MATH 410-1,2,3 Analysis:
First and Second Quarters: Real and Functional analysis. Measure spaces and integration. Convergence theorems. Lp spaces. Function spaces, including Banach and Hilbert spaces. Weak convergence. Third Quarter: Complex analysis. Holomorphic functions, Cauchy's theorem, power series, harmonic functions, conformal mapping, Riemann 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 414-0 Abstract 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.
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 420-1,2,3 Partial Differential Equations:
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 (differentiable topology): 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. Winter (intro to algebraic topology): The fundamental group of a space, covering spaces, and the Van-Kampen theorem. Singular homology, Mayer-Vietoris, degree and Euler characteristic. Spring (cohomology): de Rham cohomology, Poincare' duality, singular cohomology. Cohomology 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 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. Homological algebra: linear algebra, abelian categories, complexes and homology, projective and injective resolutions, homotopies.
MATH 477 Commutative Algebra:Ideals and modules over commutative rings; localization; primary decomposition; integral dependence;
Noetherian rings and chain conditions; discrete valuation rings and Dedekind domains; completion; dimension theory.
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: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 AnalysisAn analysis whose topics will be determined by student interest. Those of you who would like to work in
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 515-1,2,3 Topics in 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 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):
Seminar in College Teaching. Principles and practice of college and university mathematics education. Public speaking. Classroom management. Assessment. Ethics. Course planning. Syllabus writing. Issues for non-native English speakers.
MATH 590 Research:
Independent investigation of selected problems pertaining to thesis or dissertation. May be repeated for credit.