Course Schedule

APC 199/MAT 199 Math Alive Mathematics has profoundly changed our world, from the way we communicate with each other and listen to music, to banking and computers. This course is designed for those without college mathematics who want to understand the mathematical concepts behind important modern applications. The course consists of individual modules, each focusing on a particular application (e.g. compression, animation and using statistics to explain, or hide, facts). The emphasis is on ideas, not on sophisticated mathematical techniques, but there will be substantial problem-set requirements. Students will learn by doing simple examples. APC 350/CEE 350/MAT 322 Introduction to Differential Equations This course will introduce the basic theory, models and techniques for ordinary and partial differential equations. Emphasis will be placed on the connection with other disciplines of science and engineering. We will try to strike a balance between the theoretical (e.g. existence and uniqueness issues, qualitative properties) and the more practical issues such as analytical and numerical approximations. Instructor(s): Jiequn Han COS 488/MAT 474 Introduction to Analytic Combinatorics Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines. This course combines motivation for the study of the field with an introduction to underlying techniques, by covering as applications the analysis of numerous fundamental algorithms from computer science. The second half of the course introduces Analytic Combinatorics, starting from basic principles. Instructor(s): Robert Sedgewick MAE 305/MAT 391/EGR 305/CBE 305 Mathematics in Engineering I A treatment of the theory and applications of ordinary differential equations with an introduction to partial differential equations. The objective is to provide the student with an ability to solve problems in this field. Instructor(s): Pierre-Thomas Brun MAE 306/MAT 392 Mathematics in Engineering II This course covers a range of fundamental mathematical techniques and methods that can be employed to solve problems in contemporary engineering and the applied sciences. Topics include algebraic equations, numerical integration, analytical and numerical solution of ordinary and partial differential equations, harmonic functions and conformal maps, and time-series data. The course synthesizes descriptive observations, mathematical theories, numerical methods, and engineering consequences. Instructor(s): Mikko Petteri Haataja MAT 100 Calculus Foundations Introduction to limits and derivatives as preparation for further courses in calculus. Fundamental functions (polynomials, rational functions, exponential, logarithmic, trigonometric) and their graphs will be also reviewed. Other topics include tangent and normal lines, linearization, computing area and rates of change. The emphasis will be on learning to think independently and creatively in the mathematical setting. Instructor(s): Takumi Murayama MAT 103 Calculus I First semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus. The fall offering will emphasize applications to physics and engineering in preparation for MAT 104; the spring offering will emphasize applications to economics and life sciences, in preparation for MAT 175. In multi-section calculus and linear algebra courses, students register for a time slot, not a particular section. Instructor(s): Tatiana Katarzyna Howard, Ana Menezes MAT 104 Calculus II Continuation of MAT 103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers. In multi-section calculus and linear algebra courses, students register for a time slot, not a particular section. Students will be randomly allocated between available sections in their time slot prior to the beginning of classes. Instructor(s): Duncan Alexander McNicol Dauvergne, Eden Prywes, Maxime C.R Van De Moortel MAT 175 Mathematics for Economics/Life Sciences Survey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions, and constrained optimization with Lagrange multipliers. MAT 201/202 is strongly recommended for finance and math track economics. Students who complete 175 can continue in 202 if they wish. Instructor(s): Hannah Schwartz, Ian Michael Zemke, Boyu Zhang MAT 201 Multivariable Calculus Vectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields and Stokes's theorem. Instructor(s): Clark Wilmer Butler, Henry Theodore Horton, Sameer Subramanian Iyer, Chao Li MAT 202 Linear Algebra with Applications Companion course to MAT 201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems. MAT 201 and MAT 202 can be taken in either order, although most students take MAT 201 first. Instructor(s): David Boozer, Amitesh Datta, Shai Evra, Jennifer Michelle Johnson, Y. Baris Kartal, Casey Lynn Kelleher, Nicholas Fox Marshall, Mark Weaver McConnell, Joaquin Moraga, John Thomas Sheridan, Remy van Dobben de Bruyn MAT 204 Advanced Linear Algebra with Applications Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent. Instructor(s): Amitesh Datta MAT 215 Single Variable Analysis with an Introduction to Proofs An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel Theorem. The Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's Theorem. Instructor(s): Alexandru D. Ionescu MAT 217 Honors Linear Algebra A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms. Instructor(s): Kenneth Brian Ascher, Andrew V Yarmola MAT 218 Multivariable Analysis and Linear Algebra II Continuation of the rigorous introduction to analysis in MAT 216 Instructor(s): Alan Chang MAT 325 Analysis I: Fourier Series and Partial Differential Equations Basic facts about Fourier Series, Fourier Transformations, and applications to the classical partial differential equations will be covered. Also Finite Fourier Series, Dirichlet Characters, and applications to properties of primes. Instructor(s): Yakov Mordechai Shlapentokh-Rothman MAT 330 Complex Analysis with Applications The theory of functions of one complex variable, covering analyticity, contour integration, residues, power series expansions, and conformal mapping. The goal in the course is to give adequate treatment of the basic theory and also demonstrate the use of complex analysis as a tool for solving problems. Instructor(s): Michael Aizenman MAT 346 Algebra II This course is a continuation of MAT 345 and its introduction to representation theory. We will cover semi-simple algebras, application to group theory, Artin's and Brauer's theorems characterizing representations over the complex numbers, rationality questions, and Brauer's theory of representations mod p. This will lead to one or more advanced topics in finite groups or Lie algebras. Instructor(s): Gabriele Di Cerbo MAT 355 Introduction to Differential Geometry Introduction to geometry of surfaces. Surfaces in Euclidean space: first fundamental form, second fundamental form, geodesics, Gauss curvature, Gauss-Bonnet Theorem. Minimal surfaces in the Euclidean space. Instructor(s): Fernando Codá Marques MAT 375/COS 342 Introduction to Graph Theory The fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Instructor(s): Maria Chudnovsky MAT 378 Theory of Games Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Instructor(s): Jonathan Michael Fickenscher MAT 385 Probability Theory An introduction to probability theory. The course begins with the measure theoretic foundations of probability theory, expectation, distributions and limit theorems. Further topics include concentration of measure, Markov chains and martingales. Instructor(s): Allan M. Sly MAT 415 Analytic Number Theory An introduction to classical results in analytic number theory, presenting fundamental theorems with detailed proofs and highlighting the tight connections between them. Topics covered might include: the prime number theorem, Dirichlet L-functions, zero-free regions, sieve methods, representation by quadratic forms, and Gauss sums. Instructor(s): Peter Clive Sarnak MAT 419 Topics in Number Theory: Algebraic Number Theory Course on algebraic number theory. Topics covered include number fields and their integer rings, class groups, zeta and L-functions. Instructor(s): Christopher McLean Skinner MAT 429 Topics in Analysis: Distribution Theory, PDE & Basic Inequalities of Analysis Introduction to Geometric Partial Differential Equations. The course will review some basic topics in Elliptic theory and give a comprehensive introduction to linear and nonlinear wave equations with applications to relativistic filed theories including General Relativity. Instructor(s): Sergiu Klainerman MAT 449 Topics in Algebra: Representation Theory An introduction to representation theory of Lie groups and Lie algebras. The goal is to cover roughly the first half of Knapp's book. Instructor(s): Shou-Wu Zhang MAT 457 Algebraic Geometry Introduction to affine and projective algebraic varieties over fields. Instructor(s): Chenyang Xu MAT 478 Topics In Combinatorics: Extremal Combinatorics This course will cover topics in Extremal Combinatorics including ones motivated by questions in other areas like Computer Science, Information Theory, Number Theory and Geometry. The subjects that will be covered include Graph powers, the Shannon capacity and the Witsenhausen rate of graphs, Szemeredi's Regularity Lemma and its applications in graph property testing and in the study of sets with no 3 term arithmetic progressions, the Combinatorial Nullstellensatz and its applications, the capset problem, Containers and list coloring, and related topics as time permits. Instructor(s): Noga Mordechai Alon MAT 490/APC 490 Mathematical Introduction to Machine Learning This course gives a mathematical introduction to machine learning. There are three major components in this course. (1) Machine learning models (kernel methods, shallow and deep neural network models) for both supervised and unsupervised learning problems. (2) Optimization algorithms (gradient descent, stochastic gradient descent, EM). (3) Mathematical analysis of these models and algorithms. Instructor(s): Weinan E ORF 309/EGR 309/MAT 380 Probability and Stochastic Systems An introduction to probability and its applications. Topics include: basic principles of probability; Lifetimes and reliability, Poisson processes; random walks; Brownian motion; branching processes; Markov chains. Instructor(s): Ramon van Handel PHY 403/MAT 493 Mathematical Methods of Physics Mathematical methods and terminology which are essential for modern theoretical physics. These include some of the traditional techniques of mathematical analysis, but also more modern tools such as group theory, functional analysis, calculus of variations, non-linear operator theory and differential geometry. Mathematical theories are not treated as ends in themselves; the goal is to show how mathematical tools are developed to solve physical problems. Instructor(s): Bogdan Andrei Bernevig