The construction problem for Hodge numbers

Stefan Schreieder
University of Bonn
October 8, 2014
What are the possible Hodge numbers of a smooth complex projective variety? We construct enough varieties to show that many of the Hodge numbers can take all possible values satisfying the constraints given by Hodge theory. For example, there are varieties such that a Hodge number \(h^{p,0}\) is big and the intermediate Hodge numbers \(h^{i,p-i}\) are small.

Cool with a Gaussian: an \(O^*(n^3)\) volume algorithm

Santosh Vempala
Georgia Institute of Technology
October 13, 2014
Computing the volume of a convex body in n-dimensional space is an ancient, basic and difficult problem (#P-hard for explicit polytopes and exponential lower bounds for deterministic algorithms in the oracle model). We present a new algorithm, whose complexity grows as \(n^3\) for any well-rounded convex body (any body can be rounded by an affine transformation). This improves the previous best Lo-Ve algorithm from 2003 by a factor of \(n\), and bypasses the famous KLS hyperplane conjecture, which appeared essential to achieving such complexity.