Abstract:

Linnik pioneered, by his dispersion method, the study of shifted convolution in the kth fold divisor function, which reveals fascinating connections between the theory of zeta functions and analytic properties of the Eisenstein series.  In this talk, we will present an approach to the shifted convolution involving the cubic or higher integral powers, and the Whittaker-Fourier coefficients of cuspidal automorphic forms on GL(n). It leads to strong asymptotic bounds, based on among other things the Weyl inequality and the Vinogradov mean value conjecture, the latter recently proved by Bourgain-Demeter-Guth via decoupling techniques in harmonic analysis. 

Brief Bio-sketch of the Speaker:  

Wenzhi Luo received his Ph.D. in 1993 from Rutgers University under the guidance of Henryk Iwaniec. He was a member of  IAS Princeton in 1993-1994, and a member of MSRI Berkeley during 1994-1995. From 1995 to 1999 he worked in Princeton University, first as an Instructor, and then an Assistant Professor. He was a Sloan Research Fellow during 1999-2001. He joined the faculty of the Ohio State University in 1999, and is currently a Professor of Mathematics.  Professor Luo  works in the area of analytic number theory, especially the spectral theory of automorphic forms. His research interests are focused on analytic properties of eigenfunctions of the Laplacian on locally symmetric spaces, and their connections to the Langlands automorphic L-functions. Jointly with Z.Rudnick and P.Sarnak, he has made contributions to the Selberg eigenvalue conjecture and the generalized Ramanujan conjecture. His research works have been published in Annals of Math,  Inventiones Math, IHES Publ. Math and Duke Math Journal, among others.  He was a Co-PI on an NSF Focus Research Group (FRG) grant with M. Einsiedler, E. Lindenstrauss, P.Sarnak, A.Venkatesh in 2006-2009.