# Applied Mathematics Courses

**For more information on specific courses, including prerequisites, registration details and any last-minute changes, visit my.harvard**

## Mathematical Methods in the Sciences

Multivariate calculus: functions of two or three variables, approximating functions, partial differentiation, directional derivatives, multiple integration. Vectors: dot and cross products, parameterized curves, line and surface integrals. Vector calculus: gradient, divergence and curl, Green's, divergence and Stokes' theorems. Complex numbers.

## Mathematical Methods in the Sciences

Linear algebra: matrices, vector spaces, linear maps, determinants, eigenvalues, eigenvectors, inner products, and singular values. Ordinary differential equations, difference equations, Markov chains, least squares, and Fourier series. Examples draw upon everyday experience, economics, engineering, natural science, and statistics.

## Introduction to Applied Mathematics

This course provides an introduction to the problems and issues of applied mathematics, focusing on areas where mathematical ideas have had a major impact on diverse fields of human inquiry. The course is organized around two-week topics drawn from a variety of fields, and involves reading classic mathematical papers in each topic. The course also provides an introduction to mathematical modeling and programming.

## Supervised Reading and Research

Sarah Iams

An individual project of guided reading and research culminating in a substantial paper or other piece of work which can be meaningfully evaluated to assign a letter grade; may not be taken on a PA/FL basis. Students engaged in preparation of a senior thesis ordinarily should take Applied Mathematics 99r instead.

## Supervised Reading and Research

Sarah Iams

An individual project of guided reading and research culminating in a substantial paper or other piece of work which can be meaningfully evaluated to assign a letter grade; may not be taken on a PA/FL basis. Students engaged in preparation of a senior thesis ordinarily should take Applied Mathematics 99r instead.

## Thesis Research

Sarah Iams

Provides an opportunity for students to engage in preparatory research and the writing of a senior thesis. Graded on a SAT/UNS basis as recommended by the thesis supervisor. The thesis is evaluated by the supervisor and by two additional readers.

## Thesis Research

Sarah Iams

Provides an opportunity for students to engage in preparatory research and the writing of a senior thesis. Graded on a SAT/UNS basis as recommended by the thesis supervisor. The thesis is evaluated by the supervisor and by two additional readers.

## Statistical Inference for Scientists and Engineers

Introductory statistical methods for students in the applied sciences and engineering. Random variables and probability distributions; the concept of random sampling, including random samples, statistics, and sampling distributions; the Central Limit Theorem and its role in statistical inference; parameter estimation, including point estimation and maximum likelihood methods; confidence intervals; hypothesis testing; simple linear regression; and multiple linear regression. Introduction to more advanced techniques as time permits.

## Complex Analysis and Series Expansions for Applications to Science, Engineering and Finance

Introduces fundamental concepts for solving real-world problems and emphasizes their applications through motivational examples drawn from science, engineering and finance. Topics: special distribution functions; series expansions of functions and their convergence; functions of complex variables, Taylor and Laurent expansions; wave (Fourier) and wavelet expansions and transforms, and their uses in solving differential equations and in signal analysis; connections to machine learning (neural networks), probabilities, random numbers and stochastic optimization methods.

## Ordinary and Partial Differential Equations

Ordinary differential equations: power series solutions; special functions; eigenfunction expansions. Review of vector calculus. Elementary partial differential equations: separation of variables and series solutions; diffusion, wave and Laplace equations. Brief introduction to nonlinear dynamical systems and to numerical methods.

## Applied Algebra

Introduction to abstract algebra and its applications. Sets, subsets, and partitions; mappings, operations, and equivalence relations; groups, rings, and fields, polynomials, encryption, computer coding, application of modular arithmetic, combinatorial designs, lattices, application of trellis representation of lattices, fast algorithms.

## Graph Theory and Combinatorics

Topics in combinatorial mathematics that find frequent application in computer science, engineering, and general applied mathematics. Specific topics taken from graph theory, enumeration techniques, optimization theory, combinatorial algorithms, and discrete probability.

## Nonlinear Dynamical Systems

An introduction to nonlinear dynamical phenomena, covering the behavior of systems described by ordinary differential equations. Topics include: stability; bifurcations; chaos; routes to chaos and universality; approximations by maps; strange attractors; fractals. Techniques for analyzing nonlinear systems are introduced with applications to physical, chemical, and biological systems such as forced oscillators, chaotic reactions, and population dynamics.

## Introduction to Scientific Computing

Many complex physical problems defy simple analytical solutions or even accurate analytical approximations. Scientific computing can address certain of these problems successfully, providing unique insight. This course introduces some of the widely used techniques in scientific computing through examples chosen from physics, chemistry, and biology. The purpose of the course is to introduce methods that are useful in applications and research and to give the students hands-on experience with these methods.

## Mathematical Modeling

Abstracting the essential components and mechanisms from a natural system to produce a mathematical model, which can be analyzed with a variety of formal mathematical methods, is perhaps the most important, but least understood, task in applied mathematics. This course approaches a number of problems without the prejudice of trying to apply a particular method of solution. Topics drawn from biology, economics, engineering, physical and social sciences.

## Mathematical Modeling

Abstracting the essential components and mechanisms from a natural system to produce a mathematical model, which can be analyzed with a variety of formal mathematical methods, is perhaps the most important, but least understood, task in applied mathematics. This course approaches a number of problems without the prejudice of trying to apply a particular method of solution. Topics drawn from biology, economics, engineering, physical and social sciences.

## Applied Linear Algebra and Big Data

Topics in linear algebra which arise frequently in applications, especially in the analysis of large data sets: linear equations, eigenvalue problems, linear differential equations, principal component analysis, singular value decomposition, data mining methods including frequent pattern analysis, clustering, outlier detection, classification, machine learning, modeling and prediction. Examples will be given from physical sciences, biology, climate, commerce, internet, image processing, economics and more.

## Introduction to Optimization: Models and Methods

Introduction to basic mathematical ideas and computational methods for solving deterministic and stochastic optimization problems. Topics covered: linear programming, integer programming, branch-and-bound, branch-and-cut, Markov chains, Markov decision processes. Emphasis on modeling. Examples from business, society, engineering, sports, e-commerce. Exercises in AMPL, complemented by Maple or Matlab.

## Feedback Control Systems: Analysis and Design

This course provides an introduction to feedback and control in physical, biological, engineering, information, financial, and social sciences. The focus is on the basic principles of feedback and its use as a tool for inferring and/or altering the dynamics of systems under uncertainty. Key themes throughout the course will include linear system analysis, state/output feedback, frequency response, reference tracking, PID controller, dynamic programming, and limit of performance. This includes both the practical and theoretical aspects of the topic.

## Physical Mathematics I

Introduction to methods for developing accurate approximate solutions for problems in the sciences that cannot be solved exactly, and integration with numerical methods and solutions. Topics include: dimensional analysis, algebraic equations, complex analysis, perturbation theory, matched asymptotic expansions, approximate solution of integrals.

## Physical Mathematics II

Theory and techniques for finding exact and approximate analytical solutions of partial differential equations: eigenfunction expansions, Green functions, variational calculus, transform techniques, perturbation methods, characteristics, integral equations, selected nonlinear PDEs including pattern formation and solitons, introduction to numerical methods.

## Introduction to Disordered Systems and Stochastic Processes

The course will familiarize the students with various applications of probability theory, stochastic modeling and random processes, using examples from various disciplines, including physics, biology and economics.

## Advanced Scientific Computing: Numerical Methods

An examination of the mathematical foundations of a range of well-established numerical algorithms, exploring their use through practical examples drawn from a range of scientific and engineering disciplines. Emphasizes theory and numerical analysis to elucidate the concepts that underpin each algorithm. There will be a significant programming component. Students will be expected to implement a range of numerical methods through individual and group-based project work to get hands-on experience with modern scientific computing.

## Advanced Scientific Computing: Stochastic Methods for Data Analysis, Inference and Optimization

Develops skills for computational research with focus on stochastic approaches, emphasizing implementation and examples. Stochastic methods make it feasible to tackle very diverse problems when the solution space is too large to explore systematically, or when microscopic rules are known, but not the macroscopic behavior of a complex system. Methods will be illustrated with examples from a wide variety of fields, like biology, finance, and physics.

## Inverse Problems in Science and Engineering

Many problems in science and engineering are inverse problems. Any experiment that requires an explanation can be couched thus - given the data, what is the theory/model that provides it - this is an inverse problem. In engineering, a given function (in a product/software …. ) requires a design - again an inverse problem. This course will introduce a wide array of features of inverse problems from science and engineering - from oil prospecting and seismology to cognitive science, from particle physics to engineering design. We will then introduce deterministic and probabilistic algorithms for solving these problems. Much of the class will be spent studying how the recent revolution in deep neural networks can (and cannot) be used to solve such inverse problems. The class will have a substantial computational component -- part of every class session will contain instruction and computer implementation of the algorithms in question. Students will carry out final projects in their own area of interest. Programming will be taught and carried out in Python and Tensorflow.

## Instabilities and Patterns in Soft Matter and Biophysics

The course discusses various kinds of instabilities in soft matter systems including active swarms, gels, droplets, filament assemblies, and disordered solids. After an introduction to bifurcation theory and spatio-temporal instabilities, we discuss in each lecture a new class of systems where an instability occurs, often leading to patterns, failure, or sudden shape, flow or transport changes. Each lecture consists of an introductory segment and then bridges to current research topics. We will offer mini research projects the student can complete until the end of the term.

## Advanced Optimization

This is a graduate level course on optimization which provides a foundation for applications such as statistical machine learning, signal processing, finance, and approximation algorithms. The course will cover fundamental concepts in optimization theory, modeling, and algorithmic techniques for solving large-scale optimization problems. Topics include elements of convex analysis, linear programming, Lagrangian duality, optimality conditions, and discrete and combinatorial optimization. Exercises and the class project will involve developing and implementing optimization algorithms.

## Advanced Scientific Computing: Numerical Methods for Partial Differential Equations

This course examines a variety of advanced numerical methods, with a focus on those relevant to solving partial differential equations that arise in physical problems. Topics include the finite volume method, finite element method, discontinuous Galerkin method, and interface tracking methods. Associated problems in numerical linear algebra and optimization will be discussed. The course will examine the mathematical underpinnings of each method, as well as look at their practical usage, paying particular attention to efficient implementations on modern multithreaded and parallel computer architectures.

## Decision Theory

Mathematical analysis of decision making. Bayesian inference and risk. Maximum likelihood and nonparametric methods. Algorithmic methods for decision rules: perceptrons, neural nets, and back propagation. Hidden Markov models, Blum-Welch, principal and independent components.

## Special Topics in Applied Mathematics

Supervision of experimental or theoretical research on acceptable applied mathematics problems and supervision of reading on topics not covered by regular courses of instruction.

## Special Topics in Applied Mathematics

Supervision of experimental or theoretical research on acceptable applied mathematics problems and supervision of reading on topics not covered by regular courses of instruction.