
What’s the big deal with Data Science?
This seminar serves as a practical introduction to the interdisciplinary field of data science. Over the course of the semester, students will be exposed to the diversity of questions that data science can address by reading current scholarly works from leading researchers. Through handson labs and experiences, students will gain facility with computational and visualization techniques for uncovering meaning from large numerical and textbased data sets. Ultimately, students will gain fluency with data science vocabulary and ideas. There are no prerequisites for this course. FYS WRIT
 Primary Instructor
 Kinnaird

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.

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
 Akopian

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.

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

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

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.
 Primary Instructor
 Rozovsky

Applied Partial Differential Equations II
Mathematical methods based on functions of a complex variable. Fournier series and its applications to the solution of onedimensional heat conduction equations and vibrating strings. Series solution and special functions. Vibrating membrance. SturmLiouville problem and eigenfunction expansions. Fournier transform and wave propagations.
 Primary Instructor
 SanzAlonso

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
 Kunsberg

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 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
 Garcia Trillos

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.

Approximations for Piecewise Smooth Functions
We will discuss approximation methods for piecewise smooth functions
with isolated discontinuities. Such piecewise
smooth functions appear often in applications, most notably in
computational fluid dynamics of high speed flows. The basic
background required is APMA 03300340, and some knowledge of
programming (e.g. MATLAB or FORTRAN or C). APMA 1170 and/or
APMA 1180 are helpful but not required. Students will be asked to
participate actively in the class, and perform individual or group
projects which may be designed to fit the interest of each student
or group.

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
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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
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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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 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.
 Primary Instructor
 Dafermos

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 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.
 Primary Instructor
 Ainsworth

Theory of Probability
A onesemester course that provides an introduction to probability theory based on measure theory. The course covers the following topics: probability spaces, random variables and measurable functions, independence and infinite product spaces, expectation and conditional expectation, weak convergence of measures, laws of large numbers and the Central Limit Theorem, discrete time martingale theory and applications.
 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.

Convex Analysis and Minimization Algorithms
This course provides a solid mathematical presentation of modern convex analysis and convex optimization algorithms for large scale problems. Topics include: subdifferential calculus, duality and FenchelLegendre transform, proximal operators and Moreau's regularization, optimal firstorder methods, Augmented Lagrangian methods and alternating direction method of multipliers, network flows. The course will provide the mathematical and algorithmical underpinnings. It will also explore some applications in signal and image processing, optimal control and machine learning.

Finitite Element Exterior Calculus
In this course we will cover finite elements for the Hodge Laplacian. We start in three dimensions and discuss the Nedelec finite element spaces for H^1, H(curl) and H(div) and discuss the corresponding de Rham complex. We discuss how they can be applied to the Stokes problem and electromagnetic problems. We then generalize these spaces to higher dimensions and show how to use them to approximate the Hodge Laplacian. We will mostly follow the review paper: [Finite Element Exterior Calculus: from Hodge Theory to Numerical Stability].

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
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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
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 I: Independent Study/Research
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 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
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 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
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 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