# Applied Mathematics Courses

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

## Computing for Science and Engineering

This course is a systematic introduction to a computing environment suited for applications in science and engineering. It is designed as a tutorial, which will last through an entire semester, and will involve approximately half of the amount of work of a regular 4-credit course. It consists of three Modules: In the first Module (“Basics”), the basics of using computing to work with variables and functions are introduced, including arrays, basic operations (loops) and file handling and plotting. In the second Module (“Elementary”), elementary applications are presented, including integration using random numbers, function root finding, and function propagation through series (Taylor) expansions. In the third Module (“Advanced”) more advanced applications are explored, including numerical differentiation and integration, solution of differential equations, and simple visualization tools.

## Computing for Science and Engineering

This course is a systematic introduction to a computing environment suited for applications in science and engineering. It is designed as a tutorial, which will last through an entire semester, and will involve approximately half of the amount of work of a regular 4-credit course. It consists of three Modules: In the first Module (“Basics”), the basics of using computing to work with variables and functions are introduced, including arrays, basic operations (loops) and file handling and plotting. In the second Module (“Elementary”), elementary applications are presented, including integration using random numbers, function root finding, and function propagation through series (Taylor) expansions. In the third Module (“Advanced”) more advanced applications are explored, including numerical differentiation and integration, solution of differential equations, and simple visualization tools.

## 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.

## Integrating and Approximating

Multivariable and vector calculus, supplemented with numerical methods. Multivariate calculus: multiple integration, functions of two or three variables, approximating functions. Parameterized curves, line and surface integrals. Vector calculus: gradient, divergence and curl, Green’s, divergence theorems. Complex numbers. Select differential equations topics.

## 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

Supervised reading or research on topics not covered by regular courses. For AM concentrators, work may be supervised by faculty in other departments. For non-concentrators, work must be supervised by an AM faculty member. Students must receive the approval of an (Associate) Director of Undergraduate Studies and obtain their signature before submitting AM91r forms.

## Supervised Reading and Research

Sarah Iams

Supervised reading or research on topics not covered by regular courses. For AM concentrators, work may be supervised by faculty in other departments. For non-concentrators, work must be supervised by an AM faculty member. Students must receive the approval of an (Associate) Director of Undergraduate Studies and obtain their signature before submitting AM91r forms.

## 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 one additional reader.

## 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 one additional reader.

## 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; functions of complex variables, Taylor and Laurent series expansions and their convergence; contour integration; wave (Fourier) and wavelet expansions and transforms, and their uses in signal analysis and in solving partial differential equations, focusing on the evolution equations (the diffusion equation and the wave equation) and their applications.

## 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. Course focuses on graph theory on one hand, and enumeration on the other. Specific topics include graph matching and graph coloring, generating functions and recurrence relations, combinatorial algorithms, and discrete probability. Emphasis on problem solving and proofs.

## Nonlinear Dynamical Systems

An introduction to nonlinear dynamical phenomena, focused on identifying the long term behavior of systems described by ordinary differential equations. The emphasis is on stability and parameter dependence (bifurcations). Other topics include: chaos; routes to chaos and universality; 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, and machine learning, including neural networks and random forests. 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.

## 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.

## Neural Computation

This course introduces advanced mathematical methods and models used in theoretical neuroscience. We will explore computations and functions performed by the brain, and how they are implemented by neurons and their networks. We will cover selected topics from spiking neuron models; population codes; normative theories of sensory representations; learning with synaptic plasticity; computing with dynamics in recurrent neural networks; attractor network models of memory and spatial maps; neural models of probabilistic inference in the brain and drift-diffusion models of decision making. Concrete examples of applications of these ideas to the brain will be discussed. Topics at the research frontier will be emphasized.

## 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 : Classical and Quantum Fields on Grids

Part II : Mesoscale 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.

In addition, we shall provide the opportunity of hands-on on a multi scale codes for X (extreme) simulations at the interface between physics and molecular biology.

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.

## 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.