in the semester. Prerequisites: APMA 0360, or APMA 0340, or written permission; APMA 1650 is recommended.

**APMA 1930U. Introduction to Stochastic Differential Equations**.

This seminar course serves as an introduction to stochastic differential equations at the senior undergraduate level. Topics covered include Brownian motion and white noise, stochastic integrals, the Itô calculus, existence and uniqueness of solutions to Itô stochastic differential equations, and the Feynman-Kac formula. More advanced topics, such as fractional Brownian motion, Lévy processes, and stochastic control theory, may be addressed depending on the interests of the class and time restrictions.

Fall | APMA1930U | S01 | 17622 | TTh | 2:30-3:50(12) | (A. Matzavinos) |

**APMA 1940A. Coding and Information Theory**.

In a host of applications, from satellite communication to compact disc technology, the storage, retrieval, and transmission of digital data relies upon the theory of coding and information for efficient and error-free performance. This course is about choosing representations that minimize the amount of data (compression) and the probability of an error in data handling (error-correcting codes). Prerequisite: A knowledge of basic probability theory at the level of APMA 1650 or MATH 1610.

**APMA 1940B. Information and Coding Theory**.

Originally developed by C.E. Shannon in the 1940s for describing bounds on information rates across telecommunication channels, information and coding theory is now employed in a large number of disciplines for modeling and analysis of problems that are statistical in nature. This course provides a general introduction to the field. Main topics include entropy, error correcting codes, source coding, data compression. Of special interest will be the connection to problems in pattern recognition. Includes a number of projects relevant to neuroscience, cognitive and linguistic sciences, and computer vision. Prerequisites: High school algebra, calculus. MATLAB or other computer experience helpful. Prior exposure to probability theory/statistics helpful.

**APMA 1940C. Introduction to Mathematics of Fluids**.

Equations that arise from the description of fluid motion are born in physics, yet are interesting from a more mathematical point of view as well. Selected topics from fluid dynamics introduce various problems and techniques in the analysis of partial differential equations. Possible topics include stability, existence and uniqueness of solutions, variational problems, and active scalar equations. No prior knowledge of fluid dynamics is necessary.

**APMA 1940D. Iterative Methods**.

Large, sparse systems of equations arise in many areas of mathematical application and in this course we explore the popular numerical solution techniques being used to efficiently solve these problems. Throughout the course we will study preconditioning strategies, Krylov subspace acceleration methods, and other projection methods. In particular, we will develop a working knowledge of the Conjugate Gradient and Minimum Residual (and Generalized Minimum Residual) algorithms. Multigrid and Domain Decomposition Methods will also be studied as well as parallel implementation, if time permits.

**APMA 1940E. Mathematical Biology**.

This course is designed for undergraduate students in mathematics who have an interest in the life sciences. No biological experience is necessary, as we begin by a review of the relevant topics. We then examine a number of case studies where mathematical tools have been successfully applied to biological systems. Mathematical subjects include differential equations, topology and geometry.

**APMA 1940F. Mathematics of Physical Plasmas**.

Plasmas can be big, as in the solar wind, or small, as in fluorescent bulbs. Both kinds are described by the same mathematics. Similar mathematics describes semiconducting materials, the movement of galaxies, and the re-entry of satellites. We consider how all of these physical systems are described by certain partial differential equations. Then we invoke the power of mathematics. The course is primarily mathematical. Prerequisites: APMA 0340 or 0360, MATH 0180 or 0200 or 0350, and PHYS 0060 or PHYS 0080 or ENGN 0510.

**APMA 1940G. Multigrid Methods**.

Mulitgrid methods are a very active area of research in Applied Mathematics. An introduction to these techniques will expose the student to cutting-edge mathematics and perhaps pique further interest in the field of scientific computation.

**APMA 1940H. Numerical Linear Algebra**.

This course will deal with advanced concepts in numerical linear algebra. Among the topics covered: Singular Value Decompositions (SVD) QR factorization, Conditioning and Stability and Iterative Methods.

**APMA 1940I. The Mathematics of Finance**.

The mathematics of speculation as reflected in the securities and commodities markets. Particular emphasis placed on the evaluation of risk and its role in decision-making under uncertainty. Prerequisite: basic probability.

**APMA 1940J. The Mathematics of Speculation**.

The course will deal with the mathematics of speculation as reflected in the securities and commodities markets. Particular emphasis will be placed on the evaluation of risk and its role in decision making under uncertainty. Prerequisite: basic probability.

**APMA 1940K. Fluid Dynamics and Physical Oceanography**.

Introduction to fluid dynamics as applied to the mathematical modeling and simulation of ocean dynamics and near-shore processes. Oceanography topics include: overview of atmospheric and thermal forcing of the oceans, ocean circulation, effects of topography and Earth's rotation, wind-driven currents in upper ocean, coastal upwelling, the Gulf Stream, tidal flows, wave propagation, tsunamis.

**APMA 1940L. Mathematical Models in Biophysics**.

Development mathematical descriptions of biological systems aid in understanding cell function and physiology. The course will explore a range of topics including: biomechanics of blood flow in arteries and capillaries, motile cells and chemotaxis, cell signaling and quorum sensing, and additional topics. Formulating and using numerical simulations will be a further component. Students will develop individual projects. Prerequisites: APMA 0360, or APMA 0340, or written permission.

**APMA 1940M. The History of Mathematics**.

The course will not be a systematic survey but will focus on specific topics in the history of mathematics such as Archimedes and integration. Oresme and graphing, Newton and infinitesimals, simple harmonic motion, the discovery of 'Fourier' series, the Monte Carlo method, reading and analyzing the original texts. A basic knowledge of calculus will be assumed.

**APMA 1940N. Introduction to Mathematical Models in Computational Biology**.

This course is designed to introduce students to the use of mathematical models in biology as well as some more recent topics in computational biology. Mathematical techniques will involve difference equations and dynamical systems theory, ordinary differential equations and some partial differential equations. These techinques will be applied in the study of many biological applications such as: (i) Difference Equations: population dynamics, red blood cell production, population genetics; (ii) Ordinary Differential Equations: predator/prey models, Lotka/Volterra model, modeling the evolution of the genome, heart beat model/cycle, tranmission dynamics of HIV and gonorrhea; (iii) Partial Differential Equations: tumor growth, modeling evolution of the genome, pattern formation. Prerequisites: APMA 0330 and 0340.

**APMA 1940O. Approaches to Problem Solving in Applied Mathematics**.

The aim of the course is to illustrate through the examination of unsolved (but elementary) problems the ways in which professional applied mathematicians approach the solution of such questions. Ideas considered include: choosing the "simplest" nontrivial example; generalization; and specification. Ways to think outside convention. Some knowledge of probability and linear algebra helpful.

Suggested reading.

"How to solve it", G. Polya

"Nonplussed", Julian Havil

**APMA 1940P. Biodynamics of Block Flow and Cell Locomotion**.

**APMA 1940Q. Filtering Theory**.

Filtering (estimation of a "state process" from noisy data) is an important area of modern statistics. It is of central importance in navigation, signal and image processing, control theory and other areas of engineering and science. Filtering is one of the exemplary areas where the application of modern mathematics and statistics leads to substantial advances in engineering. This course will provide a student with the working knowledge sufficient for cutting edge research in the field of nonlinear filtering and its practical applications. Topics will include: hidden Markov models, Kalman and Wiener filters, optimal nonlinear filtering, elements of Ito calculus and Wiener chaos, Zakai and Kushner equations, spectral separating filters and wavelet based filters, numerical implementation of filters. We will consider numerous applications of filtering to speech recognition, analysis of financial data, target tracking and image processing. No prior knowledge in the field is required but a good understanding of the basic Probability Theory (APMA1200 or APMA2630) is important.

**APMA 1940R. Linear and Nonlinear Waves**.

From sound and light waves to water waves and traffic jams, wave phenomena are everywhere around us. In this seminar, we will discuss linear and nonlinear waves as well as the propagation of wave packets. Among the tools we shall use and learn about are numerical simulations in Matlab and analytical techniques from ordinary and partial differential equations. We will also explore applications in nonlinear optics and to traffic flow problems. Prerequisites: MATH 0180 and either APMA 0330-0340 or APMA 0350-0360. No background in partial differential equations is required.

**APMA 1941B. Graphs and Networks**.

Title: Graphs and Networks Summary: Selected topics about the mathematics of graphs and networks with an emphasis on random graph models and the dynamics of processes operating on these graphs. Topics may include: empirical properties of biological, social, and technological networks (small-world effects, scale-free properties, transitivity, community structure); mathematical and statistical models of random graphs and their properties (Bernoulli random graphs, preferential attachment models, stochastic block models, phase transitions); dynamical processes on graphs and networks (percolation, cascades, epidemics, queuing, synchronization).

**APMA 1941C. Wavelets & Applications**.

Introduce you to: the relatively new and interdisciplinary area of wavelets; the efficient and elegant algorithms to which they give rise including the wavelet transform; and mathematical tools that can be used to gain a rigorous understanding of wavelets. We will also cover some of the applications of these tools including the compression of video streams, approximation of solution of PDE's, and signal analysis. Students will select one topic related to wavelets to pursue in more depth based on applying the ideas covered in the course. There will be final presentations for these projects and students will submit an extended summary.

**APMA 1970. Independent Study**.

Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course.

**APMA 1971. Independent Study - WRIT**.

Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course. This course should be taken in place of APMA 1970 if it is to be used to satisfy the WRIT requirement.

**APMA 2050. Mathematical Methods of Applied Science**.

Introduces science and engineering graduate students to a variety of fundamental mathematical methods. Topics include linear algebra, complex variables, Fourier series, Fourier and Laplace transforms and their applications, ordinary differential equations, tensors, curvilinear coordinates, partial differential equations, and calculus of variations.

**APMA 2060. Mathematical Methods of Applied Science**.

Introduces science and engineering graduate students to a variety of fundamental mathematical methods. Topics include linear algebra, complex variables, Fourier series, Fourier and Laplace transforms and their applications, ordinary differential equations, tensors, curvilinear coordinates, partial differential equations, and calculus of variations.

**APMA 2080. Inference in Genomics and Molecular Biology**.

Sequencing of genomes has generated a massive quantity of fundamental biological data. We focus on drawing traditional and Bayesian statistical inferences from these data, including: motif finding; hidden Markov models; other probabilistic models, significances in high dimensions; and functional genomics. Emphasis is on the application of probability theory to inferences on data sequence with the goal of enabling students to independently construct probabilistic models in setting novel to them. Statistical topics: Bayesian inference, estimation, hypothesis testing and false discovery rates, statistical decision theory. For 2,000-level credit enroll in 2080; for 1,000-level credit enroll in 1080.

Fall | APMA2080 | S01 | 17037 | TTh | 10:30-11:50(13) | (C. Lawrence) |

**APMA 2110. Real Analysis**.

Provides the basis of real analysis which is fundamental to many of the other courses in the program: metric spaces, measure theory, and the theory of integration and differentiation.

Fall | APMA2110 | S01 | 16987 | MWF | 10:00-10:50(14) | (H. Dong) |

**APMA 2110A. Real Function Theory (MATH 2210)**.

Interested students must register for MATH 2210.

**APMA 2120. Hilbert Spaces and Their Applications**.

A continuation of APMA 2110: metric spaces, Banach spaces, Hilbert spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential equations.

**APMA 2120A. Real Function Theory (MATH 2210)**.

Interested students must register for MATH 2220.

**APMA 2130. Methods of Applied Mathematics: Partial Differential Equations**.

Solution methods and basic theory for first and second order partial differential equations. Geometrical interpretation and solution of linear and nonlinear first order equations by characteristics; formation of caustics and propagation of discontinuities. Classification of second order equations and issues of well-posed problems. Green's functions and maximum principles for elliptic systems. Characteristic methods and discontinuous solutions for hyperbolic systems.

**APMA 2140. Methods of Applied Mathematics: Integral Equations**.

Integral equations. Fredholm and Volterra theory, expansions in orthogonal functions, theory of Hilbert-Schmidt. Singular integral equations, method of Wiener-Hopf. Calculus of variations and direct methods.

**APMA 2160. Methods of Applied Mathematics: Asymptotics**.

Calculus of asymptotic expansions, evaluation of integrals. Solution of linear ordinary differential equations in the complex plane, WKB method, special functions.

**APMA 2170. Functional Analysis and Applications**.

Topics vary according to interest of instructor and class.

**APMA 2190. Nonlinear Dynamical Systems: Theory and Applications**.

Basic theory of ordinary differential equations, flows, and maps. Two-dimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.

Fall | APMA2190 | S01 | 16988 | TTh | 9:00-10:20(02) | (G. Menon) |

**APMA 2200. Nonlinear Dynamical Systems: Theory and Applications**.

Basic theory of ordinary differential equations, flows, and maps. Two-dimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.

**APMA 2210. Topics in Nonlinear Dynamical Systems**.

Topics to be covered in this course may vary depending on the audiences. One of the goals that is planned for this course is to discuss the boundary layers and/or the boundary value problems that appear and play a very important role in the kinetic theory of gases; in particular, in the theory of the Boltzmann equations. Students are encouraged to attend and participate in the kinetic theory program offered by the ICERM institute in the Fall 2011 semester. This course may be taken twice for credit.

**APMA 2230. Partial Differential Equations**.

The theory of the classical partial differential equations, as well as the method of characteristics and general first order theory. Basic analytic tools include the Fourier transform, the theory of distributions, Sobolev spaces, and techniques of harmonic and functional analysis. More general linear and nonlinear elliptic, hyperbolic, and parabolic equations and properties of their solutions, with examples drawn from physics, differential geometry, and the applied sciences. Generally, semester II of this course concentrates in depth on several special topics chosen by the instructor.

**APMA 2240. Partial Differential Equations**.

The theory of the classical partial differential equations, as well as the method of characteristics and general first order theory. Basic analytic tools include the Fourier transform, the theory of distributions, Sobolev spaces, and techniques of harmonic and functional analysis. More general linear and nonlinear elliptic, hyperbolic, and parabolic equations and properties of their solutions, with examples drawn from physics, differential geometry, and the applied sciences. Generally, semester II of this course concentrates in depth on several special topics chosen by the instructor.

**APMA 2260. Introduction to Stochastic Control Theory**.

The course serves as an introduction to the theory of stochastic control and dynamic programming technique. Optimal stopping, total expected (discounted) cost problems, and long-run average cost problems will be discussed in discrete time setting. The last part of the course deals with continuous time determinstic control and game problems. The course requires some familiarity with the probability theory.

**APMA 2410. Fluid Dynamics I**.

Formulation of the basic conservation laws for a viscous, heat conducting, compressible fluid. Molecular basis for thermodynamic and transport properties. Kinematics of vorticity and its transport and diffusion. Introduction to potential flow theory. Viscous flow theory; the application of dimensional analysis and scaling to obtain low and high Reynolds number limits.

**APMA 2420. Fluid Mechanics II**.

Introduction to concepts basic to current fluid mechanics research: hydrodynamic stability, the concept of average fluid mechanics, introduction to turbulence and to multiphase flow, wave motion, and topics in inviscid and compressible flow.

**APMA 2450. Exchange Scholar Program**.

Fall | APMA2450 | S01 | 15446 | Arranged | 'To Be Arranged' |

**APMA 2470. Topics in Fluid Dynamics**.

Initial review of topics selected from flow stability, turbulence, turbulent mixing, surface tension effects, and thermal convection. Followed by focussed attention on the dynamics of dispersed two-phase flow and complex fluids.

**APMA 2480. Topics in Fluid Dynamics**.

No description available.

**APMA 2550. Numerical Solution of Partial Differential Equations I**.

Finite difference methods for solving time-dependent initial value problems of partial differential equations. Fundamental concepts of consistency, accuracy, stability and convergence of finite difference methods will be covered. Associated well-posedness theory for linear time-dependent PDEs will also be covered. Some knowledge of computer programming expected.

Fall | APMA2550 | S01 | 17129 | MW | 3:00-4:20(10) | (J. Guzman) |

**APMA 2560. Numerical Solution of Partial Differential Equations II**.

An introduction to weighted residual methods, specifically spectral, finite element and spectral element methods. Topics include a review of variational calculus, the Rayleigh-Ritz method, approximation properties of spectral end finite element methods, and solution techniques. Homework will include both theoretical and computational problems.

**APMA 2570A. Numerical Solution of Partial Differential Equations III**.

We will cover spectral methods for partial differential equations. Algorithm formulation, analysis, and efficient implementation issues will be addressed. Prerequisite: APMA 2550 or equivalent knowledge in numerical methods.

Fall | APMA2570A | S01 | 17130 | M | 3:00-5:30 | (M. Ainsworth) |

**APMA 2570B. Numerical Solution of Partial Differential Equations III**.

We will cover finite element methods for ordinary differential equations and for elliptic, parabolic and hyperbolic partial differential equations. Algorithm development, analysis, and computer implementation issues will be addressed. In particular, we will discuss in depth the discontinuous Galerkin finite element method. Prerequisite: APMA 2550 or equivalent knowledge in numerical methods.

**APMA 2580A. Computational Fluid Dynamics**.

The course will focus primarily on finite difference methods for viscous incompressible flows. Other topics will include multiscale methods, e.g. molecular dynamics, dissipative particle dynamics and lattice Boltzmann methods. We will start with the mathematical nature of the Navier-Stokes equations and their simplified models, learn about high-order explicit and implicit methods, time stepping, and fast solvers. We will then cover advection-diffusion equations and various forms of the Navier-Stokes equations in primitive variables and in vorticity/streamfunction formulations. In addition to the homeworks the students are required to develop a Navier-Stokes solver as a final project.

**APMA 2580B. Computational Fluid Dynamics for Compressible Flows**.

An introduction to computational fluid dynamics with emphasis on compressible flows. We will cover finite difference, finite volume and finite element methods for compressible Euler and Navier-Stokes equations and for general hyperbolic conservation laws. Background material in hyperbolic partial differential equations will also be covered. Algorithm development, analysis, implementation and application issues will be addressed. Prerequisite: APMA 2550 or equivalent knowledge in numerical methods.

**APMA 2610. Recent Applications of Probability and Statistics**.

This course develops the mathematical foundations of modern applications of statistics to the computational, cognitive, engineering, and neural sciences. The course is rigorous, but the emphasis is on application. Topics include: Gibbs ensembles and their relation to maximum entropy, large deviations, exponential models, and information theory; statistical estimation and the generative, discriminative and algorithmic approaches to classification; graphical models, dynamic programming, MCMC computing, parameter estimation, and the EM algorithm. For 2,000-level credit enroll in 2610; for 1,000-level credit enroll in 1740. Rigorous calculus-based statistics, programming experience, and strong mathematical background are essential. For 2610, some graduate level analysis is strongly suggested.

**APMA 2630. Theory of Probability**.

Part one of a two semester course that provides an introduction to probability theory based on measure theory. The first semester (APMA 2630) covers the following topics: countable state Markov chains, review of real analysis and metric spaces, probability spaces, random variables and measurable functions, Borel-Cantelli lemmas, weak and strong laws of large numbers, conditional expectation and beginning of discrete time martingale theory. Prerequisites—undergraduate probability and analysis, co-requisite—graduate real analysis.

Fall | APMA2630 | S01 | 16989 | TTh | 1:00-2:20(08) | (H. Wang) |

**APMA 2640. Theory of Probability II**.

Part two of a two semester course that provides an introduction to probability theory based on measure theory. Standard topics covered in the second-semester (APMA 2640) include the following: discrete time martingale theory, weak convergence (also called convergence in distribution) and the central limit theorem, and a study of Brownian motion. Optional topics include the ergodic theorem and large deviation theory. Prerequisites—undergraduate probability and analysis, co-requisite—graduate real analysis.

**APMA 2660. Stochastic Processes**.

Review of the theory of stochastic differential equations and reflected SDEs, and of the ergodic and stability theory of these processes. Introduction to the theory of weak convergence of probability measures and processes. Concentrates on applications to the probabilistic modeling, control, and approximation of modern communications and queuing networks; emphasizes the basic methods, which are fundamental tools throughout applications of probability.

**APMA 2670. Mathematical Statistics I**.

This course presents advanced statistical inference methods. Topics include: foundations of statistical inference and comparison of classical, Bayesian, and minimax approaches, point and set estimation, hypothesis testing, linear regression, linear classification and principal component analysis, MRF, consistency and asymptotic normality of Maximum Likelihood and estimators, statistical inference from noisy or degraded data, and computational methods (E-M Algorithm, Markov Chain Monte Carlo, Bootstrap). Prerequisite: APMA 2630 or equivalent.

Fall | APMA2670 | S01 | 17014 | Th | 4:00-6:30 | (B. Gidas) |

**APMA 2680. Mathematical Statistics II**.

The course covers modern nonparametric statistical methods. Topics include: density estimation, multiple regression, adaptive smoothing, cross-validation, bootstrap, classification and regression trees, nonlinear discriminant analysis, projection pursuit, the ACE algorithm for time series prediction, support vector machines, and neural networks. The course will provide the mathematical underpinnings, but it will also touch upon some applications in computer vision/speech recognition, and biological, neural, and cognitive sciences. Prerequisite: APMA 2670.

**APMA 2720. Information Theory**.

Information theory and its relationship with probability, statistics, and data compression. Entropy. The Shannon-McMillan-Breiman theorem. Shannon's source coding theorems. Statistical inference; hypothesis testing; model selection; the minimum description length principle. Information-theoretic proofs of limit theorems in probability: Law of large numbers, central limit theorem, large deviations, Markov chain convergence, Poisson approximation, Hewitt-Savage 0-1 law. Prerequisites: APMA 2630, 1710.

**APMA 2810A. Computational Biology**.

Provides an up-to-date presentation of the main problems and algorithms in bioinformatics. Emphasis is given to statistical/ probabilistic methods for various molecular biology tasks, including: comparison of genomes of different species, finding genes and motifs, understanding transcription control mechanisms, analyzing microarray data for gene clustering, and predicting RNA structure.

**APMA 2810B. Computational Molecular Biology**.

Provides an up-to-date presentation of problems and algorithms in bioinformatics, beginning with an introduction to biochemistry and molecular genetics. Topics include: proteins and nucleic acids, the genetic code, the central dogma, the genome, gene expression, metabolic transformations, and experimental methods (gel electrophoresis, X-ray crystallography, NMR). Also, algorithms for DNA sequence alignment, database search tools (BLAST), and DNA sequencing.

**APMA 2810C. Elements of High Performance Scientific Computing**.

No description available.

**APMA 2810D. Elements of High Performance Scientific Computing II**.

No description available.

**APMA 2810E. Far Field Boundary Conditions for Hyperbolic Equations**.

No description available.

**APMA 2810F. Introduction to Non-linear Optics**.

No description available.

**APMA 2810G. Large Deviations**.

No description available.

**APMA 2810H. Math of Finance**.

No description available.

**APMA 2810I. Mathematical Models and Numerical Analysis in Computational Quantum Chemistry**.

We shall present on some models in the quantum chemistry field (Thomas Fermi and related, Hartree Fock, Kohn Sham) the basic tools of functional analysis for the study of their solutions. Then some of the discretization methods and iterative algorithms to solve these problems will be presented and analyzed. Some of the open problems that flourish in this field will also be presented all along the lectures.

**APMA 2810J. Mathematical Techniques for Neural Modeling**.

No description available.

**APMA 2810K. Methods of Algebraic Geometry in Control Theory I**.

Develops the ideas of algebraic geometry in the context of control theory. The first semester examines scalar linear systems and affine algebraic geometry while the second semester addresses multivariable linear systems and projective algebraic geometry.

**APMA 2810L. Numerical Solution of Hyperbolic PDE's**.

No description available.

**APMA 2810M. Some Topics in Kinetic Theory**.

Nonlinear instabilities as well as boundary effects in a collisionless plasmas; Stable galaxy configurations; A nonlinear energy method in the Boltzmann theory will also be introduced. Self-contained solutions to specific concrete problems. Focus on ideas but not on technical aspects. Open problems and possible future research directions will then be discussed so that students can gain a broader perspective. Prerequisite: One semester of PDE (graduate level) is required.

**APMA 2810N. Topics in Nonlinear PDEs**.

Aspects of the theory on nonlinear evolution equations, which includes kinetic theory, nonlinear wave equations, variational problems, and dynamical stability.

**APMA 2810O. Stochastic Differential Equations**.

This course develops the theory and some applications of stochastic differential equations. Topics include: stochastic integral with respect to Brownian motion, existence and uniqueness for solutions of SDEs, Markov property of solutions, sample path properties, Girsanov's Theorem, weak existence and uniqueness, and connections with partial differential equations. Possible additional topics include stochastic stability, reflected diffusions, numerical approximation, and stochastic control. Prerequisite: APMA 2630 and 2640.

**APMA 2810P. Perturbation Methods**.

Basic concepts of asymptotic approximations with examples with examples such as evaluation of integrals and functions. Regular and singular perturbation problems for differential equations arising in fluid mechanics, wave propagation or nonlinear oscillators. Methods include matched asymptotic expansions and multiple scales. Methods and results will be discussed in the context of applications to physical problems.

**APMA 2810Q. Discontinous Galerkin Methods**.

In this seminar course we will cover the algorithm formulation, stability analysis and error estimates, and implementation and applications of discontinuous Galerkin finite element methods for solving hyperbolic conservation laws, convection diffusion equations, dispersive wave equations, and other linear and nonlinear partial differential equations. Prerequisite: APMA 2550.

**APMA 2810R. Computational Biology Methods for Gene/Protein Networks and Structural Proteomics**.

The course presents computational and statistical methods for gene and protein networks and structural proteomics; it emphasizes: (1) Probablistic models for gene regulatory networks via microarray, chromatin immunoprecipitation, and cis-regulatory data; (2) Signal transduction pathways via tandem mass spectrometry data; (3) Molecular Modeling forligand-receptor coupling and docking. The course is recommended for graduate students.

**APMA 2810S. Topics in Control**.

No description available.

**APMA 2810T. Nonlinear Partial Differential Equations**.

This course introduces techniques useful for solving many nonlinear partial differential equations, with emphasis on elliptic problems. PDE from a variety of applications will be discussed. Contact the instructor about prerequisites.

**APMA 2810U. Topics in Differnetial Equations**.

No description available.

**APMA 2810V. Topics in Partial Differential Equations**.

The course will cover an introduction of the L_p theory of second order elliptic and parabolic equations, finite difference approximations of elliptic and parabolic equations, and some recent developments in the Navier-Stokes equations and quasi-geostrophic equations. Some knowledge of real analysis will be expected.

**APMA 2810W. Advanced Topics in High Order Numerical Methods for Convection Dominated Problems**.

This is an advanced seminar course. We will cover several topics in high order numercial methods for convection dominated problems, including methods for solving Boltzman type equations, methods for solving unsteady and steady Hamilton-Jacobi equations, and methods for solving moment models in semi-conductor device simulations. Prerequisite: APMA 2550 or equivalent knowledge of numerical analysis.

**APMA 2810X. Introduction to the Theory of Large Deviations**.

The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, PDE, weak convergence, etc.), and basic examples (e.g., Sanov's and Cramer's Theorems). We then will cover the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control and Monte Carlo methods. Prerequisites: APMA 2630 and 2640.

**APMA 2810Y. Discrete high-D Inferences in Genomics**.

Genomics is revolutionizing biology and biomedicine and generated a mass of clearly relevant high-D data along with many important high-D discreet inference problems. Topics: special characteristics of discrete high-D inference including Bayesian posterior inference; point estimation; interval estimation; hypothesis tests; model selection; and statistical decision theory.

**APMA 2810Z. An Introduction to the Theory of Large Deviations**.

The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and 2640.

**APMA 2811A. Directed Methods in Control and System Theory**.

Various general techniques have been developed for control and system problems. Many of the methods are indirect. For example, control problems are reduced to a problem involving a differential equation (such as the partial differential equation of Dynamic Programming) or to a system of differential equations (such as the canonical system of the Maximum Principle). Since these indirect methods are not always effective alternative approaches are necessary. In particular, direct methods are of interest.

We deal with two general classes, namely: 1.) Integration Methods; and, 2.) Representation Methods. Integration methods deal with the integration of function space differential equations. Perhaps the most familiar is the so-called Gradient Method or curve of steepest descent approach. Representation methods utilize approximation in function spaces and include both deterministic and stochastic finite element methods. Our concentration will be on the theoretical development and less on specific numerical procedures. The material on representation methods for Levy processes is new.

**APMA 2811B. Computational Methods for Signaling Pathways and Protein Interactions**.

The course will provide presentation of the biology and mathematical models/algorithms for a variety of topics, including: (1) The analysis and interpretation of tandem mass spectrometry data for protein identification and determination of signaling pathways, (2) Identification of Phosphorylation sites and motifs and structural aspects of protein docking problems. Prerequisites: The course is recommended for graduate students. It will be self-contained; students will be able to fill in knowledge by reading material to be indicated by the instructor.

**APMA 2811C. Stochastic Partial Differential Equations**.

SPDEs is an interdisciplinary area at the crossroads of stochastic processes and partial differential equations (PDEs). The topics of the course include: geneses of SPDEs in real life applications, mathematical foundations and analysis of SPDEs, numerical and computational aspects of SPDEs, applications of SPDEs to fluid dynamics, population biology, hidden Markov models, etc. Prerequisites: familiarity with stochastic calculus and PDEs (graduate level).

**APMA 2811D. Asymptotic Problems For Differential Equations And Stochastic Processes**.

Topics that will be covered include: WKB method: zeroth and first orders; turning points; Perturbation theory: regular perturbation, singular perturbation and boundary layers; Homogenization methods for ODE's, elliptic and parabolic PDE's; Homogenization for SDE's, diffusion processes in periodic and random media; Averaging principle for ODE's and SDE's. Applications will be discussed in class and in homework problems.

**APMA 2811E. A Posteriori Estimates for Finite Element Methods**.

This course gives an introduction to the the basic concepts of a posteriori estimates of finite element methods. After an overview of different techniques the main focus will be shed on residual based estimates where as a starting point the Laplace operator is analyzed. Effectivity and reliability of the error estimator will be proven. In a second part of the course, students will either study research articles and present them or implement the error estimates for some specific problem and present their numerical results. Recommended prerequisites: basic knowledge in finite elements, APMA 2550, 2560, 2570.

**APMA 2811F. Numerical Solution of Ordinary Differential Equations: IVP Problems and PDE Related Issues**.

The purpose of the course is to lay the foundation for the development and analysis of numerical methods for solving systems of ordinary differential equations. With a dual emphasis on analysis and efficient implementations, we shall develop the theory for multistage methods (Runge-Kutta type) and multi-step methods (Adams/BDF methods). We shall also discuss efficient implementation strategies using Newton-type methods and hybrid techniques such as Rosenbruck methods. The discussion includes definitions of different notions of stability, stiffness and stability regions, global/local error estimation, and error control. Time permitting, we shall also discuss more specialized topics such as symplectic integration methods and parallel-in-time methods. A key component of the course shall be the discussion of problems and methods designed with the discretization of ODE systems originating from PDE's in mind. Topics include splitting methods, methods for differential-algebraic equations (DAE),deferred correction methods. and order reduction problems for IBVP, TVD and IMEX methods. Part of the class will consist of student presentations on more advanced topics, summarizing properties and known results based on reading journal papers.

**APMA 2811G. Topics in Averaging and Metastability with Applications**.

Topics that will be covered include: the averaging principle for stochastic dynamical systems and in particular for Hamiltonian systems; metastability and stochastic resonance. We will also discuss applications in class and in homework problems. In particular we will consider metastability issues arising in chemistry and biology, e.g. in the dynamical behavior of proteins. The course will be largely self contained, but a course in graduate probability theory and/or stochastic calculus will definitely help.

**APMA 2811H. Survival Analysis**.

**APMA 2811I. An Introduction to Turbulence Modeling**.

Turbulence is the last mystery of classical physics. It surrounds us everywhere – in the air, in the ocean, in pipes carrying fluids and even in human body arteries. The course helps to understand what makes modeling the turbulence so difficult and challenging. The course covers the following issues: The nature of turbulence, characteristics of turbulence and classical constants of turbulence; Turbulent scales; Navier-Stokes equations, Reynolds stresses and Reynolds-Averaged Navier-Stokes (RANS) equations; RANS turbulence models: algebraic models, one-equation models, two-equation models; Low-Reynolds number turbulence models; Renormalization Group (RNG) turbulence model; Large-Eddy Simulation (LES); Students will be provided with user-friendly computer codes to run different benchmark cases. The final grade is based on two take home projects - computing or published papers analysis, optionally.

**APMA 2811K. Computational/Statistical Methods for Signaling Pathways and Protein Interactions**.

The course will cover the main mathematical/computational models/algorithms for a variety of tasks in proteomics and structural proteomics, including: (1) The analysis and interpretation of tandem mass spectrometry data for protein identification and determination of signaling pathways, (2) Identification of Phosphorylation sites and motifs, and (3) structural aspects of protein docking problems on the basis of NMR data. Open to graduate students only.

**APMA 2811L. Topics in Homogenization: Theory and Computation**.

Topics that will be covered include: Homogenization methods for ODE's, for elliptic and parabolic PDE's and for stochastic differential Equations (SDE's) in both periodic and random media; Averaging principle for ODE's and SDE's. Both theoretical and computational aspects will be studied. Applications will be discussed in class and in homework problems. Prerequisites: Some background in PDE's and probability will be helpful, even though the class will be largely self contained.

**APMA 2811O. Dynamics and Stochastics**.

This course provides a synthesis of mathematical problems at the interface between stochastic problems and dynamical systems that arise in systems biology. For instance, in some biological systems some species may be modeled stochastically while other species can be modeled using deterministic dynamics. Topics will include an introduction to biological networks, multiscale analysis, analysis of network structure, among other topics. Prerequisites: probability theory (APMA 2630/2640, concurrent enrollment in APMA 2640 is acceptable).

**APMA 2811Q. Calculus of Variations**.

An introduction to modern techniques in the calculus of variations. Topics covered will include: existence of solutions and the direct method, Euler-Lagrange equations and necessary and sufficient conditions, one-dimensional problems, multidimensional nonconvex problems, relaxation and quasiconvexity, Young's measures, and singular perturbations. The emphasis of the course will be equal parts theory and applications with numerous examples drawn from topics in nonlinear elasticity, pattern formation, wrinkling thin elastic sheets, martensitic phase transitions, minimal surfaces, differential geometry and optimal control.

**APMA 2811S. Levy Processes**.

Lévy processes are the continuous-time analogues of random walks, and include Brownian motion, compound Poisson processes, and square-integrable pure-jump martingales with many small jumps. In this course we will develop the basic theory of general Lévy processes and subordinators, and discuss topics including local time, excursions, and fluctuations. Time permitting we will finish with selected applications which are of mutual interest to the instructor and students enrolled in the class. Prerequisite: APMA 2640 or equivalent.

**APMA 2811Z. Stochastic Partial Differential Equations: Theory and Numerics**.

This course introduces basic theory and numerics of stochastic partial differential equations (SPDEs). Topics include Brownian motion and stochastic calculus in Hilbert spaces, classification of SPDEs and solutions, stochastic elliptic, hyperbolic and parabolic equations, regularity of solutions, linear and nonlinear equations, analytic and numerical methods for SPDEs. Topics of particular interest will also be discussed upon agreements between the instructor and audience.

All three courses APMA 2630, APMA 2640, APMA 2550 are background recommended but not required.

**APMA 2812B. An Introduction to SPDE's**.

An introduction to the basic theory of Stochastic PDE's. Topics will likely include (time permitting) Gaussian measure theory, stochastic integration, stochastic convolutions, stochastic evolution equations in Hilbert spaces, Ito's formula, local well-posedness for semi-linear SPDE with additive noise, weak Martingale solutions to 3D Navier-Stokes, Markov processes on Polish spaces, the Krylov–Bogolyubov theorem, the Doob-Khasminskii theorem, and Bismut-Elworthy-Li formula for a class of non-degenerate SPDE. The presentation will be largely self contained, but will assume some basic knowledge in measure theory, functional analysis, and probability theory. Some familiarity with SDE and PDE is also very helpful, but not required.

**APMA 2812C. Interacting Particle Systems**.

The course will provide an introduction to both static and dynamic interacting particle systems. Topics covered will include basic constructions of interacting particle systems and their limits, including mean-field approximations, analysis of their ergodic properties, including Gibbs measures, Markov random fields and phase transitions. A broad range of applications, ranging from queueing networks to biology and statistical physics, will be used to illustrate the theory. Prerequisite: A good knowledge of measure-theoretic probability theory (at the level of APMA 2630 and 2640)

Fall | APMA2812C | S01 | 17235 | W | 3:00-5:30 | (K. Ramanan) |

**APMA 2820A. A Tutorial on Particle Methods**.

No description available.

**APMA 2820B. Advanced Topics in Information Theory**.

Explores classical and recent results in information theory. Topics chosen from: multi-terminal/network information theory; communication under channel uncertainty; side information problems (channel, source, and the duality between them); identification via channels; and multi-antenna fading channels. Prerequisite: APMA 1710 or basic knowledge of information theory.

**APMA 2820C. Computational Electromagnetics**.

No description available.

**APMA 2820D. Conventional, Real and Quantum Computing with Applications to Factoring and Root Finding**.

No description available.

**APMA 2820E. Geophysical Fluid Dynamics**.

No description available.

**APMA 2820F. Information Theory and Networks**.

No description available.

**APMA 2820G. Information Theory, Statistics and Probability**.

No description available.

**APMA 2820H. Kinetic Theory**.

We will focus on two main topics in mathematical study of the kinetic theory: (1) The new goal method to study the trend to Maxwellians; (2) various hydrodynamical (fluids) limits to Euler and Navier-Stokes equations. Main emphasis will be on the ideas behind proofs, but not on technical details.

**APMA 2820I. Multiscale Methods and Computer Vision**.

Course will address some basic multiscale computational methods such as: multigrid solvers for physical systems, including both geometric and algebraic multigrid, fast integral transforms of various kinds (including a fast Radon transform), and fast inverse integral transforms. Basic problems in computer vision such as global contour detection and their completion over gaps, image segmentation for textural images and perceptual grouping tasks in general will be explained in more details.

**APMA 2820J. Numerical Linear Algebra**.

Solving large systems of linear equations: The course will use the text of Treften and BAO that includes all the modern concepts of solving linear equations.

**APMA 2820K. Numerical Solution of Ordinary Differential Equations**.

We discuss the construction and general theory of multistep and multistage methods for numerically solving systems of ODE's, including stiff and nonlinear problems. Different notions to stability and error estimation and control. As time permits we shall discuss more advanced topics such as order reduction, general linear and additive methods, symplectic methods, and methods for DAE. Prerequisites: APMA 2190 and APMA 2550 or equivalent. Some programming experience is expected.

**APMA 2820L. Random Processes in Mechanics**.

No description available.

**APMA 2820M. Singularities in Eliptic Problems and their Treatment by High-Order Finite Element Methods**.

Singular solutions for elliptic problems (elasticity and heat transfer) are discussed. These may arise around corners in 2-D and along edges and vertices in 3-D domains. Derivation of singular solutions, charactized by eigenpairs and generalized stress/flux intensity factors (GSIF/GFIFs) are a major engineering importance (because of failure initiation and propagation). High-order FE methods are introduced, and special algorithms for extracting eigenpairs and GSIF/GFIFs are studied (Steklov, dual-function, ERR method, and others).

**APMA 2820N. Topics in Scientific Computing**.

No description available.

**APMA 2820O. The Mathematics of Shape with Applications to Computer Vision**.

Methods of representing shape, the geometry of the space of shapes, warping and matching of shapes, and some applications to problems in computer vision and medical imaging. Prerequsite: See instructor for prerequisites.

**APMA 2820P. Foundations in Statistical Inference in Molecular Biology**.

In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferrnces in this setting including: sequence alignment, RNA secondary structure prediction, database search, and functional genomics. Statistical topics: parameter estimation, hypothesis testing, and characterization of posterior spaces. Core course in proposed PhD program in computational molecular biology.

**APMA 2820Q. Topics in Kinetic Theory**.

This course will introduce current mathematical study for Boltzmann equation and Vlasov equation. We will study large time behavior and hydrodynamic limits for Boltzmann theory and instabilities in the Vlasov theory. Graduate PDE course is required.

**APMA 2820R. Structure Theory of Control Systems**.

The course deals with the following problems: given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S' ? Most of the course will deal with the families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

**APMA 2820S. Topics in Differential Equations**.

A sequel to APMA 2210 concentrating on similar material.

**APMA 2820T. Foundations in Statistical Inference in Molecular Biology**.

In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferences in this setting including: sequence alignment. RNA secondary structure prediction, database search, and tiled arrays. Statistical topics: parameter estimation, hypothesis testing, recursions, and characterization of posterior spaces. Core course in proposed PhD program in computational molecular biology.

**APMA 2820U. Structure Theory of Control Systems**.

The course deals with the following problems: given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S' ? Most of the course will deal with the families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

**APMA 2820V. Progress in the Theory of Shock Waves**.

Course begins with self-contained introduction to theory of "hyperbolic conservation laws", that is quasilinear first order systems of partial differential equations whose solutions spontaneously develop singularities that propagate as shock waves. A number of recent developments will be discussed. Aim is to familiarize the students with current status of the theory as well as with the expanding areas of applications of the subject.

**APMA 2820W. An Introduction to the Theory of Large Deviations**.

The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel¿Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk¿sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and 2640.

**APMA 2820X. Boundary Conditions for Hyperbolic Systems: Numerical and Far Field**.

**APMA 2820Y. Approaches to Problem Solving in Applied Mathematics**.

TBA

**APMA 2820Z. Topics in Discontinuous Galerkin Methods**.

We will cover discontinuous Galerkin methods for time-dependent and steady state problems. Stability and error estimates of different discontinuous Galerkin methods will be discussed. In particular, we will discuss in depth the local discontinuous Galerkin method. Prerequisite: APMA 2550 or equivalent knowledge of numerical analysis.

**APMA 2821A. Parallel Scientific Computing: Algorithms and Tools**.

No description available.

**APMA 2821B. To Be Determined**.

**APMA 2821C. Topics in Partial Differential Equations**.

The course will start by reviewing the theory of elliptic and parabolic equations in Holder spaces. Then we will discuss several topics in nonlinear elliptic and parabolic equations, for instance, the Navier-Stokes equation and Monge-Ampere type equations. This course is a sequel to APMA 2810V, but APMA 2810V is not a prerequisite.

**APMA 2821D. Random Processes and Random Variables**.

**APMA 2821E. Topics in Variational Methods**.

This course consists of two parts: a general introduction to variational methods in PDE, and a more focused foray into some special topics. For the former we will cover the direct method in the calculus of variations, various notions of convexity, Noether's theorem, minimax methods, index theory, and gamma-convergence. For the latter we will focus on several specific problems of recent interest, with emphasis on the Ginzburg-Landau energy functional.

**APMA 2821F. Computational Linear Algebra**.

The course will cover basic and advanced algorithms for solution of linear and nonlinear systems as well as eigenvalue problems.

**APMA 2821G. High-Performance Discontinuous Galerkin Solvers**.

Addresses strategies and algorithms in devising efficient discontinuous Galerkin solvers for fluid flow equations such as Euler and Navier-Stokes. The course starts with an introduction to discontinuous Galerkin methods for elliptic and hyperbolic equations and then focuses on the following topics: 1) Serial and parallel implementations of various discontinuous Galerkin operators for curvilinear ele- ments in multiple space dimensions. 2) Explicit, semi-explicit and implicit time discretizations. 3) Multigrid (multi-level) solvers and preconditioners for systems arising from discontinuous Galerkin approximations of the partial differential equations.

**APMA 2821H. Introduction to High Performance Computing: Tools and Algorithms**.

This course will cover fundamental concepts of parallel computing: shared and distributed memory models; metrics for performance measuring; roof-line model for analysis of computational kernels, prediction and improving their performance on different processors; code optimization. We will analyze algorithms maximizing data reuse, and memory bandwidth utilization. Prior experience in coding is a plus. One course meeting will take place at IBM/Research, students will interact with experts in areas of HPC, visualization, social media and more. There will be bi-weekly homework assignments and a final project. Students are encouraged to suggest final project relevant to their research and level of expertise.

**APMA 2821I. Formulation and Approximation of Linear and Non-linear Problems of Solid Mechanics**.

Presents the formulation and approximation by the Finite Element Method (FEM) of linear and non-linear problems of Solid Mechanics. The formulation of problems is based on the Virtual Work Principle (VWP). Increasing complexity problems will be considered such as simple bar under traction, beams, plates, plane problems and solids with linear and hyperelastic materials. All problems are formulated using the same sequence of presentation which includes kinematics, strain measure, rigid body deformation, internal work, external work, VWP and constitutive equations. The approximation of the given problems is based on the High-order FEM. Examples will be presented using a Matlab code.

**APMA 2821J. Some Topics in Kinetic Theory**.

In this advanced topic course, we will go over several aspects of recent mathematical work on kinetic theory. Graduate level PDE is required.

**APMA 2821K. Probabilistic and Statistical Models for Graphs and Networks**.

Many modern data sets involve observations about a network of interacting components. Probabilistic and statistical models for graphs and networks play a central role in understanding these data sets. This is an area of active research across many disciplines. Students will read and discuss primary research papers and complete a final project.

**APMA 2821L. Introduction to Malliavin Calculus**.

The Malliavin calculus is a stochastic calculus for random variables on Gaussian probability spaces, in particular the classical Wiener space. It was originally introduced in the 1970s by the French mathematician Paul Malliavin as a probabilistic approach to the regularity theory of second-order deterministic partial differential equations. Since its introduction, Malliavin's calculus has been extended beyond its original scope and has found applications in many branches of stochastic analysis; e.g. filtering and optimal control, mathematical finance, numerical methods for stochastic differential equations. This course will introduce, starting in a simple setting, the basic concepts and operations of the Malliavin calculus, which will then be applied to the study of regularity of stochastic differential equations and their associated partial differential equations. In addition, applications from optimal control and finance, including the Clark-Ocone foruma and its connection with hedging, will be presented.

**APMA 2821M. Some Mathematical Problems in Materials Science**.

We will study a variety of mathematical models for problems in materials science. Mainly we will consider models of phase transformation, static and dynamic. Some of the topics to be treated are: (1) models of phase transformation; (2) gradient flows; (3) kinetic theories of domain growth; (4) stochastic models; (5) free boundary problems. A working familiarity with partial differential equations is required.

**APMA 2821N. Numerical Solution of Ordinary Differential Equations: IVP Problems and PDE Related Issues**.

The course seeks to lay the foundation for the development and analysis of numerical methods for solving systems of ordinary differential equations. With a dual emphasis on analysis and efficient implementations, we shall develop the theory for multistage methods (Runge-Kutta type) and multi-step methods (Adams/BDF methods). The discussion includes definitions of different notions of stability, stiffness and stability regions, global/local error estimation, and error control. We also discuss more specialized topics such as symplectic integration methods, parallel-in-time methods, include splitting methods, methods for differential-algebraic equations (DAE), deferred correction methods, and order reduction problems for IBVP, TVD and IMEX methods.

**APMA 2821O. Topics in Posteriori Error Estimations: Finite Element and Reduced Basis Methods**.

The course will contain two related parts. An introduction of different types of a posteriori error estimations for various finite element methods, certified reduced basis method, where a posteriori error estimations play an important role. Emphasize both the theory and implementation. Related Matlab programs. Residual-type, local-problem type, and recovery-type error estimators for conforming, mixed, non-conforming, and discontinuous galerkin finite element methods for different types of equations. Reduced basis methods, offline-online procedure, greedy algorithm, error estimator, empirical interpolation method, and successive constraint method will be discussed. Goal-Oriented primal-dual approach for both FEM and RBM will be covered. Objective: To learn various theoretical and practical results of adaptive finite element methods and reduced basis methods.

**APMA 2821P. Topics in the Atomistic-to-Continuum Coupling Methods for Material Science**.

Atomistic-to-continuum coupling methods (a/c methods) have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields. This course provides an introduction to the fundamentals required to understand modeling and computer simulation of material behavior. This course will first briefly review material from continuum mechanics, materials science including crystals and defects and then move on to advanced topics in development and analysis of a/c coupling methods both in static and dynamic cases. I will also select topics from statistical mechanics and temporal multiscale accelerated molecular dynamics methods (hyperdynamics, parallel replica dynamics).

**APMA 2821R. Topics in the Atomistic-to-Continuum Coupling Methods for Material Science**.

Atomistic-to-continuum coupling methods (a/c methods) have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields.

This is an advanced topics course for graduate students. Provides an introduction to the fundamentals required to understand modeling and computer simulation of material behavior. First briefly review material from continuum mechanics, materials science including crystals and defects and then move on to advanced topics in development and analysis of a/c coupling methods both in static and dynamic cases. select topics from statistical mechanics and temporal multiscale accelerated molecular dynamics methods (hyperdynamics, parallel replica dynamics).

**APMA 2821T. Theory of Large Deviations**.

The theory of large deviations is concerned with the probabilities of very rare events. There are many applications where a rare event can have a significant impact (think of the lottery) and it is of interest to know when and how these events occur. The course will begin with a review of the general framework, standard techniques, and elementary examples (e.g., Cramer's and Sanov's Theorems) before proceeding with general theory and applications. If time permits, the course will end with a study of large deviations for diffusion processes.

**APMA 2821U. Kinetic Theory**.

Topics in kinetic theory, particularly concerning Boltzmann equations and related but simpler models (e.g.the Kac model). Key issues include the mathematical derivation of the Boltzmann equation, the Cauchy problem, Boltzmann's $H$ theorem, and hydrodynamic limits yielding the equations of fluid mechanics. We will be most interested in rigorous results, but will not turn away from formal calculations when these are the only things available. A probabilistic viewpoint will be emphasized. In addition to these "traditional'' topics, we will also introduce the Smoluchowski coagulation equation and a similar equation, and some microscopic models described by these in the kinetic limit or exactly. Students should have PDE background equivalent to or exceeding MATH 2370/APMA 2230. Familiarity with probability will be helpful, but we will review this according to the audience's needs.

**APMA 2821V. Neural Dynamics: Theory and Modeling**.

Our thoughts and actions are mediated by the dynamic activity of the brain’s neurons. This course will use mathematics and computational modeling as a tool to study neural dynamics at the level of signal neurons and in more complicated networks. We will focus on relevance to modern day neuroscience problems with a goal of linking dynamics to function. Topics will include biophysically detailed and reduced representations of neurons, bifurcation and phase plane analysis of neural activity, neural rhythms and coupled oscillator theory. Audience: advanced undergraduate or graduate students. Prerequisite: APMA 0350-0360 and Matlab programming course. Instructor permission required.

**APMA 2822B. Introduction to Parallel Computing on Heterogeneous (CPU+GPU) Systems**.

This course we will learn fundamental aspects of parallel computing on heterogeneous systems composed of multi-core CPUs and GPUs. This course will contain lectures and hands-on parts. We will cover the following topics: shared memory and distributed memory programming models, parallelization strategies and nested parallelism. We will also learn techniques for managing memory and data on systems with heterogeneous memories (DDR for CPUs and HBM for GPUs); parallelization strategies using OpenMP and MPI; programming GPUs using OpenMP4.5 directives and CUDA. We will focus on programming strategies and application performance. Grading will be based on home-work assignments, and final project.

**APMA 2980. Research in Applied Mathematics**.

Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course.

**APMA 2990. Thesis Preparation**.

For graduate students who have met the residency requirement and are continuing research on a full time basis.

Fall | APMA2990 | S01 | 15447 | Arranged | 'To Be Arranged' |

### Chair

Bjorn Sandstede

### Associate Chair

Kavita Ramanan

### Professor

Mark Ainsworth

Francis Wayland Professor of Applied Mathematics

Constantine Michael Dafermos

Alumni-Alumnae University Professor of Applied Mathematics

Hongjie Dong

Professor of Applied Mathematics

Paul G. Dupuis

IBM Professor of Applied Mathematics

Peter L. Falb

Professor Emeritus of Applied Mathematics

Wendell H. Fleming

University Professor Emeritus, Professor Emeritus of Applied Mathematics and Mathematics

Stuart Geman

James Manning Professor of Applied Mathematics

Basilis Gidas

Professor of Applied Mathematics

Yan Guo

Professor of Applied Mathematics

Din-Yu Hsieh

Professor Emeritus of Applied Mathematics

George E. Karniadakis

Charles Pitts Robinson and John Palmer Barstow Professor of Applied Mathematics

Harold J. Kushner

Professor Emeritus of Applied Mathematics and Engineering and L. Herbert Ballou University Professor Emeritus

Charles Lawrence

Professor of Applied Mathematics

John Mallet-Paret

George Ide Chase Professor of Physical Science, Professor of Applied Mathematics

Martin R. Maxey

Professor of Applied Mathematics; Professor of Engineering

Donald E. McClure

Professor Emeritus of Applied Mathematics

Govind Menon

Professor of Applied Mathematics

George W. Morgan

Professor Emeritus of Applied Mathematics

David Mumford

Professor Emeritus of Applied Mathematics

Kavita Ramanan

Roland George Dwight Richardson University Professor of Applied Mathematics

Boris L. Rozovsky

Ford Foundation Professor Emeritus of Applied Mathematics

Bjorn Sandstede

Royce Family Professor of Teaching Excellence, Professor of Applied Mathematics

Chi-Wang Shu

Stowell University Professor of Applied Mathematics

Lawrence Sirovich

Professor Emeritus of Applied Mathematics

Walter A. Strauss

L. Herbert Ballou University Professor of Mathematics

Chau-Hsing Su

Professor Emeritus of Applied Mathematics

### Associate Professor

Lucien J. E. Bienenstock

Associate Professor of Applied Mathematics; Associate Professor of Neuroscience

Johnny Guzman

Associate Professor of Applied Mathematics

Matthew T. Harrison

Associate Professor of Applied Mathematics

Caroline J. Klivans

Associate Professor of Applied Mathematics

Hui Wang

Associate Professor of Applied Mathematics

### Assistant Professor

Jerome B. Darbon

Assistant Professor of Applied Mathematics

Nidhi Kaihnsa

Prager Assistant Professor of Applied Mathematics

Anastasios Matzavinos

Assistant Professor of Applied Mathematics

Samuel Punshon-Smith

Prager Assistant Professor of Applied Mathematics

Yeonjong Shin

Prager Assistant Professor of Applied Mathematics

### Visiting Scholar

Catherine A. Roberts

Visiting Scholar in Applied Mathematics

Yanjin Wang

Visiting Scholar in Applied Mathematics

Haiqiong Zhao

Visiting Scholar in Applied Mathematics

## Applied Mathematics

The concentration in Applied Mathematics allows students to investigate the mathematics of problems arising in the physical, life and social sciences as well as in engineering. The basic mathematical skills of Applied Mathematics come from a variety of sources, which depend on the problems of interest: the theory of ordinary and partial differential equations, matrix theory, statistical sciences, probability and decision theory, risk and insurance analysis, among others. Applied Mathematics appeals to people with a variety of different interests, ranging from those with a desire to obtain a good quantitative background for use in some future career, to those who are interested in the basic techniques and approaches in themselves. The standard concentration leads to either the A.B. or Sc.B. degree. Students may also choose to pursue a joint program with biology, computer science or economics. The undergraduate concentration guide is available here.

Both the A.B. and Sc.B. concentrations in Applied Mathematics require certain basic courses to be taken, but beyond this there is a great deal of flexibility as to which areas of application are pursued. Students are encouraged to take courses in applied mathematics, mathematics and one or more of the application areas in the natural sciences, social sciences or engineering. Whichever areas are chosen should be studied in some depth.

### Standard program for the A.B. degree.

Prerequisites | ||

Introductory Calculus, Part I and Introductory Calculus, Part II | ||

Or their equivalent | ||

Program | ||

Ten additional semester courses approved by the Division of Applied Mathematics. These classes must include: ^{1} | ||

MATH 0180 | Intermediate Calculus | 1 |

MATH 0520 | Linear Algebra ^{2} | 1 |

APMA 0350 & APMA 0360 | Applied Ordinary Differential Equations and Applied Partial Differential Equations I ^{3} | 2 |

Select one course on programming from the following: ^{4} | 1 | |

Introduction to Mathematical Modeling | ||

Introduction to Scientific Computing | ||

Introduction to Scientific Computing and Problem Solving | ||

Computing Foundations: Data | ||

Introduction to Object-Oriented Programming and Computer Science | ||

Computer Science: An Integrated Introduction | ||

Five additional courses, of which four should be chosen from the 1000-level courses taught by the Division of Applied Mathematics. APMA 1910 cannot be used as an elective. | 5 | |

Total Credits | 10 |

^{1} | Substitution of alternate courses for the specific requirements is subject to approval by the division. |

^{2} | Concentrators are urged to consider MATH 0540 as an alternative to MATH 0520. |

^{3} | APMA 0330, APMA 0340 will sometimes be accepted as substitutes for APMA 0350, APMA 0360. APMA 1910 cannot be used as an elective. |

^{4} | Concentrators are urged to complete their introductory programming course before the end of their sophomore year. |

### Standard program for the Sc.B. degree.

Program | ||

Eighteen approved semester courses in mathematics, applied mathematics, engineering, the natural or social sciences. These classes must include: ^{1} | ||

MATH 0090 & MATH 0100 | Introductory Calculus, Part I and Introductory Calculus, Part II | 2 |

MATH 0180 | Intermediate Calculus | 1 |

MATH 0520 | Linear Algebra ^{2} | 1 |

APMA 0350 & APMA 0360 | Applied Ordinary Differential Equations and Applied Partial Differential Equations I ^{3} | 2 |

Select one senior seminar from the APMA 1930 or APMA 1940 series, or an approved equivalent. | 1 | |

Select one course on programming from the following: ^{4} | 1 | |

Introduction to Mathematical Modeling | ||

Introduction to Scientific Computing | ||

Introduction to Scientific Computing and Problem Solving | ||

Computing Foundations: Data | ||

Introduction to Object-Oriented Programming and Computer Science | ||

Computer Science: An Integrated Introduction | ||

Ten additional courses, of which six should be chosed from the 1000-level or higher level courses taught by the Division of Applied Mathematics. APMA 1910 cannot be used as an elective. | 10 | |

Total Credits | 18 |

^{1} | Substitution of alternate courses for the specific requirements is subject to approval by the division. |

^{2} | Concentrators are urged to consider MATH 0540 as an alternative to MATH 0520. |

^{3} | APMA 0330, APMA 0340 will sometimes be accepted as substitutes for APMA 0350, APMA 0360. |

^{4} | Concentrators are urged to complete their introductory programming course before the end of their sophomore year. |

## Applied Mathematics-Biology

The Applied Math - Biology concentration recognizes that mathematics is essential to address many modern biological problems in the post genomic era. Specifically, high throughput technologies have rendered vast new biological data sets that require novel analytical skills for the most basic analyses. These technologies are spawning a new "data-driven" paradigm in the biological sciences and the fields of bioinformatics and systems biology. The foundations of these new fields are inherently mathematical, with a focus on probability, statistical inference, and systems dynamics. These mathematical methods apply very broadly in many biological fields including some like population growth, spread of disease, that predate the genomics revolution. Nevertheless, the application of these methods in areas of biology from molecular genetics to evolutionary biology has grown very rapidly in with the availability of vast amounts of genomic sequence data. Required coursework in this program aims at ensuring expertise in mathematical and statistical sciences, and their application in biology. The students will focus in particular areas of biology. The program culminates in a senior capstone experience that pairs student and faculty in creative research collaborations.

### Standard program for the Sc.B. degree

Required coursework in this program aims at ensuring expertise in mathematical and statistical sciences, and their application in biology. The students will focus in particular areas of biology. The program culminates in a senior capstone experience that pairs student and faculty in creative research collaborations. Applied Math – Biology concentrators are prepared for careers in medicine, public health, industry and academic research.

**Required Courses**:

*Students are required to take all of the following courses*.

MATH 0090 | Introductory Calculus, Part I | 1 |

MATH 0100 | Introductory Calculus, Part II | 1 |

or MATH 0170 | Advanced Placement Calculus | |

MATH 0180 | Intermediate Calculus (or equivalent placement) | 1 |

MATH 0520 | Linear Algebra | 1 |

or MATH 0540 | Honors Linear Algebra | |

CHEM 0330 | Equilibrium, Rate, and Structure ^{1} | 1 |

PHYS 0030 | Basic Physics A | 1 |

or PHYS 0050 | Foundations of Mechanics | |

Select one of the following sequences: | 2 | |

Applied Ordinary Differential Equations and Applied Partial Differential Equations I | ||

Methods of Applied Mathematics I, II and Methods of Applied Mathematics I, II | ||

APMA 1650 | Statistical Inference I | 1 |

or APMA 1655 | Statistical Inference I | |

APMA 1070 | Quantitative Models of Biological Systems | 1 |

APMA 1080 | Inference in Genomics and Molecular Biology | 1 |

or NEUR 2110 | Statistical Neuroscience | |

BIOL 0200 | The Foundation of Living Systems (or equivalent) | 1 |

Additional Courses | ||

In addition to required courses listed above, students must take the following: | ||

Two additional courses in Applied Math or Biology. At least one of these must be a directed research course, e.g. a senior seminar or independent study in Applied Math or a directed research/independent study in Biology. For example: ^{1} | 2 | |

A course from the APMA 1930 series | ||

A course from the APMA 1940 series. | ||

Independent Study | ||

Directed Research/Independent Study | ||

Directed Research/Independent Study | ||

We strongly recommend that Applied Mathematics-Biology concentrators take one of the following programming courses on or before their first semester as a concentrator: APMA 0160, CSCI 0040, CSCI 0150, CSCI 0170, CSCI 0190, CLPS 0950. Those who do can use it as their second Applied Math or Biology course. | ||

Four classes in the biological sciences agreed upon by the student and advisor. These four courses should form a cohesive grouping ina specific area of emphasis, at least two of which should be at the 1000-level. Some example groupings are below: | 4 | |

Areas of Emphasis and Suggested Courses: | ||

Some areas of possible emphasis for focusing of elective courses are listed below. Given the large number of course offerings in the biosciences and neuroscience, students are free to explore classes in these areas that are not listed below. However, all classes must be approved by the concentration advisor. APMA 1910 cannot be used as an elective. | ||

Biochemistry | ||

Biochemistry | ||

Advanced Biochemistry | ||

Organic Chemistry | ||

Chemical Biology | ||

Biotechnology and Physiology | ||

Principles of Physiology | ||

Cell Physiology and Biophysics | ||

and/or appropriate bioengineering courses, such as: | ||

Polymer Science for Biomaterials | ||

Biomaterials | ||

Tissue Engineering | ||

Stem Cell Engineering | ||

Synthetic Biological Systems | ||

Ecology, Evolution, and Genetics | ||

Invertebrate Zoology and Evolutionary Biology | ||

Principles of Ecology and The Evolution of Plant Diversity | ||

Genetics | ||

Experimental Design in Ecology | ||

Population Genetics | ||

Human Population Genomics | ||

Molecular Genetics | ||

Neuroscience | ||

Mathematical Methods in the Brain Sciences | ||

Neurosciences courses: See https://www.brown.edu/academics/neuroscience/undergraduate/neuroscience-concentration-requirements | ||

Cell Physiology and Biophysics | ||

Topics in Signal Transduction | ||

Synaptic Transmission and Plasticity | ||

Total Credits | 18 |

^{1} | Students whose independent study is expected to be in an experimental field are strongly encouraged to take APMA 1660, which covers experimental design and the analysis of variance (ANOVA), a method commonly used in the analysis of experimental data. |

### Honors

Requirements and Process: Honors in the Applied Math-Biology concentration is based primarily upon an in-depth, original research project carried out under the guidance of a Brown (and usually Applied Math or BioMed) affiliated faculty advisor. Projects must be conducted for no less than two full semesters, and student smust register for credit for the project via APMA 1970 or BIOL 1950/BIOL 1960 or similar independent study courses. The project culminates in the writing of a thesis which is reviewed by the thesis advisor and a second reader. It is essential that the student have one advisor from the biological sciences and one in Applied Mathematics. The thesis work must be presented in the form of an oral presentation (arranged with the primary thesis advisor) or posted at the annual Undergraduate Research Day in either Applied Mathematics or Biology. For information on registering for BIOL 1950/BIOL 1960, please see https://www.brown.edu/academics/biology/undergraduate-education/undergraduate-research

Excellence in grades within the concentration as well as a satisfactory evaluation by the advsors are also required for Honors. The student's grades must place them within the upper 20% of their cohort, in accordance with the university policy on honors. Honors recipients typically maintain a Grade Point Average of 3.4 or higher in the concentration. However, in the case of outstanding independent research as demonstrated in the thesis and supported by the Thesis Committee, candidates with a GPA between 3.0 an 3.4 will be considered and are encouraged to apply.

The deadline for applying to graduate with honors in the concentration are the same as those of the biology concentrations. However, students in the joint concentration must inform the undergraduate chair in Applied Mathematics of their intention to apply for honors by these dates.

## Applied Mathematics-Computer Science

The Sc.B. concentration in Applied Math-Computer Science provides a foundation of basic concepts and methodology of mathematical analysis and computation and prepares students for advanced work in computer science, applied mathematics, and scientific computation. Concentrators must complete courses in mathematics, applied math, computer science, and an approved English writing course. While the concentration in Applied Math-Computer Science allows students to develop the use of quantitative methods in thinking about and solving problems, knowledge that is valuable in all walks of life, students who have completed the concentration have pursued graduate study, computer consulting and information industries, and scientific and statistical analysis careers in industry or government. This degree offers a standard track and a professional track.

### Requirements for the Standard Track of the Sc.B. degree.

Prerequisites - two semesters of Calculus, for example | ||

Introductory Calculus, Part I and Introductory Calculus, Part II | ||

Advanced Placement Calculus | ||

Concentration Requirements (17 courses) | ||

Core-Math: | ||

MATH 0180 | Intermediate Calculus | 1 |

or MATH 0350 | Honors Calculus | |

MATH 0520 | Linear Algebra | 1 |

or MATH 0540 | Honors Linear Algebra | |

or CSCI 0530 | Coding the Matrix: An Introduction to Linear Algebra for Computer Science | |

Core-Applied Mathematics: | ||

APMA 0350 | Applied Ordinary Differential Equations | 1 |

APMA 0360 | Applied Partial Differential Equations I | 1 |

APMA 1170 | Introduction to Computational Linear Algebra | 1 |

or APMA 1180 | Introduction to Numerical Solution of Differential Equations | |

Core-Computer Science: | ||

Select one of the following Series: | 2 | |

Series A | ||

Introduction to Object-Oriented Programming and Computer Science and Introduction to Algorithms and Data Structures | ||

Series B | ||

Computer Science: An Integrated Introduction and Computer Science: An Integrated Introduction | ||

Series C | ||

Accelerated Introduction to Computer Science (and an additional CS course not otherwise used to satisfy a concentration requirement; (this course may be CSCI 0180, an intermediate-level CS course, or a 1000-level course) ) | ||

Select three of the following intermediate-level courses, one of which must be math-oriented and one systems-oriented. The intermediate courses must cover the requirements of the pathway chosen under additional requirements for CS. | 3 | |

Introduction to Discrete Structures and Probability (math) | ||

Introduction to Software Engineering (systems) | ||

Introduction to Computer Systems (systems) | ||

Theory of Computation (math) | ||

Probability for Computing and Data Analysis (math) ^{1} | ||

or APMA 1650 | Statistical Inference I | |

Three 1000-level Computer Science courses. Two of these courses and the intermediate courses must satisfy one of the CS pathways. ^{2} | 3 | |

Three 1000-level Applied Mathematics courses approved by the concentration advisor, of which two should constitute a standard sequence or address a common theme. Typical sequences include: APMA 1200/1210 and APMA 1650 or 1655/1660. APMA 1910 cannot be used as an elective. | 3 | |

A capstone course: a one-semester course, taken in the student's last undergraduate year, in which the student (or group of students) use a significant portion of their undergraduate education, broadly interpreted, in studying some current topic in depth, to produce a culminating artifact such as a paper or software project. The title and abstract of the artifact, along with the student's and faculty-sponsor's names, will be placed in the CS website. The inclusion of a relevant image or system diagram is strongly encouraged. The complete text of the best artifacts of each class will be featured on the CS website. A senior thesis, which involves two semesters of work, may count as a capstone. ^{3} | 1 | |

Note: CSCI 1010 and 1450 may be used either as a math-oriented intermediate courses or as advanced courses. CSCI 1010 was formerly known as CSCI 510: they are the same course and hence only one may be taken for credit. CSCI 1450 was formerly known as CSCI 450: they are the same course and hence only one may be taken for credit. Applied Math 1650 or 1655 may be used in place of CSCI 1450. However, concentration credit will be given for only one of Applied Math 1650, 1655, and CSCI 1450. | ||

Total Credits | 17 |

^{1} | APMA 1650 may only be used if not being used as an Applied Math course. |

^{2} | Pathways may be viewed here: https://cs.brown.edu/degrees/undergrad/concentrating-in-cs/concentration-requirements-2020/pathways-for-undergraduate-and-masters-students/ |

^{3} | Capstone Options may be found here: https://cs.brown.edu/degrees/undergrad/concentrating-in-cs/concentration-requirements-2020/capstone/ |

### Requirements for the Professional Track of the Sc.B. degree.

The requirements for the professional track include all those of the standard track, as well as the following:

Students must complete two two-to-four-month full-time professional experiences, doing work that is related to their concentration programs. Such work is normally done within an industrial organization, but may also be at a university under the supervision of a faculty member.

On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience addressing the following prompts, to be approved by the student's concentration advisor:

- Which courses were put to use in your summer's work? Which topics, in particular, were important?
- In retrospect, which courses should you have taken before embarking on your summer experience? What are the topics from these courses that would have helped you over the summer if you had been more familiar with them?
- Are there topics you should have been familiar with in preparation for your summer experience, but are not taught at Brown? What are these topics?
- What did you learn from the experience that probably could not have been picked up from course work?
- Is the sort of work you did over the summer something you would like to continue doing once you graduate? Explain.
- Would you recommend your summer experience to other Brown students? Explain.

## Applied Mathematics-Economics

The Applied Mathematics-Economics concentration is designed to reflect the mathematical and statistical nature of modern economic theory and empirical research. This concentration has two tracks. The first is the advanced economics track, which is intended to prepare students for graduate study in economics. The second is the mathematical finance track, which is intended to prepare students for graduate study in finance, or for careers in finance or financial engineering. Both tracks have A.B. degree versions and Sc.B. degree versions, as well as a Professional track option.

### Standard Program for the A.B. degree (Advanced Economics track):

Prerequisites: | ||

Introductory Calculus, Part II | ||

Linear Algebra | ||

Course Requirements: | ||

Applied Mathematics Requirements | ||

(a) ^{1} | ||

APMA 0350 & APMA 0360 | Applied Ordinary Differential Equations and Applied Partial Differential Equations I ^{2} | 2 |

Select one of the following: | 1 | |

Introduction to Scientific Computing (preferred) | ||

Introduction to Scientific Computing and Problem Solving | ||

Computing Foundations: Data | ||

Introduction to Object-Oriented Programming and Computer Science | ||

Computer Science: An Integrated Introduction | ||

Select one of the following: | 1 | |

Operations Research: Probabilistic Models | ||

Operations Research: Deterministic Models | ||

APMA 1650 | Statistical Inference I | 1 |

or APMA 1655 | Statistical Inference I | |

(b) ^{1} | ||

Select one of the following: | 1 | |

Operations Research: Probabilistic Models | ||

Operations Research: Deterministic Models | ||

Methods of Applied Mathematics | ||

Applied Dynamical Systems | ||

Statistical Inference II | ||

Computational Probability and Statistics | ||

Monte Carlo Simulation with Applications to Finance | ||

Recent Applications of Probability and Statistics | ||

Analysis: Functions of One Variable | ||

Economics Requirements: | ||

ECON 1130 | Intermediate Microeconomics (Mathematical) ^{3} | 1 |

ECON 1210 | Intermediate Macroeconomics | 1 |

ECON 1630 | Mathematical Econometrics I | 1 |

Two 1000-level courses from the "mathematical-economics" group: ^{4} | 2 | |

Welfare Economics and Social Choice Theory | ||

Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies | ||

Industrial Organization (Mathematical) | ||

Bargaining Theory and Applications | ||

Designing Internet Marketplaces | ||

Mathematical Econometrics II | ||

Big Data | ||

Advanced Topics in Econometrics | ||

Investments II | ||

Theory of Behavioral Economics | ||

Theory of Economic Growth | ||

The Theory of General Equilibrium | ||

Game Theory and Applications to Economics | ||

One 1000-level course from the "data methods" group: ^{4} | 1 | |

Economics of Education I | ||

Labor Economics | ||

Health, Education, and Social Policy | ||

Economics of Global Warming | ||

Environmental Issues in Development Economics | ||

Health Economics | ||

Inequality of Opportunity in the US | ||

The Economics of Mass Media | ||

The Economics of Social Policy | ||

Public Economics | ||

Economic Development | ||

Health, Hunger and the Household in Developing Countries | ||

Applied Research Methods for Economists | ||

Mathematical Econometrics II | ||

Big Data | ||

Advanced Topics in Econometrics | ||

ECON 1765 | Finance, Regulation, and the Economy | |

ECON 1825 | Behavioral Economics and Public Policy | |

ECON 1830 | Behavioral Finance | |

One additional 1000-level economics course. ^{5} | 1 | |

Total Credits | 13 |

^{1} | No course may be used to simultaneously satisfy (a) and (b). |

^{2} | APMA 0330 and APMA 0340 may be substituted with advisor approval. APMA 1910 cannot be used as an elective. |

^{3} | Or ECON 1110 with permission. |

^{4} | No course may be used to simultaneously satisfy the "mathematical economics" and the "data methods" requirements. |

^{5} | Note that ECON 1620, ECON 1960, and ECON 1970 (independent study) cannot be used for concentration credit. However, 1620 and 1960 can be used for university credit and up to two 1970s may be used for university credit. |

### Standard program for the Sc.B. degree (Advanced Economics track):

Prerequisites: | ||

Introductory Calculus, Part II | ||

Linear Algebra | ||

Course Requirements: | ||

Applied Mathematics Requirements | ||

(a) ^{1} | ||

APMA 0350 & APMA 0360 | Applied Ordinary Differential Equations and Applied Partial Differential Equations I ^{2} | 2 |

Select one of the following: | 1 | |

Introduction to Scientific Computing (preferred) | ||

Introduction to Scientific Computing and Problem Solving | ||

Computing Foundations: Data | ||

Introduction to Object-Oriented Programming and Computer Science | ||

Computer Science: An Integrated Introduction | ||

Select one of the following: | 1 | |

Operations Research: Probabilistic Models | ||

Operations Research: Deterministic Models | ||

APMA 1650 | Statistical Inference I | 1 |

or APMA 1655 | Statistical Inference I | |

(b) ^{1} | ||

Select two of the following: | 2 | |

Operations Research: Probabilistic Models | ||

Operations Research: Deterministic Models | ||

Methods of Applied Mathematics | ||

Applied Dynamical Systems | ||

Statistical Inference II | ||

Computational Probability and Statistics | ||

Monte Carlo Simulation with Applications to Finance | ||

Recent Applications of Probability and Statistics | ||

Analysis: Functions of One Variable | ||

Economics Requirements: | ||

ECON 1130 | Intermediate Microeconomics (Mathematical) ^{3} | 1 |

ECON 1210 | Intermediate Macroeconomics | 1 |

ECON 1630 | Mathematical Econometrics I | 1 |

Three 1000-level courses from the "mathematical-economics" group: ^{4} | 3 | |

Welfare Economics and Social Choice Theory | ||

Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies | ||

Industrial Organization (Mathematical) | ||

Bargaining Theory and Applications | ||

Designing Internet Marketplaces | ||

Mathematical Econometrics II | ||

Big Data | ||

Advanced Topics in Econometrics | ||

Investments II | ||

Theory of Behavioral Economics | ||

Theory of Economic Growth | ||

The Theory of General Equilibrium | ||

Game Theory and Applications to Economics | ||

One 1000-level course from the "data methods" group: ^{4} | 1 | |

Economics of Education I | ||

Labor Economics | ||

Health, Education, and Social Policy | ||

Economics of Global Warming | ||

Environmental Issues in Development Economics | ||

Health Economics | ||

Inequality of Opportunity in the US | ||

The Economics of Mass Media | ||

The Economics of Social Policy | ||

Public Economics | ||

Economic Development | ||

Health, Hunger and the Household in Developing Countries | ||

Applied Research Methods for Economists | ||

Mathematical Econometrics II | ||

Big Data | ||

ECON 1765 | Finance, Regulation, and the Economy | |

ECON 1825 | Behavioral Economics and Public Policy | |

ECON 1830 | Behavioral Finance | |

Two additional 1000-level economics courses ^{5} | 2 | |

Total Credits | 16 |

^{1} | No course may be used to simultaneously satisfy (a) and (b). |

^{2} | APMA 0330 and APMA 0340 may be substituted with advisor approval. APMA 1910 cannot be used as an elective. |

^{3} | Or ECON 1110 with permission. |

^{4} | No course may be used to simultaneously satisfy the "mathematical economics" and the "data methods" requirements. |

^{5} | Students may use either ECON 1070 or ECON 1090 toward the concentration, but not both. Note that ECON 1620, ECON 1960, and ECON 1970 (independent study) cannot be used for concentration credit. However, 1620 and 1960 can be used for university credit and up to two 1970s may be used for university credit. |

### Standard program for the A.B. degree (Mathematical Finance track):

Prerequisites: | ||

Introductory Calculus, Part II | ||

Linear Algebra | ||

Course Requirements: 13 Courses: 6 Applied Math and 7 Economics | ||

Applied Mathematics Requirements | ||

(a) | ||

APMA 0350 & APMA 0360 | Applied Ordinary Differential Equations and Applied Partial Differential Equations I ^{1} | 2 |

Select one of the following: | 1 | |

Introduction to Scientific Computing (preferred) | ||

Introduction to Scientific Computing and Problem Solving | ||

Computing Foundations: Data | ||

Introduction to Object-Oriented Programming and Computer Science | ||

Computer Science: An Integrated Introduction | ||

APMA 1200 | Operations Research: Probabilistic Models | 1 |

APMA 1650 | Statistical Inference I | 1 |

or APMA 1655 | Statistical Inference I | |

(b) | ||

Select one of the following: | 1 | |

Introduction to Numerical Solution of Differential Equations | ||

Operations Research: Deterministic Models | ||

Methods of Applied Mathematics | ||

Applied Dynamical Systems | ||

Statistical Inference II | ||

Statistical Inference I | ||

Computational Probability and Statistics | ||

Monte Carlo Simulation with Applications to Finance (preferred) | ||

Recent Applications of Probability and Statistics | ||

Analysis: Functions of One Variable | ||

Economics Requirements: | ||

ECON 1130 | Intermediate Microeconomics (Mathematical) ^{3} | 1 |

ECON 1210 | Intermediate Macroeconomics | 1 |

ECON 1630 | Mathematical Econometrics I | 1 |

Select two 1000-level courses from the "financial economics" group: ^{2} | 2 | |

Investments I | ||

Corporate Finance | ||

Venture Capital, Private Equity, and Entrepreneurship | ||

Investments II | ||

Financial Institutions | ||

ECON 1765 | Finance, Regulation, and the Economy | |

Advanced Topics in Corporate Finance | ||

ECON 1830 | Behavioral Finance | |

Select one 1000-level course from the "mathematical economics" group: ^{2} | 1 | |

Welfare Economics and Social Choice Theory | ||

Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies | ||

Industrial Organization (Mathematical) | ||

Bargaining Theory and Applications | ||

Designing Internet Marketplaces | ||

Mathematical Econometrics II | ||

Big Data | ||

Advanced Topics in Econometrics | ||

Investments II | ||

Theory of Behavioral Economics | ||

Theory of Economic Growth | ||

The Theory of General Equilibrium | ||

Game Theory and Applications to Economics | ||

Select one 1000-level course from the "data methods" group: ^{2} | 1 | |

Economics of Education I | ||

Labor Economics | ||

Health, Education, and Social Policy | ||

Economics of Global Warming | ||

Environmental Issues in Development Economics | ||

Health Economics | ||

Inequality of Opportunity in the US | ||

The Economics of Mass Media | ||

The Economics of Social Policy | ||

Public Economics | ||

Economic Development | ||

Health, Hunger and the Household in Developing Countries | ||

Applied Research Methods for Economists | ||

Mathematical Econometrics II | ||

Big Data | ||

ECON 1765 | Finance, Regulation, and the Economy | |

ECON 1825 | Behavioral Economics and Public Policy | |

ECON 1830 | Behavioral Finance | |

Total Credits | 13 |

^{1} | APMA 0330 and APMA 0340 may be substituted with advisor approval. APMA 1910 cannot be used as an elective. |

^{2} | No course may be used to simultaneously satisfy any two or more of the "financial economics," "mathematical economics," and "data methods" requirements. |

^{3} | Or ECON 1110 with permission. |

^{4} | Note that ECON 1620, ECON 1960, and ECON 1970 (independent study) cannot be used for concentration credit. However, 1620 and 1960 can be used for university credit and up to two 1970s may be used for university credit. |

### Standard program for the Sc.B. degree (Mathematical Finance track):

Prerequisites: | ||

Introductory Calculus, Part II | ||

Linear Algebra | ||

Course Requirements: 16 courses: 7 Applied Math and 9 Economics | ||

Applied Mathematics requirements: | ||

(a) | ||

APMA 0350 & APMA 0360 | Applied Ordinary Differential Equations and Applied Partial Differential Equations I ^{1} | 2 |

Select one of the following: | 1 | |

Introduction to Scientific Computing (preferred) | ||

Introduction to Scientific Computing and Problem Solving | ||

Computing Foundations: Data | ||

Introduction to Object-Oriented Programming and Computer Science | ||

Computer Science: An Integrated Introduction | ||

APMA 1200 | Operations Research: Probabilistic Models | 1 |

APMA 1650 | Statistical Inference I | 1 |

or APMA 1655 | Statistical Inference I | |

(b) | ||

Select two of the following: | 2 | |

Introduction to Numerical Solution of Differential Equations | ||

Operations Research: Deterministic Models | ||

Methods of Applied Mathematics | ||

Applied Dynamical Systems | ||

Statistical Inference II | ||

Computational Probability and Statistics | ||

Monte Carlo Simulation with Applications to Finance (preferred) | ||

Recent Applications of Probability and Statistics | ||

Analysis: Functions of One Variable | ||

Economics Requirements: | ||

ECON 1130 | Intermediate Microeconomics (Mathematical) ^{3} | 1 |

ECON 1210 | Intermediate Macroeconomics | 1 |

ECON 1630 | Mathematical Econometrics I | 1 |

Select three 1000-level courses from the "financial economics" group: ^{2} | 3 | |

Investments I | ||

Corporate Finance | ||

Venture Capital, Private Equity, and Entrepreneurship | ||

Investments II | ||

Financial Institutions | ||

ECON 1765 | Finance, Regulation, and the Economy | |

Advanced Topics in Corporate Finance | ||

ECON 1830 | Behavioral Finance | |

Select two 1000-level courses from the "mathematical economics" group: ^{2} | 2 | |

Welfare Economics and Social Choice Theory | ||

Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies | ||

Industrial Organization (Mathematical) | ||

Bargaining Theory and Applications | ||

Designing Internet Marketplaces | ||

Mathematical Econometrics II | ||

Big Data | ||

Advanced Topics in Econometrics | ||

Investments II | ||

Theory of Behavioral Economics | ||

Theory of Economic Growth | ||

The Theory of General Equilibrium | ||

Game Theory and Applications to Economics | ||

Select one 1000-level course from the "data methods" group: ^{2} | 1 | |

Economics of Education I | ||

Labor Economics | ||

Health, Education, and Social Policy | ||

Economics of Global Warming | ||

Environmental Issues in Development Economics | ||

Health Economics | ||

Inequality of Opportunity in the US | ||

The Economics of Mass Media | ||

The Economics of Social Policy | ||

Public Economics | ||

Economic Development | ||

Health, Hunger and the Household in Developing Countries | ||

Applied Research Methods for Economists | ||

Mathematical Econometrics II | ||

Big Data | ||

ECON 1765 | Finance, Regulation, and the Economy | |

ECON 1825 | Behavioral Economics and Public Policy | |

ECON 1830 | Behavioral Finance | |

Total Credits | 16 |

^{1} | |

^{2} | No course may be used to simultaneously satisfy any two or more of the "financial economics," "mathematical economics," and "data methods" requirements. |

^{3} | Or ECON 1110 with permission. |

^{4} | Note that ECON 1620, ECON 1960, and ECON 1970 (independent study) cannot be used for concentration credit. However, 1620 and 1960 can be used for university credit and up to two 1970s may be used for university credit. |

### Honors and Capstone Requirement

Admission to candidacy for honors in the concentration is granted on the following basis: 3.7 GPA for Economics courses, and a 3.5 GPA overall. To graduate with honors, a student must write an honors thesis in the senior year following the procedures specified by the concentration (see Economics Department website).

### Professional Track

The requirements for the professional track include all those of the standard track, as well as the following:

Students must complete two two-to-four month full-time professional experiences, doing work that is related to their concentration programs. Such work is normally done within an industrial organization, but may also be at a university under the supervision of a faculty member.

On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience addressing the following prompts, to be approved by the student's concentration advisor:

- Which courses were put to use in your summer's work? Which topics, in particular, were important?
- In retrospect, which courses should you have taken before embarking on your summer experience? What are the topics from these courses that would have helped you over the summer if you had been more familiar with them?
- Are there topics you should have been familiar with in preparation for your summer experience, but are not taught at Brown? What are these topics?
- What did you learn from the experience that probably could not have been picked up from course work?
- Is the sort of work you did over the summer something you would like to continue doing once you graduate? Explain.
- Would you recommend your summer experience to other Brown students? Explain.

## Data Fluency

The Certificate in Data Fluency provides a curricular structure to undergraduates in concentrations other than applied mathematics, computational biology, computer science, math, and statistics who wish to gain fluency and facility with the tools of data analysis and its conceptual framework. The driving intellectual question is how we can infer meaning from data whilst avoiding false predictions. The required experiential learning component provides students with the opportunity to apply their data-science skills in their concentration, engage in research that uses data science, teach data science as an undergraduate teaching assistant, or undertake an internship that has a substantive data-science component.

As with all undergraduate certificates, students must be enrolled in or have completed at least two courses toward the certificate at the time they declare in ASK, which must be no earlier than the beginning of the fifth semester and no later than the last day of classes of the antepenultimate (typically the sixth) semester, in order to facilitate planning for the experiential learning opportunity. Students must submit a proposal for their experiential learning opportunity by the end of the sixth semester.

Excluded Concentrations: Applied Mathematics, Computational Biology, Computer Science, Mathematics, and Statistics (including joint concentrations in these areas)

**Certificate Requirements**

Core Courses: | ||

DATA 0080 | Data, Ethics and Society | 1 |

CSCI 0111 | Computing Foundations: Data | 1 |

or CSCI 0150 | Introduction to Object-Oriented Programming and Computer Science | |

or CSCI 0170 | Computer Science: An Integrated Introduction | |

or CSCI 0190 | Accelerated Introduction to Computer Science | |

DATA 0200 | Data Science Fluency | 1 |

Elective Course: Select one follow-up Applied Math, Biostatistics, Computer Science or domain-specific course with a significant data component from the following list (or another course with approval from the certificate advisor): | 1 | |

Introduction to Geographic Information Systems and Spatial Analysis | ||

Statistical Inference I | ||

Statistical Analysis of Biological Data | ||

Methods in Informatics and Data Science for Health | ||

Survey of Biomedical Informatics | ||

Statistical Methods | ||

Computational Methods for Mind, Brain and Behavior | ||

Probability for Computing and Data Analysis | ||

Deep Learning | ||

Data Science | ||

Data Science Fellows | ||

Introduction to Econometrics | ||

Big Data | ||

Introductory Statistics for Education Research and Policy Analysis | ||

Introduction to Environmental GIS | ||

Introduction to Geographic Information Systems for Environmental Applications | ||

Global Environmental Remote Sensing | ||

Probability | ||

Essentials of Data Analysis | ||

Principles of Biostatistics and Data Analysis | ||

Methods of Social Research | ||

Introductory Statistics for Social Research | ||

Principles and Methods of Geographic Information Systems | ||

Capstone: | 0-1 | |

The required experiential learning component provides students with the opportunity to apply their data-science skills in their concentration, engage in research that uses data science, teach data science as UTAs, or undertake an internship that has a data-science component. The capstone may be completed for credit via an independent study course or not for credit. ^{1} | ||

Total Credits | 4-5 |

^{1} | Students must submit a proposal for their practicum project by the end of the sixth semester. |

## Applied Mathematics

The department of Applied Mathematics offers graduate programs leading to the Master of Science (Sc.M.) degree and the Doctor of Philosophy (Ph.D.) degree.

For more information on admission and program requirements, please visit the following website:

http://www.brown.edu/academics/gradschool/programs/applied-mathematics