Speaker :	Prof. Wen-Ching Li (National Center for Theoretical Sciences and Pennsylvania State University)
Title :	Zeta functions in combinatorics and number theory
Time :	2011-05-12 (Thu) 15:00 - 16:00
Place :	Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)
Abstract:	Roughly speaking, a zeta function is a counting function. Well-known zeta functions in number theory include the Riemann zeta function and the zeta function attached to an algebraic variety defined over a finite field. The former counts integral ideals of a given norm, while the latter counts solutions over a finite field. A combinatorial zeta function counts tailless geodesic cycles of a given length in a finite simplicial complex. One-dimensional complexes are graphs; attached to graphs are the well-studied Ihara zeta functions. Zeta functions attached to 2-dimensional complexes are recently obtained by myself and students Ming-Hsuan Kang and Yang Fang by considering finite quotients of the Bruhat-Tits buildings associated to SL(3) and Sp(4) over a p-adic field. The purpose of this talk is to show connections between combinatorics and number theory, using zeta functions as a theme. We shall give closed form expressions of the combinatorial zeta functions mentioned above, and compare their features, in particular, the role of the Riemann Hypothesis, with those of the zeta functions for varieties over finite fields.