Rigorous introduction to probability, using techniques of measure theory. Includes limit theorems, martingales, and stochastic processes. Prerequisite: 7310 or equivalent.
Continuation of Probability Theory I. Elements of stochastic processes, including Brownian motion, continuous time martingales, and Markov processes.
Studies the basic principles of linear analysis, including spectral theory of compact and selfadjoint operators. Prerequisite: MATH 7340 and 7310, or equivalent.
This course provides the opportunity to offer a new topic in the subject of mathematics.
Examines categories, functors, abelian catqegories, limits and colimits, chain complexes, homology and cohomology, homological dimension, derived functors, Tor and Ext, group homology, Lie algebra homology, spectral sequences, and calculations. Prerequisite: …
A continuation of the theory presented in MATH 5770 and 7800 intensively training students to apply the theory to proving theorems and solving problems in topology, especially in preparation for …
Studies groups, rings, fields, modules, tensor products, and multilinear functions. Prerequisite: MATH 5651, 5652, or equivalent.
Studies groups, rings, fields, modules, tensor products, and multilinear functions. Prerequisite: MATH 5651, 5652, or equivalent.
Further topics in algebra.
A continuation of the theory presented in MATH 7751 and 7752 intensively training students to apply the theory to proving theorems in algebra, especially in preparation for the General Examination …
Topics include the fundamental group, covering spaces, covering transformations, the universal covering spaces, graphs and subgroups of free groups, and the fundamental groups of surfaces. Additional topics will be from …
Devoted to chomology theory: cohomology groups, the universal coefficient theorem, the Kunneth formula, cup products, the cohomology ring of manifolds, Poincare duality, and other topics if time permits. Prerequisite: MATH …
Topics include smooth manifolds and functions, tangent bundles and vector fields, embeddings, immersions, transversality, regular values, critical points, degree of maps, differential forms, de Rham cohomology, and connections. Prerequisite: MATH …
Examines fiber bundles; induced bundles, principal bundles, classifying spaces, vector bundles, and characteristic classes, and introduces K-theory and Bott periodicity. Prerequisite: MATH 7800.
Definition of homotopy groups, homotopy theory of CW complexes, Huriewich theorem and Whitehead's theorem, Eilenberg-Maclane spaces, fibration and cofibration sequences, Postnikov towers, and obstruction theory. Prerequisite: MATH 7800.
Studies algebraic properties of RNA secondary structures, primarily using techniques from algebraic topology and homological algebra.
Theory of distributions. Sobolev spaces and their properties (trace and embedding theorems). Theory of elliptic equations. Time-dependent partial differential equations: parabolic and hyperbolic equations. Topics in nonlinear partial differential equations. …
Topics in the theory of operators on a Hilbert space and related areas of function theory.
Topics in the theory of operators on a Hilbert space and related areas of function theory.
This course presents the basic theory of stochastic differential equations and provides examples of its applications. It is an essential topic for students preparing to do research in probability. Topics …