Title: Positivity and L-functions Speaker: Prof. Wei Zhang (MIT) Time: 2018-6-25, 9:15-10:15 Place: N818 Abstract: We discuss several examples of positivity in arithmetic geometry related to L-functions.

 Title: Defects of Poisson's formula and zeros of Riemann's zeta function Speaker: Prof. Xianjin Li (Brigham Young University) Time: 2018-6-25, 10:30-11:30 Place: N818 Abstract: According to Weil's explicit formula, RH measures the failure of the Poisson summation formula. In this talk, a sequence of functions will be given which are precisely the defects coming out the failure of the Poisson summation formula. By using these functions, nontrivial zeros of the Riemann zeta-function are shown to have geometric multiplicity one. For every nontrivial zero rho of the zeta-function, an eigenfunction with eigenvalue rho of the fundamental differential operator on the multiplicative group of positive real numbers is constructed. It is also shown that this eigenfunction is the eigenvector of the convolution operator v(h) with eigenvalue \hat h(rho) as appearing in Weil's explicit formula.