Welcome to my home page!
This academic year I'm a postdoc at Regensburg University,
in the research group of Marc Hoyois.
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.
We are partly supported by SFB 1085 "Higher invariants".
"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: The motive of the Hilbert scheme of infinite affine
Abstract: The Hilbert schemes (of points) of surfaces have many nice properties, however they are hard to study for higher dimensional schemes. In particular the Hilbert scheme of the two-dimensional affine space has a pure Tate motive, but we don’t know the motives of Hilbert schemes of An for higher n. Surprisingly, we observe that the ind-scheme Hilb(A∞) has a pure Tate motive, at least rationally. Moreover, if we let both the dimension of affine space and the degree of points go to infinity, we get that colimd Hilbd(A∞) is A1-homotopy equivalent to the infinite Grassmanian, which has a well-known pure Tate motive thanks to its cellular structure. Time permitting, we will discuss applications of this result to algebraic K-theory and its version for hermitian K-theory. This is joint work with Marc Hoyois, Joachim Jelisiejew, and Denis Nardin.
Title: Motivic generalized cohomology theories from framed
Abstract: All motivic generalized cohomology theories acquire unique structure of so called framed transfers. If one takes framed transfers into account, it turns out that many interesting cohomology theories can be constructed simply as suspension spectra on certain moduli stacks (and their variations). This way important cohomology theories on schemes get new geometric interpretations, and so do canonical maps between different cohomology theories. In the talk we will explain the general formalism of framed transfers and show how it works for various cohomology theories. This is a summary of joint projects with Tom Bachmann, Elden Elmanto, Marc Hoyois, Joachim Jelisiejew, Adeel Khan, Denis Nardin and Vladimir Sosnilo.