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, numerical differentiation, fitting of curves and error analysis, plotting and visualization tools in higher dimensions. 2. Advanced: root finding, series expansions, numerical integration, solving simple ordinary and partial differential equations, use of random numbers for sampling and simulations, such as Monte Carlo integration and random walks. 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.

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.

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.  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 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.

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 and other discrete transforms, wavelets. Applications to image, audio and morphological 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.

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.

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, 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 basic mathematical ideas and computational methods for solving deterministic optimization problems. Topics covered: linear programming, integer programming, branch-and-bound, branch-and-cut. Emphasis on modeling. Examples from business, society, engineering, sports, e-commerce. Exercises in AMPL, complemented by Mathematica 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.

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.

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 biological and artificial 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 training of neural networks and models of synaptic plasticity; reinforcement learning in the brain; neural population codes; normative theories of sensory representations; computing with dynamics in recurrent neural networks; attractor network models of memory and spatial maps. Concrete examples of applications of these ideas to the brain will be discussed.

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.

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) in these models. We will play particular attention to sparsity-induced regression models, that arise for instance in compressed sensing, 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  generative models of deep networks called model-based deep learning. At present, we lack a principled way to design artificial neural networks, the workhorses of modern AI systems. Moreover, modern AI systems lack the ability to explain how they reach their decisions. In other words, we cannot yet call AI explainable or interpretable which, as a society, poses important questions as to the responsible use of such technology. Model-based deep learning provides a framework to develop and constrain neural-network architectures in a principled fashion. 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. We will invite an exciting line up of speakers. Speakers will suggest papers that a group of students will present at the beginning of lecture, which will build up to a final project/paper that utilizes/on model-based deep learning applied to problems of interest to students.

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.