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
Chapter 4 | Basic Iterative Methods |
Chapter 5 | Projection Methods |
Chapters 6 and 7 | Krylov Subspace Methods |
Chapters 9 and 10 | Preconditioning |
The focus of this course will be on cohomology theory, dual to the homology theory developed previously. 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, or other elements of algebraic topology, such as homotopy theory.
This class will provide an introduction to differential Galois theory. We will discuss Picard-Vessiot extensions (the differential analogue of Galois extensions) and give several different constructions of the differential Galois group. We will prove the fundamental theorem and consider when differential equations are solvable by quadratures--the differential analogue of solvability by radicals. We will further discuss the monodromy group of linear differential equations and its relationship to the differential Galois group.
The topics for this course are the following:
Prerequisites: Students should know basic elliptic PDE theory (e.g. Poisson equation, convection/diffusion equation, weak formulations, etc.) as well as some numerical methods for solving PDEs, e.g. finite difference or finite element methods, etc.; however, some discussion on numerical methods will be given in the class. Knowledge of continuum mechanics is a plus, but not required (necessary concepts will be reviewed).
In this course we will study the algebraic structure of mapping class groups and the detailed description of its individual elements. This includes the Nielsen – Thurston classification theorem, which gives a particularly nice representative for each element of the group. Furthermore, we will introduce spaces on which mapping class groups act, such as the complex of curves and Teichmüller space. An important theme throughout the course will be the relationships between the geometry of these spaces, the algebra of the mapping class group, and the topology of the surface.