## Mathematics

- 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 standard problems in this field.
- 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 203: Advanced Vector CalculusVector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 218. Recommended for prospective physics majors and others with a strong interest in applied mathematics.
- MAT 214: Numbers, Equations, and ProofsAn introduction to classical number theory, to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions and quadratic reciprocity. There will be a topic, chosen by the instructor, from more advanced or more applied number theory: possibilities include p-adic numbers, cryptography, and Fermat's Last Theorem. This course is suitable both for students preparing to enter the Mathematics Department and for non-majors interested in exposure to higher mathematics.
- 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 216: Multivariable Analysis and Linear Algebra IRigorous theoretical introduction to the foundations of analysis in one and several variables: basic set theory, vector spaces, metric and topological spaces, continuous and differential mapping between n-dimensional real vector spaces. Normally followed by MAT 218.
- MAT 300: Multivariable Analysis ITo familiarize the student with functions in many variables and higher dimensional generalization of curves and surfaces. Topics include: point set topology and metric spaces; continuous and differentiable maps in several variables; smooth manifolds and maps between them; Sard's theorem; vector fields and flows; differential forms and Stokes' theorem; differential equations; multiple integrals and surface integrals. An introduction to more advanced courses in analysis, differential equations, differential geometry, topology.
- MAT 320: Introduction to Real AnalysisIntroduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space, and the theory of Fourier series and Hilbert spaces.
- MAT 321/APC 321: Numerical MethodsIntroduction to numerical methods with emphasis on algorithms, applications and numerical analysis. Topics covered include solution of nonlinear equations; numerical differentiation, integration, and interpolation; direct and iterative methods for solving linear systems; computation of eigenvectors and eigenvalues; and approximation theory. Lectures include mathematical proofs where they provide insight and are supplemented with numerical demos using MATLAB.
- MAT 335: Analysis II: Complex AnalysisStudy of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. This course is the second semester of a four-semester sequence, but may be taken independently of the other semesters.
- MAT 340: Applied AlgebraAn applied algebra course that integrates the basics of theory and modern applications for students in MAT, APC, PHY, CBE, COS, ELE. This course is intended for students who have taken a semester of linear algebra and who have an interest in a course that treats the structures, properties and application of groups, rings, and fields. Applications and algorithmic aspects of algebra will be emphasized throughout.
- MAT 345: Algebra IThis course will cover the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions, and the representation theory of finite groups, rings and modules.
- MAT 365: TopologyIntroduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications.
- MAT 377/APC 377: Combinatorial MathematicsThe course covers the basic combinatorial techniques as well as introduction to more advanced ones. The topics discussed include elementary counting, the pigeonhole principle, counting spanning trees, Inclusion-Exclusion, generating functions, Ramsey Theory, Extremal Combinatorics, Linear Algebra in Combinatorics, introduction to the probabilistic method, spectral graph theory, topological methods in combinatorics.
- 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 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 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 457: Algebraic GeometryIntroduction to affine and projective algebraic varieties over fields.
- MAT 477: Advanced Graph TheoryAdvanced course in Graph Theory. Further study of graph coloring, graph minors, perfect graphs, graph matching theory. Topics covered include: stable matching theorem, list coloring, chi-boundedness, excluded minors and average degree, Hadwiger's conjecture, the weak perfect graph theorem, operations on perfect graphs, and other topics as time permits.
- 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 517: Topics in Arithmetic Geometry: Rational Points on CurvesIn this course, we first study the original proof of Mordell-Lang conjecture following Vojta and Faltings.Then, we discuss recent work of uniform Mordell-Lang by Dimitrov-Gao-Habbeger and Xinyi Yuan.
- 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 522/APC 522: Introduction to PDEThe course is an introduction to partial differential equations, problems associated to them and methods of their analysis. Topics may include: basic properties of elliptic equations, wave equation, heat equation, Schr\"{o}dinger equation, hyperbolic conservation laws, Fokker-Planck equation, basic function spaces and inequalities, regularity theory for linear PDE, De Giorgi method, basic harmonic analysis methods, existence results and long time behavior for classes of nonlinear PDE including the Navier-Stokes equations.
- MAT 529: Topics in Analysis: Interpolation and ApproximationFor each point x in R^n, suppose we're given a convex set K(x) in R^D. For fixed m, we want to find a C^m function F:R^n->R^D such that F(x) lies in K(x) for each x. Under mild restrictions on the family K(x), a recent result of Jiang, Luli and O'Neill gives necessary and sufficient conditions for the existence of such an F. The course starts with elementary background and builds up to the proof of that result. No special background is assumed.
- MAT 547: Topics in Algebraic Geometry: Algebraic K-stability TheoryWe discuss the algebraic K-stability theory of Fano varieties, specifically 1) the foundational theory of K-stability, including the original definition of K-stability, the valuative criterion of K-stability, etc., 2) the construction of general K-moduli stacks/spaces, including various necessary ingredients, and 3) verifying explicit examples of Fano varieties to be K-stable, including explicit K-moduli construction and estimate of stability of thresholds.
- MAT 549: Topics in Algebra: Geometric Aspects of p-adic Hodge TheoryThis course covers some recent work on recasting concepts in p-adic Hodge theory in the language of (derived) algebraic stacks, and then using this reformulation to better understand the former.
- MAT 550: Differential GeometryThis is an introductory graduate course covering questions and methods in differential geometry. As time permits, more specialized topics are covered as well, including minimal submanifolds, curvature and the topology of manifolds, the equations of geometric analysis and its main applications, and other topics of current interest.
- MAT 555: Topics in Differential Geometry: Minimal SurfacesOver the course of the semester, we discuss the theory of minimal surfaces, including some recent developments and their applications.
- MAT 567: Topics in Low Dimensional Topology: Smooth Surfaces in 4-manifoldsThis course is about techniques to understand diffeomorphism groups of manifolds, e.g. through the work of Cerf, Palais, Haefliger, Dax, Hatcher and Wagoner.
- MAT 572/APC 572: Topics in Combinatorial Optimization: Introduction to Combinatorial OptimizationThis course surveys the theory of combinatorial optimization. We cover the elementary min-max theorems of graph theory, such as Konig's theorems and Tutte's matching theorem, network flows, linear programming and polyhedral optimization, hypergraph packing and covering problems, perfect graphs, polyhedral methods to prove min-max theorems, packing directed cuts, the Lucchesi-Younger theorem, Packing T-cuts, T-joins and circuits, Edmonds' matching polytope theorem, relations with the four-color theorem, Lehman's results on ideal clutters, various further topics as time permits.
- MAT 579: Topics in Discrete Mathematics: Hadwiger's ConjectureA graph is a "minor" of another if it can be obtained from a subgraph of the second by contracting edges. Perhaps the most well-known open question about minors is Hadwiger's conjecture, extending the four-color theorem, that for all t, every graph with no K_{t+1} minor can be t-colored. After an introduction to minors of graphs, the course focuses on aspects of Hadwiger's conjecture, particularly on recent results giving upper bounds on the chromatic number of graphs with no K_{t+1} minor.
- 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