# 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, determinants, eigenvalues, eigenvectors, Markov processes. Optimization and least-squares analysis. Ordinary differential equations. Infinite series and Fourier series. Orthogonality and completeness. Introduction to partial differential equations. Applications in electrical and mechanical engineering.

## 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

Margo Levine,

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

Margo Levine,

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.

## Series Expansions and Complex Analysis

Introduces fundamental concepts for solving real-world problems and emphasizes their applications through examples from the physical and social sciences. Topics: series expansions and their convergence; complex functions, mappings, differentiation, integration, residues, Taylor and McLaurin expansions; wave (Fourier) and wavelet expansions and transformations, and their uses in signal and image analysis and solving differential equations.

## 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

Sarah Iams

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.

## Computational Geometry

An inquiry based and hands on exploration in computational geometry. Topics include: projective geometry (duality between points/lines, symmetry among spheric/planar/hyperbolic geometry), linear algebra (vectors, matrices, symmetry groups) and recursion. We will draw pretty pictures (fractals, tesselations, algebraic curves, etc.). We will write computer programs in Mathematica (and possibly Java, if time permits).

## Computational Geometry

An inquiry based and hands on exploration in computational geometry. Topics include: projective geometry (duality between points/lines, symmetry among spheric/planar/hyperbolic geometry), linear algebra (vectors, matrices, symmetry groups) and recursion. We will draw pretty pictures (fractals, tesselations, algebraic curves, etc.). We will write computer programs in Mathematica (and possibly Java, if time permits).

## Computational Geometry

An inquiry based and hands on exploration in computational geometry. Topics include: projective geometry (duality between points/lines, symmetry among spheric/planar/hyperbolic geometry), linear algebra (vectors, matrices, symmetry groups) and recursion. We will draw pretty pictures (fractals, tesselations, algebraic curves, etc.). We will write computer programs in Mathematica (and possibly Java, if time permits).

## Computational Geometry

## Computational Music Theory

An inquiry based and hands on exploration in computational music theory, combining mathematics, computer programming and aesthetics. Math topics: vector space model of music theory, binary tree model of scale theory. Programming in Mathematica: converting between notes and numbers, output to music notation, input from audio. Aesthetics: build your own musical instruments, invent your own music notation, compose pieces. You need to bring your laptop to class every day. Be sure to install & register Mathematica before the first class. Generally, we will program on Mondays and Wednesdays, and build/test instruments on Fridays. Grading based on final project, in-class assignments, in-class participation. No written exams or written homework outside class. You will present your finished programs, instruments and beautiful music to the class.

## Computational Music Theory

An inquiry based and hands on exploration in computational music theory, combining mathematics, computer programming and aesthetics. Math topics: vector space model of music theory, binary tree model of scale theory. Programming in Mathematica: converting between notes and numbers, output to music notation, input from audio. Aesthetics: build your own musical instruments, invent your own music notation, compose pieces. You need to bring your laptop to class every day. Be sure to install & register Mathematica before the first class. Generally, we will program on Mondays and Wednesdays, and build/test instruments on Fridays. Grading based on final project, in-class assignments, in-class participation. No written exams or written homework outside class. You will present your finished programs, instruments and beautiful music to the class.

## Computational Music Theory

An inquiry based and hands on exploration in computational music theory, combining mathematics, computer programming and aesthetics. Math topics: vector space model of music theory, binary tree model of scale theory. Programming in Mathematica: converting between notes and numbers, output to music notation, input from audio. Aesthetics: build your own musical instruments, invent your own music notation, compose pieces. You need to bring your laptop to class every day. Be sure to install & register Mathematica before the first class. Generally, we will program on Mondays and Wednesdays, and build/test instruments on Fridays. Grading based on final project, in-class assignments, in-class participation. No written exams or written homework outside class. You will present your finished programs, instruments and beautiful music to the class.

## Computational Music Theory

## Computational Music Theory

## Computational Music Theory

## 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

Christopher Rycroft,

Ariel Amir,

Margo Levine,

Sarah Iams,

Thomas Fai

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.

## 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 Applied Algebra

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; selected readings.

## 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.

## Fundamentals of Biological Signal Processing

The course will introduce Bayesian analysis, maximum entropy principles, hidden markov models and pattern theory. These concepts will be used to understand information processing in biology. The relevant biological background will be covered in depth.

## 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.

## Information Processing and Statistical Physics

This course introduces students to several fundamental notions and methods in statistical physics that have been successfully applied to the analysis of information processing systems. Discussions will be focused on studying such systems in the infinite-size limit, on analyzing the emergence of phase transitions, and on understanding the behaviors of efficient algorithms. This course seeks to start from basics, assuming just undergraduate probability and analysis, and in particular assuming no knowledge of statistical physics. Students will take an active role by applying what they learn from the course to their preferred applications.

## 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.