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
Additional topics may be introduced depending on the interests of the class.
Prerequisites: Some basic functional analysis and PDE theory (MATH 4340, or MATH 7386 or equivalent) will be reviewed. Prior exposure to numerical analysis (MATH 4065 or 4066) and Python would be useful but not mandatory.
The goal of this course is to give a gentle introduction to some basic properties of elliptic curves, modular forms, and their connection. In the later part of the semester, we will discuss some applications, advanced topics, and open problems, depending on audiences' interests.
In the first part of the course, we will study the Weyl algebra and its modules, i.e. D-modules on C^{n}. We will then discuss D-modules on smooth complex algebraic varieties; here, it is necessary to introduce sheaf-theoretic methods. A main goal of the course is to discuss the Riemann-Hilbert correspondence--a vast generalization of Hilbert's 21st problem on the existence of linear differential equations with a prescribed monodromy group--in some special cases. Time permitting, we will discuss some applications to representation theory and to the geometric Langlands program.
The course starts with basic definitions of Lie groups and Lie algebras. We then discuss some examples. Then the exponential map, the Lie algebra of a closed linear groups, actions on manifolds and homogeneous spaces. Several examples will be given. The rest depends on how much time we have: finite dimensional representations, homogeneous vector bundles and the Plancherel formula for square integrable sections of homogeneous vector bundles over compact homogeneous spaces.