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).
