Math Department: Class Notes and Learning Materials

Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe

Laura Lynch — Fri, 06 Aug 2010 08:44:31 PDT

Topics include: Topological space and continuous functions (bases, the product topology, the box topology, the subspace topology, the quotient topology, the metric topology), connectedness (path connected, locally connected), compactness, completeness, countability, filters, and the fundamental group.

Class Notes for Math 918: Local Cohomology, Instructor Tom Marley

Laura Lynch — Fri, 06 Aug 2010 08:43:50 PDT

Topics include: Injective Module, Basic Properties of Local Cohomology Modules, Local Cohomology as a Cech Complex, Long exact sequences on Local Cohomology, Arithmetic Rank, Change of Rings Principle, Local Cohomology as a direct limit of Ext modules, Local Duality, Chevelley’s Theorem, Hartshorne- Lichtenbaum Vanishing Theorem, Falting’s Theorem.

Class Notes for Math 915: Homological Algebra, Instructor Tom Marley

Laura Lynch — Fri, 06 Aug 2010 08:41:11 PDT

Topics covered are: Complexes, homology, direct and inverse limits, Tor, Ext, and homological dimensions. Also, Koszul homology and cohomology.

Class Notes for Math 918: Cohen Macaulay Modules, Instructor Roger Wiegand

Laura Lynch — Fri, 06 Aug 2010 08:40:09 PDT

Topics covered are: Cohen Macaulay modules, zero-dimensional rings, one-dimensional rings, hypersurfaces of finite Cohen-Macaulay type, complete and henselian rings, Krull-Remak-Schmidt, Canonical modules and duality, AR sequences and quivers, two-dimensional rings, ascent and descent of finite Cohen Macaulay type, bounded Cohen Macaulay type.

Class Notes for Math 918: Homological Conjectures, Instructor Tom Marley

Laura Lynch — Fri, 06 Aug 2010 08:39:27 PDT

This course was an overview of what are known as the “Homological Conjectures,” in particular, the Zero Divisor Conjecture, the Rigidity Conjecture, the Intersection Conjectures, Bass’ Conjecture, the Superheight Conjecture, the Direct Summand Conjecture, the Monomial Conjecture, the Syzygy Conjecture, and the big and small Cohen Macaulay Conjectures. Many of these are shown to imply others.

This document contains notes for a course taught by Tom Marley during the 2009 spring semester at the University of Nebraska-Lincoln. The notes loosely follow the treatment given in Chapters 8 and 9 of Cohen-Macaulay Rings, by W. Bruns and J. Herzog, although many other sources, including articles and monographs by Peskine, Szpiro, Hochster, Huneke, Grith, Evans, Lyubeznik, and Roberts (to name a few), were used. Special thanks to Laura Lynch for putting these notes into LaTeX.

Class Notes for Math 905: Commutative Algebra, Instructor Sylvia Wiegand

Laura Lynch — Fri, 06 Aug 2010 08:37:26 PDT

Topics include: Rings, ideals, algebraic sets and affine varieties, modules, localizations, tensor products, intersection multiplicities, primary decomposition, the Nullstellensatz

Class Notes for Math 921/922: Real Analysis, Instructor Mikil Foss

Laura Lynch — Fri, 06 Aug 2010 08:36:32 PDT

Topics include: Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, L^p spaces, general measure and integration theory, Radon- Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration, Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, L^p spaces, general measure and integration theory, Radon-Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration.

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Laura Lynch — Fri, 06 Aug 2010 08:33:09 PDT

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's p^aq^b Theorem; Injective modules.

Class Notes for Math 953: Algebraic Geometry, Instructor Roger Wiegand

Laura Lynch — Fri, 06 Aug 2010 08:31:19 PDT

Topics include: Affine schemes and sheaves, morphisms, dimension theory, projective varieties, graded rings, Artin rings