## Mathematics

- COS 487/MAT 407: Theory of ComputationIntroduction to computability and complexity theory. Topics will include models of computation such as automata, and Turing machines; decidability and decidability; computational complexity; P, NP, and NP completeness; others.
- 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 210: One Variable Calculus with ProofsMAT 210 will survey the main ideas of calculus in a single variable incorporating an introduction to formal mathematical proofs. The course will place equal emphasis on theory (how to construct formal mathematical definitions and rigorous, logical proofs) and on practice (concrete computational examples involving integration and infinite sequences and series). This course provides a more theoretical foundation in single variable calculus than MAT104, intended to prepare students better for a first course in real analysis (MAT215), but it covers all the computational tools needed to continue to multivariable calculus (MAT201 or MAT203).
- 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 Analysis and Scientific ComputingIntroduction 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 or Python.
- 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: Modular Forms and L-functionsIntroduction to the theory of modular forms, with an emphasis on geometric aspects of the theory. Topics include modular curves and their moduli interpretation, q-expansions, elliptic curves, Hecke correspondences, the Eichler-Shimura relation.
- MAT 449: Topics in Algebra: Representation TheoryAn introduction to representation theory of Lie groups and semisimple Lie algebras.
- MAT 457: Algebraic GeometryIntroduction to affine and projective algebraic varieties over fields.
- 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 518: Topics in Automorphic Forms: Automorphic Forms and Special Values of L-functionsThis course coverd some recent applications of the arithmetic of automorphic forms to questions about special values of L-functions. Particular attention is given to automorphic constructions of cohomology classes and their arithmetic applications.
- 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 526: Topics in Geometric Analysis: Rectifiable sets and rectifiable measuresThis is a monographic course on the topic of rectifiability, a concept introduced first by Besicovitch in the particular case of 1-dimensional sets in the plane. After introducing the Hausdorff dimension, the Hausdorff measures, the course covers Besicovitch's theory and touches upon the famous Besicovitch 1/2 conjecture, which is almost a century old and still unsolved. It then defines rectifiable sets and rectifiable measures in general dimension and codimension and covers the most basic and widely used rectifiability criteria.
- MAT 527: Topics in Differential Equations: Local and global regularity of nonlinear evolution equationsWe focus on several basic questions concerning the dynamics of solutions of evolution equations. The main topics to be discussed are local regularity, global regularity, regularity criteria, and asymptotic stability of stationary solutions. While many examples are possible, we are mostly interested in these topics in the context of plasma models, such as the Vlasov-Poisson and the Vlasov-Maxwell systems.
- MAT 529: Topics in Analysis: Interpolation and ApproximationLet X be the Sobolev space of functions on R^n whose derivatives up to order m belong to L^p. For a subset E of R^n, let X(E) denote the space of restrictions to E of functions in X. If p>n there exists a bounded linear operator T:X(E)->X such that Tf=f on E (i.e. T is an "extension operator"). The corresponding result for p<n is unknown. The course is an attempt to find a proof.
- MAT 547: Topics in Algebraic Geometry: Introduction to the Theory of SchemesA gentle introduction to the theory of schemes, emphasizing examples (continuing the course from Spring 2023.) We assume familiarity with quasi-projective varieties, maps between quasi-projective varieties, some basic awareness about manifolds and their homology, the language of commutative algebra, sheaves, locally ringed spaces, Spec and Proj, quasicoherent and coherent sheaves, and base change. Time permitting, we formulate the main results of intersection theory and discuss their applications. The primary references are Vakil's note 'The Rising Sea' and the book of Hartshorne 'Algebraic geometry'.
- MAT 549: Topics in Algebra: Introduction to \ell-adic etale cohomologyThe course is devoted to the general equidistribution theorem of Forey, Fresan, and Kowalski.
- 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 558: Topics in Conformal and Cauchy-Rieman (CR) Geometry: Conformal and CR InvariantsThe course covers CR geometry in dimension three: we study two natural linear operators, the CR conformal Laplacian and the CR P-prime operators. Each one is associated with a global invariant of the CR structures. We develop the analysis of the extremal problems associated with these curvature invariants.
- MAT 559: Topics in Geometry: Metric Geometry and Degenerations of Einstein ManifoldsThe first part of the course introduces basic tools in the geometry of the Ricci curvature, including volume comparison and splitting theorems, structure theorems of nonnegative Ricci, elliptic PDEs and lower Ricci bound, etc. The second part focuses on the degenerations of Einstein metrics, such as the structure of Einstein limit spaces, quantitative differentiation theory, the Codimension-4 Conjecture, regularity theory for collapsed Einstein manifolds, etc. Time permitting, we also introduce the collapsing geometry of special holonomy spaces.
- MAT 566: Topics in Differential Topology: Symplectic Methods in Low-dimensional TopologyHeegaard Floer homology is an invariant for low-dimensional manifolds constructed using methods in symplectic geometry (Lagrangian Floer homology). A related invariant for knots can also be constructed, whose Euler characteristic, in a suitable sense, is the Alexander polynomial of that knot. This course gives the construction of Heegaard Floer homology and the knot invariant, and with a view towards topological applications, and a special emphasis on modern computational tools.
- MAT 567: Topics in Low Dimensional Topology: Smooth Surfaces in 4-manifoldsManifold theory is the branch of topology concerned with understanding objects that locally appear as standard Euclidean spaces. Central questions are concerned with determining these spaces and special submanifolds and understanding the geometric structures that they support. This course addresses these questions from the point of view of surfaces in 4-manifolds.
- MAT 579: Topics in Discrete Mathematics: Structural Graph Minor TheoryThis course covers the Graph Minors project of Robertson and the speaker, and related topics. Two highlights of this are: (a) in any infinite set of graphs, one of them is a minor of another, and (b) the k vertex-disjoint paths problem in graphs is solvable in polynomial time, for all fixed k. On the way we cover graph path-width and tree-width, graph well-quasi-ordering, containment relations and some structure theorems for tournaments, and other topics 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
- PHI 323/MAT 306: Advanced LogicAn introduction to axiomatic set theory, up to the proof of the consistency of the axiom of choice.