MATH 599 - Independent Study

Institution:

University of Pennsylvania

Subject:

Description:

Topology and Geometric Analysis. (A) Staff. Prerequisite(s): Math 500/501 or with the permission of the instructor. Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions. Topology and Geometric Analysis. (B) Staff. Prerequisite(s): Math 600 or with the permission of the instructor. Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem. Algebra. (A) Staff. Prerequisite(s): Math 370/371 or Math 502/503. Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal ideals, localization, Hilbert basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated to affine varieties, semisimple rings, Wedderburn's theorem, elementary representation theory. Linear algebra: Diagonalization and canonical form of matrices, elementary representation theory, bilinear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras. Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's Lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations. Algebra. (B) Staff. Prerequisite(s): Math 602 or with the permission of the instructor. Continuation of Math 602. First Year Seminar in Mathematics. (A) Staff. Prerequisite(s): Open to first year Mathematics graduate students. Others need permission of the instructor. This is a seminar for first year Mathematics graduate student, supervised by faculty. Students give talks on topics from all areas of mathematics at a level appropriate for first year graduate students. Attendance and preparation will be expected by all participants, and learning how to present mathematics effectively is an important part of the seminar. First Year Seminar in Mathematics. (B) Staff. Prerequisite(s): Open to first year Mathematics graduate students. Ohters need permission of the instructor. Continuation of Math 604.

Credits:

3.00

Credit Hours:

Prerequisites:

Corequisites:

Exclusions:

Level:

Instructional Type:

Lecture

Notes:

Additional Information:

Historical Version(s):

Institution Website:

www.upenn.edu

Phone Number:

(215) 898-5000

Regional Accreditation:

Middle States Association of Colleges and Schools

Calendar System:

Semester

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