
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
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dafermos
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dupuis
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Rozovsky
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Ramanan
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Morrow
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Geman
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Gidas
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Darbon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Matzavinos
 Schedule Code
 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Karniadakis
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Lawrence
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Maxey
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 McClure
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Shu
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Klivans
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Harrison
 Schedule Code
 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Wang
 Schedule Code
 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dong
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Guo
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Guzman
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 MalletParet
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Menon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Sandstede
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Su
 Schedule Code
 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Crawford
 Schedule Code
 I: Independent Study/Research

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
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dafermos
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dupuis
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Rozovsky
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Ramanan
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Matzavinos
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Geman
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Gidas
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Harrison
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Wang
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Crawford
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Karniadakis
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Lawrence
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Maxey
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 McClure
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Shu
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Darbon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Menon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Guo
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Klivans
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 PunshonSmith
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dong
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Guzman
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 MalletParet
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Sandstede
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Su
 Schedule Code
 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 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.
 Schedule Code
 E: Grad Enrollment Fee/Dist Prep

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
 Dobrushkin

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
 Akopian

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).
 Primary Instructor
 Sandstede

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.

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 nonrepresentative 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.
 Primary Instructor
 Lawrence

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.
 Primary Instructor
 Bienenstock

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.

Operations Research: Probabilistic Models
Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birthdeath 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.
 Primary Instructor
 Matzavinos

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.

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
 PunshonSmith

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.

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 likelihoodratio tests, nonparametric tests, introduction to statistical computing, matrix approach to simplelinear and multiple regression, analysis of variance, and design of experiments. Prerequisite: APMA 1650, 1655 or equivalent, basic linear algebra.
 Primary Instructor
 Kaihnsa

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.
 Primary Instructor
 Dafermos

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.

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 RayleighRitz method, approximation properties of spectral end finite element methods, and solution techniques. Homework will include both theoretical and computational problems.

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.

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 2,000level
credit enroll in 2610; for 1,000level credit enroll in
1740. Rigorous calculusbased statistics,
programming experience, and strong mathematical
background are essential. For 2610, some graduate level
analysis is strongly suggested.
 Primary Instructor
 Harrison

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 2,000level
credit enroll in 2610; for 1,000level credit enroll in
1740. Rigorous calculusbased statistics,
programming experience, and strong mathematical
background are essential. For 2610, some graduate level
analysis is strongly suggested.
 Primary Instructor
 Harrison

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 secondsemester (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, corequisite—graduate real analysis.
 Primary Instructor
 Ramanan

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 curvefitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. Prerequisite: MATH 0100 or its equivalent.

Combinatorial Theory
An introduction to combinatorial theory at the graduate level. Areas to be covered include: posets and lattice theory, enumeration and generating functions, matroids and simplicial complexes, followed by selected topics in the field.
 Primary Instructor
 Klivans

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 multistep and multistage methods, error analysis, stepsize control for ordinary differential equations. Solution of twopoint 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.

Real Function Theory (MATH 2210)

An Introduction to Pattern Theory
Study Bayesian methods for inference that were pioneered by Grenander. The emphasis is mainly on models and algorithms, but the class also introduces theoretical tools such as Gaussian processes, stochastic gradient descent and mass transportation.
Class proceeds from Shannon’s model of text to patterns of increased complexity. The order of difficulty is text, music, character recognition, images, faces and natural scenes. The goal of pattern theory is to develop Bayesian models to synthesize and decode such patterns. There is not enough time to cover all these topics, so the class will progress by the development of theory in parallel with examples.

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.
 Primary Instructor
 Bienenstock
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dafermos
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dupuis
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Rozovsky
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Ramanan
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Morrow
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Geman
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Gidas
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Darbon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Matzavinos
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Karniadakis
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Lawrence
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Maxey
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 McClure
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Shu
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Klivans
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Harrison
 Schedule Code
 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Wang
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Dong
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Guo
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Guzman
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 MalletParet
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Menon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Sandstede
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Su
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Crawford
 Schedule Code
 I: Independent Study/Research

Race and Gender in the Scientific Community (MATH 1910)