
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.

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 gives a comprehensive introduction to the qualitative and quantitative theory of ordinary differential equations and their applications. Specific topics covered in the course are applications of differential equations in biology, chemistry, economics, and physics; integrating factors and separable equations; techniques for solving linear systems of differential equations; numerical approaches to solving differential equations; phaseplane analysis of planar nonlinear systems; rigorous theoretical foundations of differential equations.
Format: Six hours of lectures, and two hours of recitation.
Prerequisites: MATH 0100, MATH 0170, MATH 0180, MATH 0190, MATH 0200, MATH 0350 or advanced placement. MATH 0520 (can be taken concurrently).
 Primary Instructor
 Kunsberg

Applied Partial Differential Equations I
Covers the same material as APMA 0340, albeit of greater depth. Intended primarily for students who desire a rigorous development of the mathematical foundations of the methods used, for those students considering one of the applied mathematics concentrations, and for all students in the sciences who will be taking advanced courses in applied mathematics, mathematics, physics, engineering, etc. Three hours lecture and one hour recitation. Prerequisite: MATH 0100, 0170, 0180, 0190, 0200, or 0350, or advanced placement.

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
 Kinnaird

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
 Ramanan

Topics in Chaotic Dynamics
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.
 Primary Instructor
 MalletParet

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
 SanzAlonso

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.

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
 Garcia Trillos

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 wellrepresented 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/SrSMZZfFVHWsPkbz2
 Primary Instructor
 Sandstede

Topics in Information Theory and Coding Theory
This class builds on APMA 1710, but stresses applications of information and coding theory, rather than its mathematical foundations. The class provided an overview of widely used probabilistic methods and algorithms, such as Markov Chain Monte Carlo (MCMC), hidden Markov models (HMM), dynamic programming, belief propagation, and Bayesian inference. Information theory is used in combination with these algorithms as a framework to study applications such as codebreaking, speech recognition, image analysis and the study of genetic sequences.
This class is best suited to students looking for topics for senior theses or capstone classes in applied mathematics, computer science and mathematics.

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
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 I: Independent Study/Research
 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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 I: Independent Study/Research
 Primary Instructor
 Morrow
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 I: Independent Study/Research
 Primary Instructor
 Geman
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 I: Independent Study/Research
 Primary Instructor
 Gidas
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 I: Independent Study/Research
 Primary Instructor
 SanzAlonso
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 I: Independent Study/Research
 Primary Instructor
 Matzavinos
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 I: Independent Study/Research
 Primary Instructor
 Garcia Trillos
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 I: Independent Study/Research
 Primary Instructor
 Karniadakis
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 I: Independent Study/Research
 Primary Instructor
 Lawrence
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 I: Independent Study/Research
 Primary Instructor
 Maxey
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 I: Independent Study/Research
 Primary Instructor
 McClure
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 I: Independent Study/Research
 Primary Instructor
 Shu
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 I: Independent Study/Research
 Primary Instructor
 Klivans
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 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
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 I: Independent Study/Research
 Primary Instructor
 Harrison
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Kunsberg
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 I: Independent Study/Research
 Primary Instructor
 Wang
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 I: Independent Study/Research
 Primary Instructor
 Garcia Trillos
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 I: Independent Study/Research
 Primary Instructor
 Dong
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 I: Independent Study/Research
 Primary Instructor
 Guo
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 I: Independent Study/Research
 Primary Instructor
 Guzman
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 I: Independent Study/Research
 Primary Instructor
 MalletParet
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 I: Independent Study/Research
 Primary Instructor
 Menon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Sandstede
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 I: Independent Study/Research
 Primary Instructor
 Su
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 I: Independent Study/Research
 Primary Instructor
 Garcia Trillos
 Schedule Code
 I: Independent Study/Research

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
 Matzavinos

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 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.
 Primary Instructor
 Karniadakis

Computational Fluid Dynamics
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 NavierStokes 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.

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
 Garcia Trillos

Theory of Probability
A onesemester course in probability that provides an introduction to stochastic processes. The course covers the following subjects: Markov chains, Poisson process, birth and death processes, continuoustime martingales, optional sampling theorem, martingale convergence theorem, Brownian motion, introduction to stochastic calculus and Ito's formula, stochastic differential equations, the FeynmanKac formula, Girsanov's theorem, the BlackScholes formula, basics of Gaussian and stationary processes. Prerequisite: AMPA 2630 or equivalent course.

High Order Finite Element Methods: Theory and Implementation
Traditional FEM seeks accuracy through mesh refinement whereas high order FEM seek to achieve accuracy through the use of increasingly high order
polynomials. However, computer implementation and analysis of high order
methods is considerably more complex than for traditional FEM.
We will cover theory and implementation including: rate of convergence for
problems with singularities, choice of basis functions, cost of computing
element matrices and load vectors, and efficient solution of the resulting
matrix equation.
I will assume you are familiar with material in APMA2560 and have written
your own piecewise linear FEM code.
 Primary Instructor
 Ainsworth

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
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 I: Independent Study/Research
 Primary Instructor
 Geman
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Gidas
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 I: Independent Study/Research
 Primary Instructor
 Harrison
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Wang
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 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
 Darbon
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Menon
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 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Guo
 Schedule Code
 I: Independent Study/Research
 Schedule Code
 I: Independent Study/Research
 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
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 I: Independent Study/Research
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 I: Independent Study/Research
 Primary Instructor
 Ainsworth
 Schedule Code
 I: Independent Study/Research

Thesis Preparation
For graduate students who have met the tuition requirement and are paying the registration fee to continue active enrollment while preparing a thesis.
 Schedule Code
 E: Grad Enrollment Fee/Dist Prep