Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
Title: Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics (Thesis Defense)
Speaker: Bryan Ek, Rutgers University
Date: Thursday, March 29, 2018 5:00pm
Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ
The main goal of these projects was utilizing experimental mathematics to further our knowledge of several areas in math. We begin by tweaking a proof of unimodality by O'Hara to produce many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. We then employ a generating function relation technique used by Ayyer and Zeilberger to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded planes. The method in which we do this is formulated to be highly algorithmic so that a computer can automate most, if not all, of the work. I easily recover many well-known results for simpler step sets and discover new results for more complex step sets.