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

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.

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.
 Primary Instructor
 Lawrence

An Introduction to Numerical Optimization
This course provides a thorough introduction to numerical methods and algorithms for solving nonlinear 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, trustregion methods, nonlinear conjugate gradient methods, an introduction to constrained optimization (KarushKuhnTucker conditions, minimaximization, saddlepoints 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.

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.), roundoff 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.

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. SturmLiouville problem and eigenfunction expansions.

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 universitylevel 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.
 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.

Computational Neuroscience (NEUR 1680)

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 calculusbased 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.
 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, sourcechannel separation, lossy data compression. Prerequisite: one course in probability.

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.
 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 FeynmanKac 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.
 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.
 Primary Instructor
 Bienenstock
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 I: Independent Study/Research
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 Dafermos
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 Dupuis
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 Rozovsky
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 Ramanan
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 Primary Instructor
 Morrow
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 Geman
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 Primary Instructor
 Gidas
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 Darbon
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 Matzavinos
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 Primary Instructor
 Karniadakis
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 Lawrence
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 Primary Instructor
 Maxey
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 Primary Instructor
 McClure
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 Primary Instructor
 Shu
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 Primary Instructor
 Klivans
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 Primary Instructor
 Dobrushkin
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 Harrison
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 Primary Instructor
 Kunsberg
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 Wang
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 Primary Instructor
 Dong
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 Primary Instructor
 Guo
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 Primary Instructor
 Guzman
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 Primary Instructor
 MalletParet
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 Primary Instructor
 Menon
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 Primary Instructor
 Sandstede
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 Su
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 Crawford
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Real Function Theory (MATH 2210)

Nonlinear Dynamical Systems: Theory and Applications
Basic theory of ordinary differential equations, flows, and maps. Twodimensional 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.
 Primary Instructor
 MalletParet

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.

Numerical Solution of Partial Differential Equations I
Finite difference methods for solving timedependent initial value problems of partial differential equations. Fundamental concepts of consistency, accuracy, stability and convergence of finite difference methods will be covered. Associated wellposedness theory for linear timedependent PDEs will also be covered. Some knowledge of computer programming expected.

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 NavierStokes equations and their simplified models, learn about highorder explicit and implicit methods, time stepping, and fast solvers. We will then cover advectiondiffusion equations and various forms of the NavierStokes equations in primitive variables and in vorticity/streamfunction formulations. In addition to the homeworks the students are required to develop a NavierStokes solver as a final project.
 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, BorelCantelli lemmas, weak and strong laws of large numbers, conditional expectation and beginning of discrete time martingale theory. Prerequisites—undergraduate probability and analysis, corequisite—graduate real analysis.
 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 (EM Algorithm, Markov Chain Monte Carlo, Bootstrap). Prerequisite: APMA 2630 or equivalent.

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.

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 wellposedness for semilinear SPDE with additive noise, weak Martingale solutions to 3D NavierStokes, Markov processes on Polish spaces, the Krylov–Bogolyubov theorem, the DoobKhasminskii theorem, and BismutElworthyLi formula for a class of nondegenerate 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.
 Primary Instructor
 PunshonSmith

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.
 Primary Instructor
 Bienenstock
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 I: Independent Study/Research
 Primary Instructor
 Dafermos
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 Primary Instructor
 Dupuis
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 I: Independent Study/Research
 Primary Instructor
 Rozovsky
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 I: Independent Study/Research
 Primary Instructor
 Ramanan
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 Primary Instructor
 Matzavinos
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 Primary Instructor
 Geman
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 Primary Instructor
 Gidas
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 I: Independent Study/Research
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 Harrison
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 Wang
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 Crawford
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 Karniadakis
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 Lawrence
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 Primary Instructor
 Maxey
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 Primary Instructor
 McClure
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 Primary Instructor
 Shu
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 Primary Instructor
 Darbon
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 Primary Instructor
 Menon
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 Primary Instructor
 Dobrushkin
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 Guo
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 Klivans
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 Dong
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 Primary Instructor
 Guzman
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 MalletParet
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 Primary Instructor
 Sandstede
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 Su
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 Primary Instructor
 Ainsworth
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Thesis Preparation
For graduate students who have met the residency requirement and are continuing research on a full time basis.
 Schedule Code
 E: Grad Enrollment Fee/Dist Prep