This course is a systematic introduction to computing (with Python and Jupyter notebooks) for science and engineering applications. Examples of applications are drawn from a broad range of disciplines, including physical, financial, and biological-epedemiological problems. The course consists of two Modules: 1. Basics: essential elements of computing, including types of variables, lists, arrays, basic recursive operations (for, while loops, if statement), definition of functions, 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 and circle packings. Applications to visualization, art (especially M.C. Escher) and photography.  Fourier Analysis:  orthogonality, Fourier Series, Fourier transforms. Signal processing: sampling theorems (Nyquist, Shannon), fast Fourier and other discrete transforms, wavelets and filtering. Applications to image, video, audio and morphological analysis:  filtering and cleaning images, musical analysis, fraud and authentication, filter banks for engineering.

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.

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.

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.

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. This class follows a "flipped-classroom" format; students are required to watch the lecture videos and study new materials prior to each class meeting (including the first class meeting). 

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. This class follows a "flipped-classroom" format; students are required to watch the lecture videos and study new materials prior to each class meeting (including the first class meeting). 

This class presents fundamental computing concepts for the solution of problems at interfaces of science and engineering. The course will connect dynamical systems and machine learning, juxtapose deterministic and stochastic simulation and present a unifying approach to stochastic methods for modeling, search, optimization and uncertainty quantification. Class projects will emphasize the steps necessary to transfer an idea to software in multi- and many-core computer architectures.

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.

This course introduces advanced mathematical methods and models used in theoretical neuroscience and theory of neural networks. 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 deep learning theory; spiking neuron models; population codes; normative theories of sensory representations; models of 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.

This graduate level course studies dynamic systems in time domain with inputs and outputs. Students will learn how to design estimator and controller for a system to ensure desirable properties (e.g., stability, performance, robustness) of the dynamical system. In particular, the course will focus on systems that can be modeled by linear ordinary differential equations (ODEs) and that satisfy time-invariance conditions. The course will introduces the fundamental mathematics of linear spaces, linear operator theory, and then proceeds with the analysis of the response of linear time-variant systems. Advanced topics such as robust control, model predictive control, linear quadratic games and distributed control will be presented based on allowable time and interest from the class. The material learned in this course will form a valuable foundation for further work in systems, control, estimation, identification, detection, signal processing, and communications.

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.