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
The course starts with basic definitions and examples. Then we will discuss the exponential map, closed subgroups, the Lie algebra of a closed subgroup, group actions on manifolds, homogeneous spaces, and homogeneous vector bundles. Further material will depend on time and interests of the participants.
We will begin with a review of continuous optimization in Euclidean space; the role of convexity will be emphasized and basic tools of nonsmooth analysis will be introduced. The main topic of the course is dynamic optimization. The first half will cover the classical material of the calculus of variations, including topics such as the Euler-Lagrange equation, Weierstrass maximality condition, Erdmann corner conditions, and Jacobi conjugate points. Plenty of examples will be covered. There is a natural transition into optimal control, which will be the focus of the rest of the course from a neo-classical point of view.
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 complement of an arrangement, what is left of space after the hyperplanes have been removed, is an object of fundamental interest in the topological study of arrangements. For instance, a collection of lines cuts the real plane into pieces, and understanding the topology of the complement amounts to counting the pieces. This depends on combinatorial aspects of the collection of lines such as parallelism, how many of the lines intersect at a given point, and so on. This example illustrates a central theme in the subject, the relationship between combinatorial and topological aspects of arrangements.
A complex hyperplane does not disconnect space, and the complement of an arrangement in a complex vector space has rich topological structure. One focus of this course will be on the (low-dimensional) topology of these spaces. Specific topics include braid groups, both as primary objects of study and as tools, configuration spaces, the Fox calculus, and Alexander-type invariants. In particular, we will develop general algorithms for computing these latter invariants, and investigate the extent to which these invariants of arrangements are combinatorially determined.
The topological aspects of the course will assume familiarity with the fundamental group, while other (combinatorial) aspects will be self-contained.
Michael Artin has an excellent online course at MIT:
https://ocw.mit.edu/courses/mathematics/18-701-algebra-i-fall-2010/index.htm
https://ocw.mit.edu/courses/mathematics/18-702-algebra-ii-spring-2011/
There will be one in class midterm, worth 25% of the grade. The remaining 75% will be in assigned homework problems turned in, including a take-home final.
Modern algebraic geometry is based on the fundamental notion of a scheme. This course will give an introduction to schemes and their geometry, with particular emphasis on motivating the definitions and constructions and providing many examples. Topics covered will include algebraic varieties, sheaf theory, affine schemes, and projective schemes.