## 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.
- COS 522/MAT 578: Computational ComplexityComputational complexity theory is a mathematical discipline that studies efficient computation. The course covers some of the truly beautiful ideas of modern complexity theory such as: approaches to the famous P vs NP question and why they are stuck; complexity classes and their relationship; circuit lower bounds; proof systems such as zero knowledge proofs, interactive proofs and probabilistically checkable proofs; hardness of approximation; de-randomization and the hardness vs randomness paradigm; quantum computing.
- 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.
- 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.
- 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: 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 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 IILocal Fields and the Galois theory of Local Fields.
- 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 419: Topics in Number Theory: The Arithmetic of Quadratic FormsAn introduction to the arithmetic and geometry of quadratic forms. Topics will include lattices, class number, local-global principle, composition, and other connections to algebraic number theory. Applications to the representation of integers by quadratic forms will be discussed, including various generalizations of Lagrange's Theorem which states that every positive integer is the sum of four squares.
- MAT 427: Ordinary Differential EquationsThis course offers an introduction to the study of ordinary differential equations. Topics include: Explicit solutions of some linear and non-linear equations (using tools that span: separation of variables, integrating factors, Greens functions, and Laplace transform methods); Series solutions of ODEs with analytic coefficients and with regular singular points; Fundamental existence and uniqueness theorems (Peano, Picard-Lindelof, and Osgood); Introduction to dynamical systems (Poincare-Bendixon theorem); Stability of equilibrium points and periodic orbits.
- MAT 447: Commutative AlgebraThis course will cover the standard material in a first course on commutative algebra. Topics include: ideals in and modules over commutative rings, localization, primary decomposition, integral dependence, Noetherian rings and chain conditions, discrete valuation rings and Dedekind domains, completion; and dimension theory.
- MAT 449: Topics in Algebra: Representation TheoryAn introduction to representation theory of Lie groups and semisimple Lie algebras.
- MAT 478: Topics In Combinatorics: The Probabilistic MethodThis course covers probabilistic methods in combinatorics and their applications in theoretical computer science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, VC-dimension and possibly more as time permits.
- MAT 515: Topics in Number Theory and Related Analysis: Spectral Theory of Automorphic FormsThis course covers the basics of the spectral theory of automorphic forms, including the analytic continuation of Eisenstein Series, the trace formula (in some basic cases) and the general Ramanujan Conjectures. Applications to number theory are highlighted.
- MAT 516: Topics in Algebraic Number Theory: Galois RepresentationsThis course focuses on Sen's theory in the study of p-adic Galois representations. We also discuss several generalizations, for instance: 1. Fontaine's study of B_dR-representations, 2. a relative version of Sen's theory.
- MAT 518: Topics in Automorphic Forms: Automorphic Forms and Special Values of L-functionsThis course will cover some recent applications of the arithmetic of automorphic forms to questions about special values of L-functions. Particular attention will be given to L-functions of elliptic curves and modular forms.
- MAT 520: Functional AnalysisBasic introductory course to modern methods of analysis. The possible topics may include Lp spaces, Banach spaces, uniform boundedness principle, closed graph theorem, locally convex spaces, distributions, Fourier transform, Riesz interpolation theorem, Hardy-Littlewood maximal function, Calderon-Zygmund theory, oscillatory integrals, almost orthogonality, Sobolev spaces, restriction theorems, spectral theory of compact operators, applications to partial differential equations.
- MAT 527: Topics in Differential Equations: Dynamics of Nonlinear PDEWe discuss long time behavior of solutions of nonlinear evolution equations modeling fluids and plasmas. Topics include linear and nonlinear stability, invariant manifolds, singularity formation.
- MAT 528: Topics in Nonlinear Analysis: Introduction to PDE's and General RelativityThis course coverd the following topics: 1. Fast introduction to mathematical General Relativity. 2. Kerr family 3. Wave Equation in Minkowski space. Boundedness and Decay. Vectorfield Method. 4. Wave Equation on Black Holes. Boundedness and Decay. 5. Introduction to the problem of the nonlinear stability of Kerr.
- MAT 529: Topics in Analysis: Fluid Dynamics and Related EquationsThe course studies the Muskat equation, which governs the evolution of the interface between two fluids in a porous medium, e.g. oil and water in sand. In two dimensions, when the two fluids have the same viscosity but different densities, the problem reduces to a contour dynamics equation. When the interface is the graph of a function, a lot is known; but in the "turnover regime" in which a vertical line may meet the interface in more than one point, hard problems remain. The lectures explain the state of the art as of a few years ago, then present recent work of Jia Shi.
- 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: Local Systems and Finite GroupsWe discuss relations local systems and finite groups.
- 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 568: Topics in Knot Theory: Khovanov Homology and Knot Floer HomologyThis course covers some of the modern techniques and recent developments in knot theory. The focus this semester is on Khovanov homology and knot Floer homology. We discuss applications in knot theory and smooth 4-dimensional topology. We also discuss algebraic and combinatorial methods to compute knot homologies.
- MAT 577: Topics in Combinatorics: The Probabilistic MethodThis course covers probabilistic methods in combinatorics and their applications in theoretical computer science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, VC-dimension and possibly more as time permits.
- 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. The lectures start with a brisk review of what was covered in the course in Spring 2021, and then continue beyond that with a more extended, though still self contained, discussion of quantum systems.