This course is a systematic introduction to computing (with python and jupyter notebooks) for science and engineering applications. Applications are drawn from a broad range of disciplines, including physical, financial, and biological-epidemiological problems. The course consists of two parts: 1. Basics: essential elements of computing, including types of variables, lists, arrays, iteration and control flow (for, while loops, if statement), definition of functions, recursion, file handling and simple plots, plotting and visualization tools in higher dimensions. 2. Applications: development of computational skills for problem solving, including numerical and machine learning methods, and their use in deterministic and stochastic approaches; examples include numerical differentiation and integration, fitting of curves and error analysis, solution of simple differential equations, random numbers and stochastic sampling, and advanced methods like neural networks and simulated annealing for optimization in complex systems. Course work consists of attending lectures and labs, weekly homework assignments, a mid-term project and a final project; while work is developed collaboratively, coding assignments are submitted individually.

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.

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.

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.

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.

Complex analysis: complex numbers, functions, mappings, Laurent series, differentiation, integration, contour integration and residue theory, conformal mappings. Applications to visualization, art (especially M.C. Escher). Anamorphic images. Fourier Analysis: orthogonality, Fourier Series, Fourier transforms. Signal processing: sampling theorems (Nyquist, Shannon), fast Fourier transforms. Applications to image, audio analysis: filtering and deblurring.

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.

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.

This course provides an introduction to basic mathematical ideas and computational methods for optimization. Topics include linear programming, integer programming, branch-and-bound, branch-and-cut, as well as first-order gradient-based methods with an emphasis on modeling and data science applications.

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.

Mathematical theory and implementation aspects of well-established numerical algorithms applied in various scientific and engineering disciplines. The course will cover data fitting, numerical linear algebra, numerical differentiation and integration, optimization, and numerical solvers for differential equations. There will be a significant programming component. Students will be expected to implement a range of numerical methods as part of individual and group-based projects. The material is sufficiently diverse to match each student's background and programming skills.

The class aims to highlight the process of scientific discovery under uncertainty in the age of data. The class content stresses a unifying approach to data driven modeling and inference through stochastic  simulations, optimization and Bayesian uncertainty quantification. The class projects require transferring an idea to software in multi- and many-core computer architectures.

Mathematical modeling is the essential component of the revolution in computation-based research over the past decade. While designing mathematical models is itself an art form, it is equally important to learn how to transform them into computational systems that allow robust evaluation, which is critical for real-world use cases, from modeling the spread of COVID, to designing better large language models.  This course introduces mathematical modeling ideas while teaching how to transform them into robust computational frameworks for model evaluation and deployment, as done in industry. The aim is to give both a broad view of “what a mathematical model” is and, at the same time, to teach the core computational skills for building usable state-of-the-art models. Topics drawn from biology, economics, engineering, physical and social sciences.

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.

This course is an introduction to the theory of computation with artificial and biological neural networks. We will cover selected topics from theoretical neuroscience and deep learning theory with an emphasis on topics at the research frontier. These topics include expressivity and generalization in deep learning models; infinite-width limit of neural networks and kernel machines; deep learning dynamics; biologically-plausible learning and models of synaptic plasticity; reinforcement learning in the brain; neural population codes; normative theories of sensory representations; attractor network models of memory and spatial maps; sequential processing with recurrent neural networks and transformers; generative modeling.

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.

This course introduces students to fundamental results and recently developed techniques in high-dimensional probability theory and statistical physics that have been successfully applied to the analysis of information processing and machine learning problems. Discussions will be focused on studying such problems in the high-dimensional limit, on analyzing the emergence of phase transitions, and on understanding the scaling limits of efficient algorithms. This course seeks to start from basics, assuming just a solid understanding of undergraduate probability theory. Students will take an active role by exploring and applying what they learn from the course to their own research problems.

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.

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.