#### NSF Award from the Division of Mathematical Sciences

## Theory of Motivic Sheaves and Beyond

**Principal Investigator:**Vadim Vologodsky, PhD, Assistant Professor, Department of Mathematics**Start Date:**July 1, 2009**Total Award Amount:**$35,599

### Project Description

In this project, the PI intends to use the theory of motives to create new tools for the study of problems in algebraic geometry and number theory. The PI will develop a theory of motivic sheaves. His approach is based on the idea that the category of motivic sheaves over a scheme can obtained by gluing the DG categories of Voevodsky's motives over points. The PI plans to study the functor of motivic vanishing cycles and prove that this functor commutes with the Hodge realization and l-adic realization. The PI will apply these results to the study of degenerations of algebraic varieties and, more specifically, to the integrality conjectures in mirror symmetry.

Additionally, the PI plans to use the theory of motivic sheaves to embed the category of log schemes into the category of motives providing a bridge between the Fujiwara-Kato-Nakayama Hodge and etale realizations of log schemes. The theory of motives is a central part of algebraic geometry. Algebraic geometry deals with systems of algebraic equations. The theory of motives uses ideas from algebraic topology (study of shapes by means of algebra)to understand structural properties of the solution set even when one cannot actually solve the equation explicitly. This theory provides a language to relate seemingly unrelated objects such as the space of complex solutions to a given system of algebraic equations with integral coefficient and the set of its integral solutions. The PI plans to borrow some ideas from singularity theory (a part of algebraic topology, that studies shapes by degenerating them into simpler ones), adapt these ideas to the new setting, using the language of motives, and then apply it to solve problems in algebraic geometry and number theory that can not be solved by other methods.

This award is funded under the American Recovery and Reinvestment Act of 2009, NSF Award number: 0901707