Mathematics 797AP - ST-Asymptotic Problems

This course is a graduate level introduction to certain types of extremal problems in mathematics. Though the objects we study come from disparate parts of mathematics (coding theory, graph theory, curves over finite fields, algebraic number fields, lattices, ...), there is an underlying common structure to all the problems, whereby upper bounds on the 'quality' of the objects come from zeta functions, and the lower bound on existence of optimal or near-optimal objects come from group theory (or modular forms). The problems we study are often motivated by applications to information-theory, engineering and other disciplines; though the applications are not the focus of the course, we use them as motivation for studying the intricate algebraic structures which often achieve the optimal solutions for the problems studied, especially in an asymptotic sense. We will use a number of survey papers and textbooks available on codes (Van Lint, Algebraic Coding Theory, GTM, Springer), Expander Graphs (AMS Bulletin survey by Hoory-Linial-Wigderson and by Lubotzky), as well as notes provided by the Instructor. There will be weekly problem sets and a Final Project.

Prerequisites: Math 611 and Math 612 would be useful but not crucial.