## All Courses

**MATH 410-1,2,3 Analysis:** First and Second Quarters: Real analysis. Topological spaces, metric spaces. Lebesque measure and integration. Function spaces, including Banach and Hilbert spaces. Elementary functional analysis. Weak convergence. Third Quarter: Complex analysis. Holomorphic functions, Cauchy's theorem, power series, harmonic functions, conformal mapping, analytic continuation.

**MATH 413-1,2,3 Functions of a Complex Variable:** Holomorphic functions: theorems of Cauchy, Morera, and Rouché residue and open mapping theorems; harmonic and entire functions; analytic continuation; conformal mapping. Schlicht functions, functions of several complex variables, Hp spaces, and complex manifolds.

**MATH 415-0 Riemann Surfaces:** Abstract Riemann Surfaces, differential forms, Poincare-Hopf formula, algebraic curves Riemann-Hurwitz formula, Riemann-Roch formula and applications, Jacobi variety and Abel theorem, and Uniformization theorem.

**MATH 415-1,2 Functional Analysis:** Topological groups and topological vector spaces; Banach spaces, linear functionals, and operators; applications to functional equations.

**MATH 420-1,2,3 Partial Differential Equations:** Introduction to basic differential equations, with emphasis on the theory of partial differential equations. Prerequisites: Advanced calculus and linear algebra or permission of instructor.

**MATH 425-1,2,3 Partial Differential Equations II:** Nonlinear elliptic differential equations, nonlinear hyperbolic differential equations, pseudodifferential operators, and other topics.

**MATH 428 Geometric Measure Theory and Applications:** General measure theory, Hausdorff measure, area and co-area formulas, Sobolev functions, BV functions and set of finite perimeter, Gauss-Green theorem, differentiability and approximation, applications.

**MATH 429 Fourier Analysis:** A short overview of classical Fourier analysis on the circle. Selected topics about Fourier analysis on the line and in Euclidean space. Prerequisites: Permission of instructor.

**MATH 430-1,2,3 Dynamical Systems:** Qualitative theory of differentiable dynamical systems, emphasizing global properties such as structural stability theorems.

**MATH 435 Ergodic Theory:** Introduction to abstract ergodic theory, focusing on the asymptotic behavior of measure preserving transformations. Topics to be covered include: measure preserving transformations and flows, convergence theorems, recurrence properties, isomorphism invariants, and applications to problems in number theory, probability, and combinatorics. Prerequisites: MATH 412-1.

**MATH 440-1,2,3 Geometry/Topology:** The course will emphasize examples throughout the year. Fall (topology): Topological spaces, fundamental group, covering spaces, principal bundles, vector bundles, classifying spaces. Winter (differentiable manifolds): differentiable manifolds; implicit function theorem and Sard's theorem; smooth vector bundles, tangent vectors, tensors, vector fields and flows. Lie derivatives, Lie groups and Lie algebras. Integral manifolds, Frobenius's theorem. Differential forms and the de Rham complex. Orientation, integration, Riemannian metrics, geodesics, exponential map. Spring (cohomology): de Rham cohomology, Mayer-Vietoris, Poincare' duality, singular homology and cohomology. Cohomolgy of cell complexes, simplicial cohomology, Cech cohomology. Cup product; sheaves. Prerequisites: For MATH 440-2: MATH 440-1; For MATH 440-3: MATH 440-2.

**MATH 444 Hamiltonian Dynamics and Symplectic Geometry:** Symplectic structure and cotagent bundle. Hamiltonian flow and their invariants. Integrable systems and stability. Lagrangian intersection theory and symplectic fixed points theorems. Arnold conjecture on n-torus.

**MATH 445-1,2,3 Differential Geometry:** Fall- Riemannian geometry: connections, geodesics, completeness, Jacobi fields, exponential map, constant curvature. Winter- Hodge theory: connections, curvature, de Rham complex, Hodge decomposition, Kahler manifolds, Chern-Weil theorem. Spring- Further topics: connections and curvature on principal and associated bundles; symplectic geometry, classical mechanics and geometric quantization; Dirac operators and index theorems.

**MATH 448-1,2,3 Geometry and Topology:** This is a working seminar for students with interests in geometry, topology, and related fields. Its primary aim is to introduce students to research subjects of current interest to faculty members in these areas.

**MATH 450-1,2,3 Probability Theory and Stochastic Analysis:** Probability spaces, random variables, distribution functions, conditional probability, laws of large numbers, and central limit theorem. Random walk, Markov chains, martingales, and stochastic processes. Definition and properties of standard Brownian motion. Stochastic Integration and stochastic differential calculus, with applications to diffusion processes.

**MATH 460-1,2,3 Algebraic Topology I:** MATH 460-1 Fundamental group and covering spaces, MATH 460-2 Simplical, singular, and cellular (co-) homology; universal coefficient and Kuenneth theorems, MATH 460-3 Cohomology rings and Poincare duality; Thom Isomorphism and characteristic classes. Prerequisites: For MATH 460-2: MATH 460-1; For MATH 460-3: MATH 460-2.

**MATH 465-1,2,3 Algebraic Topology II:** Cohomology theories and operations, homotopy and obstruction theory, and CW complexes; spectral sequences. Multiple registrations allowed.

**MATH 468 Homological Algebra:** Exact sequences, Ext and Tor, and homological dimensions.

**MATH 470-1,2,3 Algebra:** Free, permutation, solvable, simple, and linear groups. Actions of groups on sets; Sylow theorems. Rings and modules: polynomials and power series, Euclidean domains, PIDs, UFDs, and free and projective modules. Field and Galois theory. Extensions: algebraic, transcendental, normal, and integral. Splitting fields. Wedderburn theory. Commutative algebra: prime ideals; localization.

**MATH 477 Commutative Algebra:** Research in commutative algebra: theory of depth (regular sequences, Koszul complexes), dimension theory, completions, Hilbert functions, Cohen-Macaulay modules, excellent rings, Hensel rings, and minimal resolutions. Prerequisites: MATH 470-1,2,3 or equivalent.

**MATH 478 Representation Theory:** Topics in the representation theory and cohomology of finite and infinite groups, including compact and non-compact Lie groups.

**MATH 482-1,2 Algebraic Number Theory:** 1. The theory of global and local fields; various special topics. 2. Abelian Galois extensions of algebraic number fields (class field theory). Complex multiplication, other examples, and relations with geometry.

**MATH 483-1,2,3 Algebraic Geometry:** Introduction to classical and scheme theoretic methods of algebraic geometry. Algebraic vector bundles, sheaf cohomology, the Riemann-Roch theorem for curves, and intersection theory.

**MATH 484 Lie Theory:** Topics in the theory of Lie algebras and Lie groups including classification.

**MATH 485-1,2 Modular Forms:** First quarter: introduction to the theory of modular forms. Congruence subgroups of SL (2,Z), the definitions of modular functions and modular forms, Fourier expansions, Hecke operators, theta functions, modular curves. Second quarter: possible topics include the connections between modular forms and the representation theory of GL (2), automorphic forms, Galois representations attached to modular forms, and the relations with algebraic geometry and other areas of mathematics.

**MATH 486-1,2,3 Algebraic K-Theory:** Classical algebraic K-theory. Functors K0 and K1; origins in and relations with topology; congruence subgroup problem; techniques of computation: exact sequences, localization, resolution, and devissage; polynomial and related extensions; higher K- theories: Karoubi-Villamayor, Quillen.

**MATH 495 Statistical Phenomena in the Theory of Networks:** This interdisciplinary course combines graph theory and probability theory to develop a rigorous foundation for the study of network-related problems.

**MATH 499 Independent Study:** Permission of instructor and department required. May be repeated for credit.

**MATH 511-1,2,3 Topics in Analysis**

**MATH 512-1,2,3 Topics in Partial Differential Equations**

**MATH 513-1,2,3 Topics in Dynamical Systems**

**MATH 514-1,2,3 Topics in Geometry**

**MATH 516-1,2,3 Topics in Topology**

**MATH 517-1,2,3 Topics in Algebra**

**MATH 518-1,2,3 Topics in Number Theory**

**MATH 519 Responsible Conduct of Research Training**

**MATH 520-1,2,3 Topics in Mathematical Physics**

**MATH 521-1,2,3 Topics in Representation Theory**

**MATH 580 Seminar in College Teaching (No credit):** A weekly two-hour seminar introducing the technique, philosophy, and practice of teaching undergraduate mathematics. Student presentations are critiqued by fellow students, as well as a senior faculty member.

**MATH 590 Research:** Independent investigation of selected problems pertaining to thesis or dissertation. May be repeated for credit.