# 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. Applications to ordinary differential equations, difference equations, Markov chains, least squares. Examples draw upon everyday experience, economics, engineering, natural science, and statistics.

## Solving and Optimizing

This course covers a combination of linear algebra and multivariate calculus with an eye towards solving systems of equations and optimization problems. Students will learn how to prove some key results, and will also implement these ideas with code.

Linear algebra: matrices, vector spaces, bases and dimension, inner products, least squares problems, eigenvalues, eigenvectors, singular values, singular vectors.

Multivariate calculus: partial differentiation, gradient and Hessian, critical points, Lagrange Multipliers.

## 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. 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.

## 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, and machine learning including neural networks. Examples will be given from physical sciences, biology, climate, commerce, internet, image processing 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.

## 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: characteristics, eigenfunction expansions, transform techniques, integral relations, Green functions, variational methods, perturbation methods and asymptotic analysis.

## 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.

## Patterns

The topic for the course this year will be the dynamics of fluid flow, on scales that range from MHD in stars to active matter flows in cells to 2D electron gases, and of course fluid flows on the human scale, with which we will start. Specific content will include: Hydrodynamic Stability and Turbulence, Coherent Structures and Dynamical Systems, Methods for Numerical Simulation, Multiphase and Non-Newtonian Flows, Thin Film Flows, Biophysical, Geophysical and Astrophysical Flows, Active Matter Flows, Viscous Electronics. The course will meet for 150 minutes once a week, with 90 minutes devoted to lecture, and 60 minutes devoted to a discussion of 2-4 papers that build on the theory.

## 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, 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.

## Computational Methods in the Physical Sciences

In this Course, we shall familiarize with the main computational methods which permit to simulate and analyze the behavior of a wide range of problems involving fluids, solids, soft matter, electromagnetic and quantum systems, as well as the dynamics of (some) biological and social systems.

The course consists of three main parts,

Part I : Fields on Grids

Part II : Mesoscale Particle Methods

Part III: Statistical Data Analysis and Learning

In Part I, we shall discuss the fundamentals of grid discretization and present concrete applications to a broad variety of problems from classical and quantum physics, such as Advection-Diffusion Reaction transport, Navier-Stokes fluid-dynamics, nonlinear classical and quantum wave propagation. Both regular and complex geometrical grids will be discussed through Finite Differences, Volumes and Elements, respectively.

In Part II we shall discuss mesoscale technique based on the two basic mesoscale descriptions: probability distribution functions, as governed by Boltzmann and Fokker-Planck kinetic equations, and stochastic particle dynamics (Langevin equations). The lattice Boltzmann method will be discussed in great detail, with applications to fluids and soft matter problems. Mesoscale particle methods, such as Dissipative Particle Dynamics will also be illustrated in detail.

Finally, in Part III, we shall present data analysis & learning tools of particular relevance to complex systems with non-gaussian statistics, such as turbulence, fractional transport and extreme events in general. An introduction to Physics-Aware Machine Learning will also be presented.

## 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.