## Mathematics

- APC 199/MAT 199: Math AliveMathematics 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/MAT 322: Introduction to Differential EquationsThis 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.
- COS 488/MAT 474: Introduction to Analytic CombinatoricsAnalytic 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.
- HUM 598/MAT 564/VIS 598: Humanistic Perspectives on the Arts: Multiplicity, Problems in Graphic Design & TopologyIn this course, students explore graphic design from the vantage point of topology and topology through the practice of graphic design. We investigate topology at the junctions of surface, network, and set, illustrating the schematic nature of these configurations, as they appear in the context of certain problems in modern and contemporary graphic design. Such as, how to render figures that take multiple forms? Student design work and topological experiments are guided through class prompts, readings, and discussion. No particular experience in design or mathematics required.
- MAE 305/MAT 391/EGR 305/CBE 305: Mathematics in Engineering IA 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.
- MAE 306/MAT 392: Mathematics in Engineering IIThis 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.
- MAT 100: Calculus FoundationsIntroduction 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.
- MAT 103: Calculus IFirst 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.
- MAT 104: Calculus IIContinuation 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.
- MAT 175: Mathematics for Economics/Life SciencesSurvey 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.
- MAT 201: Multivariable CalculusVectors 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.
- MAT 202: Linear Algebra with ApplicationsCompanion 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.
- MAT 204: Advanced Linear Algebra with ApplicationsCompanion 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.
- MAT 215: Single Variable Analysis with an Introduction to ProofsAn 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.
- MAT 217: Honors Linear AlgebraA 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.
- MAT 218: Multivariable Analysis and Linear Algebra IIContinuation of the rigorous introduction to analysis in MAT 216
- MAT 325: Analysis I: Fourier Series and Partial Differential EquationsBasic 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.
- MAT 330: Complex Analysis with ApplicationsThe 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.
- MAT 346: Algebra IIThis 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.
- MAT 355: Introduction to Differential GeometryIntroduction 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.
- MAT 375/COS 342: Introduction to Graph TheoryThe 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.
- MAT 378: Theory of GamesGames in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications.
- MAT 385: Probability TheoryAn 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.
- MAT 415: Analytic Number TheoryAn 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.
- MAT 419: Topics in Number Theory: Algebraic Number TheoryCourse on algebraic number theory. Topics covered include number fields and their integer rings, class groups, zeta and L-functions.
- MAT 429: Topics in Analysis: Distribution Theory, PDE & Basic Inequalities of AnalysisIntroduction 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.
- MAT 449: Topics in Algebra: Representation TheoryAn introduction to representation theory of Lie groups and Lie algebras. The goal is to cover roughly the first half of Knapp's book.
- MAT 457: Algebraic GeometryIntroduction to affine and projective algebraic varieties over fields.
- MAT 478: Topics In Combinatorics: Extremal CombinatoricsThis 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.
- MAT 490/APC 490: Mathematical Introduction to Machine LearningThis 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.
- MAT 500: Effective Mathematical CommunicationThis course is for second-year graduate students to help them develop their writing and speaking skills for communicating mathematics in a wide variety of settings, including teaching, grant applications, teaching statement, research statement, talks aimed at a general mathematical audience, and seminars, etc. In addition, responsible conduct in research (RCR) training is an integral part of this course.
- MAT 519: Topics in Number Theory: Sieves and Algebraic Number TheoryWe discuss applications of sieves in algebraic number theory and arithmetic geometry, such as to effective versions of Hilbert irreducibility and other related counting problems.
- MAT 525: Topics in Harmonic AnalysisThis course covers current topics in Harmonic Analysis. More specific topic information will be provided when the course is offered.
- MAT 526: Topics in Geometric Analysis and General Relativity: Advanced Topics in General RelativityThis course gives an introduction to General Relativity from a mathematical point of view. Topics covered include the basic formulation of the theory, the black hole solutions of Schwarzschild and Kerr, the well-posedness of the Cauchy problem for the Einstein equations, Penrose's celebrated incompleteness theorems, and the formulation of the "cosmic censorship" conjectures. No prior background in the subject is required though basic differential geometry is assumed. Some experience with PDE's is also useful, but strictly speaking is not essential.
- MAT 527: Topics in Differential Equations: Dynamics of Nonlinear PDEWe study long-time behavior of solutions of nonlinear PDE with dynamic boundary interactions. Applications include Navier-Stokes equations and related systems.
- MAT 529: Topics in Analysis: Interpolation and ApproximationThe course attempts to prove the following simple-sounding conjecture: Let E be a subset of the plane, let X be the space of all functions on the plane with second derivatives in L^p, and let X(E) be the Banach space of all restrictions to E of functions in X. Then there exists a bounded linear map T from X(E) to X such that Tf=f on E for any f in X(E).That's known for p>=2, but not for p<2. After providing a bit of elementary background, the lectures study examples, with audience participation welcome, in hopes of eventually understanding the general case.
- MAT 531: Introduction to Riemann SurfacesThis course is an introduction to the theory of compact Riemann surfaces, including some basic properties of the topology of surfaces, differential forms and the basic existence theorems, the general uniformization theorem, and the Riemann-Roch theorem and some of its consequences.
- MAT 547: Topics in Algebraic Geometry: Introduction to \ell-adic etale cohomologyWe discuss relations between life over finite fields and finite group theory.
- MAT 558: Topics in Conformal and Cauchy-Rieman (CR) Geometry: Recent Developments in Conformal GeometryWe cover some background materials in full-nonlinear elliptic PDE, then selected problems in conformal geometry. Topics include: a brief review of some key facts of 2nd order linear and semi-linear elliptic PDE and basic theory of fully non-linear 2nd order elliptic PDE; Garding theory of fully non-linear PDE; brief survey of Yamabe problem in conformal geometry; elementary symmetric functions on tensors; the curvature on compact closed 4-manifolds; works of S. Chen and J. Case-Y. Wang on classes of fully non-linear PDE with matching (fully non-linear) boundary conditions on manifold settings, etc.
- MAT 559: Topics in Geometry: Local theory of normed and metric spacesThe local theory of normed spaces aims to understand the geometry of norms through their finite dimensional subspaces. As any two norms on R^n are equivalent, such questions are trivial unless one asks for quantitative bounds, at which point rich and subtle structures emerge whose study involves a wealth of deep insights and tools. In analogy to the linear theory, the local theory of metric spaces aims to understand the geometry of metrics through the quantitative behavior of their finite subsets. We cover examples of such investigations, with emphasis on how the linear and metric theories often mimic each other, yet sometimes diverge.
- MAT 560: Algebraic TopologyThe aim of the course is to study some of the basic algebraic techniques in Topology, such as homology groups, cohomology groups and homotopy groups of topological spaces.
- MAT 566: Topics in Differential Topology: Fukaya Categories and Floer Homology of 3-ManifoldsWe introduce the Fukaya category of a symplectic manifold, with a particular focus on the case of surfaces (where many of definitions can be made combinatorial). In this case the objects of the Fukaya category are (decorated) immersed curves in the surface and morphisms count pseudoholomorphic disks. We also survey Heegaard Floer homology as well as its variants knot Floer homology and bordered Floer homology and related invariants such as Khovanov homology. We highlight some recent applications of the interplay between immersed curves in surfaces and these 3-dimensional invariants.
- MAT 569: Topics in Topology: Contact and Symplectic TopologyThis course explores stable homotopical underpinnings of the enumerate theory of pseudo-holomorphic curves in symplectic manifolds.
- MAT 577: Topics in Combinatorics: Extremal CombinatoricsThis course covers topics in Extremal Combinatorics including ones motivated by questions in other areas like computer science, information theory, number theory and geometry. The subjects that are 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.
- MAT 586/APC 511/MOL 511/QCB 513: Computational Methods in Cryo-Electron MicroscopyThis course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and non-linear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.
- MAT 589: Topics in Probability, Statistics and Dynamics: Modern Discrete Probability TheoryThe aim of this course is to survey a range of topics on discrete probability including large deviation theory, multi-scale analysis and graph limits.
- ORF 309/EGR 309/MAT 380: Probability and Stochastic SystemsAn 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.
- PHY 403/MAT 493: Mathematical Methods of PhysicsMathematical 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.
- PHY 521/MAT 597: Introduction to Mathematical PhysicsAn introduction to the statistical mechanic of classical and quantum spin systems. Among the topics to be discussed are phase transitions, emergent structures, critical phenomena, and scaling limits. The goal is to present the physics embodied in the subject along with mathematical methods, from probability and analysis, for rigorous results concerning the phenomena displayed by, and within, the subject's essential models.