
Introduction to Scientific Computing
For student 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.
 Primary Instructor
 Kilikian

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
 Dafermos

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, 0170, 0180, 0190, 0200, 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

Inference in Genomics and Molecular Biology
Sequencing of genomes has generated a massive quantity of fundamental biological data. 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  application of probability theory to inferences on data sequence, the goal of enabling students to construct prob models. Statistical topics: Bayesian inferences, estimation, hypothesis testing and false discovery rates, statistical decision theory. Enroll in 2080 for more in depth coverage of the class.
Prerequisite: APMA 1650, 1655 or MATH 1610 or CSCI 1450; BIOL 0200 recommended, programming skills required.
 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. Prereq: functions of several variables (MATH 0180) or equivalent of this. Computational Linear Algebra (APMA 1170) or a similar course is recommended.

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.

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

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. This course can be used as a senior seminar. WRIT
 Primary Instructor
 Sandstede

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. Since 2016 is a presidential election year, examples throughout the course will be drawn from electoral politics.
Prerequisite: One year of universitylevel calculus. At Brown, this corresponds to MATH 0100, 0170, 0180, 0190, 0200, or 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.
 Primary Instructor
 Garcia Trillos

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

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/5l62uscm1Zmzz5YJ2
 Primary Instructor
 Sandstede
 Primary Instructor
 Sandstede

Topics in Coding Theory
This class covers two distinct areas: (1) algebraic coding theory; (2) examples of code breaking and design.
Part (1) stresses cryptography, data compression, error correction and sphere packings.
Part (2) will involve case studies of code breaking and code design in applications. Depending on student interest these may include decoding scripts (Ventris and Linear B), or design problems in synthetic biology (e.g. RNA folding and DNA selfassembly).

Randomized Algorithms for Counting, Integration and Optimzation
We consider the construction and analysis of random methods for approximating sums and integrals, and related questions. Example, consider the problem of counting the number of vectors with integer components that satisfy a collection of linear equality and inequality constraints. Depending on the number of constraints, this could be a problem of counting the number of needles in a haystack, and straightforward enumeration is impossible. There are now a variety of randomized methods that can attack this problem and other problems with similar difficult features. We survey some of the methods and the problems to which they apply.

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
 Li
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 I: Independent Study/Research
 Primary Instructor
 Matzavinos
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 I: Independent Study/Research
 Primary Instructor
 Hesthaven
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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
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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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 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
 Schedule Code
 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
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Garcia Trillos
 Schedule Code
 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

Hilbert Spaces and Their Applications
A continuation of APMA 2110: metric spaces, Banach spaces, Hilbert spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential equations.

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

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.
 Primary Instructor
 Ainsworth

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

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
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.

Mathematical Statistics II
The course covers modern nonparametric statistical methods. Topics include: density estimation, multiple regression, adaptive smoothing, crossvalidation, bootstrap, classification and regression trees, nonlinear discriminant analysis, projection pursuit, the ACE algorithm for time series prediction, support vector machines, and neural networks. The course will provide the mathematical underpinnings, but it will also touch upon some applications in computer vision/speech recognition, and biological, neural, and cognitive sciences. Prerequisite: APMA 2670.

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.

Dynamics and Stochastics
This course provides a synthesis of mathematical problems at the interface between stochastic problems and dynamical systems that arise in systems biology. For instance, in some biological systems some species may be modeled stochastically while other species can be modeled using deterministic dynamics. Topics will include an introduction to biological networks, multiscale analysis, analysis of network structure, among other topics. Prerequisites: probability theory (APMA 2630/2640, concurrent enrollment in APMA 2640 is acceptable).
 Primary Instructor
 Ramanan

Neural Dynamics: Theory and Modeling
Our thoughts and actions are mediated by the dynamic activity of the brain’s neurons. This course will use mathematics and computational modeling as a tool to study neural dynamics at the level of signal neurons and in more complicated networks. We will focus on relevance to modern day neuroscience problems with a goal of linking dynamics to function. Topics will include biophysically detailed and reduced representations of neurons, bifurcation and phase plane analysis of neural activity, neural rhythms and coupled oscillator theory. Audience: advanced undergraduate or graduate students. Prerequisite: APMA 03500360 and Matlab programming course. Instructor permission required.

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
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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
 Hesthaven
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 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
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 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
 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
 Schedule Code
 I: Independent Study/Research
 Primary Instructor
 Tice
 Schedule Code
 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