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 aim of this course is to cover the classical theory of classical modular forms. The course will begin with basic definitions and examples. In the later of the semester, we will discuss some applications, advanced topics, and open problems, depending on audiences' interests.
In this course, we will develop homological algebra in a modern, algebraic way with a focus on the language of derived categories. We will discuss various applications with a particular emphasis on sheaf cohomology. This course will provide much of the background material needed for the geometric representation theory class Math 7250 that will be offered in the spring.
The course will be mainly based on Hatcher's book Algebraic topology, and additional material that we will hand out.
In this course, we will begin with the necessary prerequisites about Riemannian manifolds and Lie groups, and then spend a large part of the semester getting comfortable with symmetric spaces of noncompact type (which are the simply-connected models for manifolds of non-positive curvature). We will end with the Mostow Rigidity Theorem and (hopefully) generalizations by Gromov, Ballman and Burns--Spatzier. A good reference for this material is the book Geometry of Nonpositively Curved Manifolds by Patrick Eberlein.
The linear theory is sometimes considered part of linear algebra. The overarching concept is the matrix exponential and how its structure reflects canonical forms of linear operators. I will introduce the spectral theory of linear differential operators and its intimate connection with complex analysis. I also want to include some more specialized topics, such as linear systems governed by indefinite quadratic forms.
The overarching concepts for the nonlinear theory are flows of vector fields and dynamical systems. Upon that basis, one studies diverse phenomena such as bifurcations, separation of time scales, bursting (such as in neurobiology), hysteresis, stability (this is the connection between linear and nonlinear), control systems, chaos, and strange attractors.
Obviously, a course cannot come close to doing justice to all of these topics, and neither can one person. My goal is twofold: (1) to present the foundational rigorous theory of ODEs, and (2) to introduce a broad variety of topics in ODEs that highlight what makes the field interesting.
Throughout the course, we will use numerical simulations to illustrate various features of the models built in class. Course evaluation will be project-based.