
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. Singlevariable 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.
 Primary Instructor
 Dafermos

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).
 Primary Instructor
 Sandstede

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.

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

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.

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.

Applied Partial Differential Equations II
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.
 Primary Instructor
 Matzavinos

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.

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.
 Primary Instructor
 Klivans

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.

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.
 Primary Instructor
 Harrison

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.

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
 Primary Instructor
 Dafermos
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 I: Independent Study/Research
 Primary Instructor
 Dupuis
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 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
 Darbon
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 Primary Instructor
 Matzavinos
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 Primary Instructor
 Karniadakis
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 I: Independent Study/Research
 Primary Instructor
 Lawrence
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 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
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 I: Independent Study/Research
 Primary Instructor
 Kunsberg
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 I: Independent Study/Research
 Primary Instructor
 Wang
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 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
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 I: Independent Study/Research
 Primary Instructor
 Sandstede
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 I: Independent Study/Research
 Primary Instructor
 Su
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 Primary Instructor
 Crawford
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 I: Independent Study/Research

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 2,000level credit enroll in 2080; for 1,000level credit enroll in 1080.
 Primary Instructor
 Lawrence

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.

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

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.
 Primary Instructor
 Karniadakis

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.
 Primary Instructor
 Ainsworth

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.

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.

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.

Topics in Stochastic Analysis
The course provides an introduction to the theory of continuoustime stochastic processes, stochastic integration, stochastic calculus and its applications. The course will cover Brownian motion, continuoustime martingales, stochastic integration, Ito's formula, Girsanov's theorem, and a selection of other topics such as excursion theory, reflected processes, stochastic control and stochastic filtering.
 Primary Instructor
 Ramanan

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
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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
 Matzavinos
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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
 Harrison
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 I: Independent Study/Research
 Primary Instructor
 Wang
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 I: Independent Study/Research
 Primary Instructor
 Crawford
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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
 Darbon
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 I: Independent Study/Research
 Primary Instructor
 Menon
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 I: Independent Study/Research
 Primary Instructor
 Dobrushkin
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 I: Independent Study/Research
 Primary Instructor
 Guo
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 I: Independent Study/Research
 Primary Instructor
 Klivans
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 I: Independent Study/Research
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 I: Independent Study/Research
 Primary Instructor
 Dong
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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
 Sandstede
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 I: Independent Study/Research
 Primary Instructor
 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