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.

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; parameter estimation; confidence intervals; hypothesis testing; simple linear regression; and multiple linear regression. Introduction to more advanced techniques as time permits.

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.

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.

Many science and engineering problems don’t have 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, biology, computer science and other fields. 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. The main programming language will be Python.

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, economics, 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.

This course covers optimal control and reinforcement learning for dynamical systems, with a strong emphasis on robotic applications such as quadrupeds and humanoids. The first half focuses on optimal control for systems with known, potentially nonlinear dynamics. Students will learn the fundamentals of dynamic programming and the linear quadratic regulator (LQR) before delving into trajectory optimization and model predictive control (MPC) for nonlinear systems, which emphasizes numerical optimization techniques for synthesizing complex motions.

The second half explores reinforcement learning (RL) for systems with unknown dynamics. Topics include both model-free and model-based RL algorithms such as proximal policy optimization (PPO), actor-critic methods, and model-based policy optimization, with a focus on continuous state and action spaces. Additional topics may include Lyapunov analysis, vision-based feedback control, and advanced convex optimization. The course prioritizes computational algorithms over theoretical analysis, equipping students with practical tools for solving complex control problems.

Assignments involve programming in Python and MATLAB to control simulated dynamical systems in MuJoCo and other environments.

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 methods 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.

Algebra gives mathematical abstractions that allow us to process information. Many optimization problems in data and learning are built on algebraic ideas. For example, principal component analysis finds a low rank approximation of a matrix, a problem central to linear algebra. This course builds out from this example to study the algebraic fundamentals of optimization problems to find representations of data. The course combines mathematical theory, computational experiments, and exploration of data. The focus is on current research developments and connections to open problems. By the end, students will have a unified algebraic toolbox to understand existing methods, to design new models, and to prove results on their theoretical underpinnings.

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.

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.

ES 201/AM 231 is a course in statistical inference and estimation from a signal processing perspective. The course will emphasize the entire pipeline from writing a model, estimating its parameters and performing inference utilizing real data. The first part of the course will focus on linear and nonlinear probabilistic generative/regression models (e.g., linear, logistic, Poisson regression), and algorithms for optimization (ML/MAP estimation) and Bayesian inference in these models. We will pay particular attention to sparsity-induced regression models, because of their relation to artificial neural networks, the topic of the second part of the course. The second part of the course will introduce students to the nascent and exciting research area of model-based deep learning and sparse auto-encoders. We will see, for instance, how neural-networks with ReLU nonlinearities arise from sparse probabilistic generative models introduced in the first part of the course. This will form the basis for a rigorous recipe we will teach you to build interpretable deep neural networks, from the ground up. More broadly, model-based deep learning and sparse auto-encoders have become popular approaches to reverse-engineer intelligence in both biological and artificial settings: in each case, we are able to train these systems to perform complicated tasks, but our understanding of how they do so remains opaque.

Reverse engineering intelligence—whether in the brain or in artificial neural networks—means using mathematical tools to reveal what information these systems truly represent. By moving beyond performance metrics to probe internal representations, we gain interpretability and transparency, with real-world benefits for the safety, fairness, and trustworthiness of modern AI and brain-machine systems

We will invite an exciting lineup of speakers. We encourage you to pursue a final project that could lead to prototype or production solutions to challenges businesses face around AI adoption due to lack of transparency.

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.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.

Experimental or theoretical research project on acceptable problems in applied mathematics supervised by a SEAS faculty member, and/or supervised reading on topics not covered by regular courses of instruction. The project or reading must be arranged between the student and individual SEAS faculty supervisor prior to enrolling in the course.