All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu
All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu
This course is designed to give an introduction to representation theory, with an emphasis on Lie algebras and algebraic groups. The class is designed to be suitable both for students planning to specialize in representation theory and for those who need it for applications. It will start with an outline of the representation theory of finite groups over the complex numbers. We will then introduce complex algebraic groups and their Lie algebras. After discussing the basic theory of nilpotent, solvable, and semisimple Lie algebras, we will describe the classification of semisimple Lie algebras. We will continue with the universal enveloping algebra of a Lie algebra and the Poincare-Birkhoff-Witt theorem. We will then cover highest-weight representations of Lie algebras, including Verma modules and finite-dimensional irreducible representations. We will also discuss the relationship between the representations of a semisimple algebraic group and the representations of its Lie algebra.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
The focus of this course will be on cohomology theory, dual to the homology theory developed in MATH 7512. One reason to pursue cohomology theory is that the cohomology of a space may be given a natural ring structure. This additional algebraic structure provides another topological invariant. In developing this structure, we will study several products relating homology and cohomology. These considerations will be used to study the topology of manifolds, yielding a number of duality theorems also relating homology and cohomology, with a variety of applications.
In addition to its importance within topology, cohomology theory also provides connections between topology and other subjects, including algebra and geometry. Depending on the interests of the audience, we may pursue some of these connections, such as cohomology of groups or the De Rham theorem.
Geometric group theory is a field which sits at the juncture of geometry, topology and group theory. The basic idea is to glean algebraic information about finitely generated groups via their actions on geometric or topological spaces, and we will see many examples of this in the course. We will emphasize the questions and techniques which highlight the interplay with low dimensional topology.
After starting with the basics (Cayley graphs, geometric group actions, quasi-isometries and various quasi-isometry invariants) we will study delta-hyperbolic spaces (generalizations of fundamental groups of closed hyperbolic manifolds) and non-positively curved spaces in detail. Then we will make brief forays into a few other topics, tailored to the interests of the class. Possibilities include the Mostow Rigidity theorem, the Gromov polynomial growth theorem, PL Morse theory and right-angled Artin groups, or special cube complexes and the virtual Haken conjecture.