This project aims at studying the algebraic K-theory of, and algebraic cycles on, algebraic varieties, using geometric, topological and algebraic methods. The investigator proposes to further apply the descent methods developed in earlier work in order to better understand the K-theory and Hochschild homology of singularities. This work is related to well-known conjectures in the homology theory of commutative algebras, as for example Bergers conjecture that curves with torsion free differentials are smooth. In a separate project, the investigator plans to study new height-type functions on algebraic cycles in hopes of making progress on question of divisibility of cycles modulo algebraic equivalence.

This project studies invariants of algebraic varieties, that is, objects that can be realized as sets of solutions of polynomial equations. In particular, it is proposed to study singularities (a black hole is an example of a singularity) via an invariant called K-theory that encodes the behavior of very large matrices of polynomial functions on the singularity; and to study the behavior of families of formal sums of subvarieties (the sets of solutions of even more equations).

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0813100
Program Officer
Tie Luo
Project Start
Project End
Budget Start
2007-08-16
Budget End
2009-03-31
Support Year
Fiscal Year
2008
Total Cost
$79,538
Indirect Cost
Name
University of Illinois at Chicago
Department
Type
DUNS #
City
Chicago
State
IL
Country
United States
Zip Code
60612