Welcome to my home page!
The focus of our research group is motivic homotopy theory,
which applies powerful methods of algebraic topology
to shed light on mysteries of algebraic geometry.
"The more I learned, the more conscious did I become of the fact that I was ridiculous. So that for me my years of hard work at the university seem in the end to have existed for the sole purpose of demonstrating and proving to me, the more deeply engrossed I became in my studies, that I was an utterly absurd person."
Fyodor Dostoevsky, The Dream of a Ridiculous Man
Title: Modules over algebraic cobordism
Abstract: When k is a field with resolution of singularities, it is known that Voevodsky’s category of motives DM(k) is equivalent to the category of modules over the motivic cohomology spectrum HZ. This means that a structure of an HZ-module on a motivic spectrum is equivalent to a structure of transfers in the sense of Voevodsky. In this talk, we will discuss an analogous result for modules over the algebraic cobordism spectrum MGL. Concretely, a structure of an MGL-module is equivalent to a structure of coherent transfers along finite syntomic maps, over arbitrary base scheme. Time permitting, we will see a generalization of this result to modules over other motivic Thom spectra, such as the algebraic special linear cobordism spectrum MSL. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.
Title: Higher Chow-Witt groups
Abstract: Algebraic K-theory of smooth schemes is built out of blocks given by motivic cohomology, via a spectral sequence. Motivic cohomology groups, defined as hypercohomology groups of a motivic analogue of the singular chain complex, have a beautiful geometric presentation in terms of algebraic cycles, by higher Chow groups. For hermitian K-theory there is a similar spectral sequence, however in this case the spectral sequence has additional terms, given by the Milnor-Witt motivic cohomology of Calmès-Fasel. In this talk, we provide a geometric presentation of MW-motivic cohomology in the flavour of higher Chow groups. The main ingredient is our computation of Levine’s homotopy coniveau tower for “discrete” motivic spectra. This is joint work with Tom Bachmann.