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 or research on topics not covered by regular courses. It cannot be taken as a fifth course. For AM concentrators, work may be supervised by faculty in other departments. For non-concentrators, work must be supervised by an AM faculty member. To be eligible to enroll in the course, students must receive the approval of the course instructors, including approved registration forms, prior to the start of the semester.

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.

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.

This course is an introduction to abstract algebra and its applications. Topics will include rings, polynomials, and ideals, factorization of matrices and polynomials, exact and numerical algorithms for solving equations, and applications to data analysis, modeling, and optimization.

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.

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.

This course serves as an introduction to partial differential equations (PDE) and their applications across the sciences. The course will familiarize students with the process of starting with a model, deriving the appropriate PDE, and solving it. Examples include wave equations, diffusion equations, the Laplace equation, and several nonlinear equations such as the Burgers and KdV equations. To build intuition for the analytical solutions, simple numerical simulations will be utilized.

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.

Topics in linear algebra that frequently arise 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 and machine learning methods: clustering (unsupervised learning) and classification (supervised) using neural networks and random forests. Examples from physical sciences, biology, climate, commerce, the internet, image processing, and more will be given. The approach is application-motivated, focusing on an intuitive understanding of the algorithms behind these methods obtained by analyzing small data sets. Programming assignments can be done using Python or Matlab.

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.

Recently, there has been a surge of interest in exploiting geometric structure in data and models in machine learning. This course will give an overview of this emerging research area and its mathematical foundation, with a focus on recent literature and open problems. We will cover a range of topics at the intersection of geometry and machine learning including basic differential geometry, graph representation learning, manifold learning, graph neural networks, machine learning on manifolds, and geometric deep learning. Lectures will be complemented by student-led discussions of relevant papers.

Active matter describes out of equilibrium systems that consume energy to do work and become functional. Understanding their behavior and function has implications for biology and complex systems across scales, from cells to ecosystems, e.g., morphogenesis, collective behavior of flocks and herds, neurodynamics of locomotion, etc. The tools and concepts needed include non-equilibrium statistical mechanics, kinetic theory, soft matter, and hydrodynamics; methods for the analysis of the models include scaling, coarse-graining (homogenization, renormalization) and computational algorithms (for stochastic and deterministic DE). This course will provide an introduction to the questions, techniques and successes of this exploding field that cuts across the physical and biological sciences.

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