Combinatorial zeta and L-functions


14:00:00 - 15:00:00

103 , Mathematics Research Center Building (ori. New Math. Bldg.)

Combinatorial zeta functions are discrete analogue of the Selberg zeta function. Ihara generalized the Selberg zeta function from PGL(2, R) to PGL(2, Q_p), and Serre realized that Ihara\'s zeta function can be defined for all finite graphs. Zeta functions for finite simplicial complexes were studied in a joint work with Kang for finite quotients of the building attached to PGL(3), and with Fang and Wang for finite quotients of the building attached to SL(4). Such zeta function is a rational function with a closed form expression which gives both topological and spectral information of the underlying combinatorial object.
The Artin L-functions for graphs were considered by Ihara, Hashimoto, Stark and Terras, respectively. Very recently in a joint work with Kang we obtained a closed form expression for the Artin L-functions attached to finite quotients of the building of PGL(3).
  In this talk we shall survey the progress of the combinatorial zeta and L-functions, and compare them with the zeta and L-functions in number theory.