# Courses for Fall 2019

• #### 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.
##### APMA 0330 S01
Primary Instructor
Akopian
• #### 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 0340 S01
Primary Instructor
Dobrushkin
• #### 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).
##### APMA 0350 S01
Primary Instructor
Guo
• #### 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.
##### APMA 0360 S01
Primary Instructor
Maxey
• #### 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.
##### APMA 1080 S01
Primary Instructor
Lawrence
• #### 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 1160 S01
Primary Instructor
Darbon
• #### 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.
##### APMA 1170 S01
Primary Instructor
Shin
• #### 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.
##### APMA 1330 S01
Primary Instructor
Geman
• #### 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.
##### APMA 1650 S01
Primary Instructor
Klivans
• #### 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.
##### APMA 1655 S01
Primary Instructor
Wang

• #### 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.
##### APMA 1690 S01
Primary Instructor
Harrison
• #### 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 1710 S01
Primary Instructor
Menon
• #### Waves

The seminar will discuss a diverse sample of wave phenomena encountered in physics, biology and other aspects of common experience, which are modeled through differential equations, and will demonstrate how the marvelous mathematics emerging from the study of these equations contribute to our understanding of the underlying phenomena. Students are expected to have some familiarity with the theory of partial differential equations, at the level of APMA 0360.
##### APMA 1930T S01
Primary Instructor
Dafermos
• #### 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. Several applications, including stochastic control theory and continuous MCMC optimization methods, may be addressed depending on the interests of the class and time restrictions.
##### APMA 1930U S01
Primary Instructor
Matzavinos
• #### 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 1970 S01
Primary Instructor
Bienenstock
Schedule Code
I: Independent Study/Research
##### APMA 1970 S02
Primary Instructor
Dafermos
Schedule Code
I: Independent Study/Research
##### APMA 1970 S03
Primary Instructor
Dupuis
Schedule Code
I: Independent Study/Research
##### APMA 1970 S04
Primary Instructor
Rozovsky
Schedule Code
I: Independent Study/Research
##### APMA 1970 S05
Primary Instructor
Ramanan
Schedule Code
I: Independent Study/Research
##### APMA 1970 S06
Primary Instructor
Morrow
Schedule Code
I: Independent Study/Research
##### APMA 1970 S07
Primary Instructor
Geman
Schedule Code
I: Independent Study/Research
##### APMA 1970 S08
Primary Instructor
Gidas
Schedule Code
I: Independent Study/Research
##### APMA 1970 S09
Primary Instructor
Darbon
Schedule Code
I: Independent Study/Research
##### APMA 1970 S10
Primary Instructor
Matzavinos
Schedule Code
I: Independent Study/Research
##### APMA 1970 S11
Schedule Code
I: Independent Study/Research
##### APMA 1970 S12
Primary Instructor
Karniadakis
Schedule Code
I: Independent Study/Research
##### APMA 1970 S13
Primary Instructor
Lawrence
Schedule Code
I: Independent Study/Research
##### APMA 1970 S14
Primary Instructor
Maxey
Schedule Code
I: Independent Study/Research
##### APMA 1970 S15
Primary Instructor
McClure
Schedule Code
I: Independent Study/Research
##### APMA 1970 S16
Primary Instructor
Shu
Schedule Code
I: Independent Study/Research
##### APMA 1970 S17
Primary Instructor
Klivans
Schedule Code
I: Independent Study/Research
##### APMA 1970 S18
Primary Instructor
Dobrushkin
Schedule Code
I: Independent Study/Research
##### APMA 1970 S19
Primary Instructor
Harrison
Schedule Code
I: Independent Study/Research
##### APMA 1970 S20
Primary Instructor
Kunsberg
Schedule Code
I: Independent Study/Research
##### APMA 1970 S21
Primary Instructor
Wang
Schedule Code
I: Independent Study/Research
##### APMA 1970 S22
Schedule Code
I: Independent Study/Research
##### APMA 1970 S23
Primary Instructor
Dong
Schedule Code
I: Independent Study/Research
##### APMA 1970 S24
Primary Instructor
Guo
Schedule Code
I: Independent Study/Research
##### APMA 1970 S25
Primary Instructor
Guzman
Schedule Code
I: Independent Study/Research
##### APMA 1970 S26
Primary Instructor
Mallet-Paret
Schedule Code
I: Independent Study/Research
##### APMA 1970 S27
Primary Instructor
Menon
Schedule Code
I: Independent Study/Research
##### APMA 1970 S28
Primary Instructor
Sandstede
Schedule Code
I: Independent Study/Research
##### APMA 1970 S29
Primary Instructor
Su
Schedule Code
I: Independent Study/Research
##### APMA 1970 S30
Schedule Code
I: Independent Study/Research
##### APMA 1970 S31
Primary Instructor
Crawford
Schedule Code
I: Independent Study/Research

• #### 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 2190 S01
Primary Instructor
Mallet-Paret
• #### 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 2230 S01
Primary Instructor
Dong
• #### 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.
##### APMA 2550 S01
Primary Instructor
Shu
• #### 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 2580A S01
Primary Instructor
Karniadakis
• #### 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.
##### APMA 2630 S01
Primary Instructor
Ramanan
• #### 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.
##### APMA 2670 S01
Primary Instructor
Gidas
• #### 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.
Prerequisites APMA 2630, APMA 2640; APMA 2550 (recommended).
• #### An Introduction to Stochastic Control

This is a course on the optimal control of random processes. Much of the course will focus on discrete time and optimal control of Markov chains (also called Markov Decision Theory in the context of Reinforcement Learning). Various optimality criteria are introduced and questions of existence of optimal controls and their characterization are addressed. Applications from finance, engineering and optimal stopping will be developed, and well as methods for numerical solution. Depending on interests and background, models that evolve in continuous time and/or with partial observations will also be considered. Prerequisites: APMA 2630/2640.
##### APMA 2812A S01
Primary Instructor
Dupuis
• #### 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 2812B S01
Primary Instructor
Punshon-Smith
• #### 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 2980 S01
Primary Instructor
Bienenstock
Schedule Code
I: Independent Study/Research
##### APMA 2980 S02
Primary Instructor
Dafermos
Schedule Code
I: Independent Study/Research
##### APMA 2980 S03
Primary Instructor
Dupuis
Schedule Code
I: Independent Study/Research
##### APMA 2980 S04
Primary Instructor
Rozovsky
Schedule Code
I: Independent Study/Research
##### APMA 2980 S05
Primary Instructor
Ramanan
Schedule Code
I: Independent Study/Research
##### APMA 2980 S06
Primary Instructor
Matzavinos
Schedule Code
I: Independent Study/Research
##### APMA 2980 S07
Primary Instructor
Geman
Schedule Code
I: Independent Study/Research
##### APMA 2980 S08
Primary Instructor
Gidas
Schedule Code
I: Independent Study/Research
##### APMA 2980 S09
Primary Instructor
Harrison
Schedule Code
I: Independent Study/Research
##### APMA 2980 S10
Primary Instructor
Wang
Schedule Code
I: Independent Study/Research
##### APMA 2980 S11
Primary Instructor
Crawford
Schedule Code
I: Independent Study/Research
##### APMA 2980 S12
Primary Instructor
Karniadakis
Schedule Code
I: Independent Study/Research
##### APMA 2980 S13
Primary Instructor
Lawrence
Schedule Code
I: Independent Study/Research
##### APMA 2980 S14
Primary Instructor
Maxey
Schedule Code
I: Independent Study/Research
##### APMA 2980 S15
Primary Instructor
McClure
Schedule Code
I: Independent Study/Research
##### APMA 2980 S16
Primary Instructor
Shu
Schedule Code
I: Independent Study/Research
##### APMA 2980 S17
Primary Instructor
Darbon
Schedule Code
I: Independent Study/Research
##### APMA 2980 S18
Primary Instructor
Menon
Schedule Code
I: Independent Study/Research
##### APMA 2980 S19
Primary Instructor
Dobrushkin
Schedule Code
I: Independent Study/Research
##### APMA 2980 S20
Primary Instructor
Guo
Schedule Code
I: Independent Study/Research
##### APMA 2980 S21
Primary Instructor
Klivans
Schedule Code
I: Independent Study/Research
##### APMA 2980 S22
Schedule Code
I: Independent Study/Research
##### APMA 2980 S23
Primary Instructor
Dong
Schedule Code
I: Independent Study/Research
##### APMA 2980 S24
Primary Instructor
Guzman
Schedule Code
I: Independent Study/Research
##### APMA 2980 S25
Primary Instructor
Mallet-Paret
Schedule Code
I: Independent Study/Research
##### APMA 2980 S26
Primary Instructor
Sandstede
Schedule Code
I: Independent Study/Research
##### APMA 2980 S27
Primary Instructor
Su
Schedule Code
I: Independent Study/Research
##### APMA 2980 S28
Schedule Code
I: Independent Study/Research
##### APMA 2980 S29
Primary Instructor
Ainsworth
Schedule Code
I: Independent Study/Research
• #### Thesis Preparation

For graduate students who have met the residency requirement and are continuing research on a full time basis.
##### APMA 2990 S01
Schedule Code
E: Grad Enrollment Fee/Dist Prep