Welcome to my home page!
I'm a 4th year math PhD student in Marc Levine's group
at the University of Duisburg-Essen, in Germany.
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: The space of algebraic cobordism
Abstract: Classically, the universal oriented cohomology theory is represented by the complex cobordism spectrum MU, which has a description
in terms of cobordism spaces of compact smooth manifolds with an extra structure. In motivic settings, the analogous spectrum MGL was constructed by Voevodsky, and over a base field of
characteristic 0, its corresponding homotopy groups can be expressed via algebraic “cobordism groups” of varieties, as defined by Levine-Morel. In this talk we will discuss a model for the
underlying space of MGL as an algebro-geometric incarnation of the cobordism space. As an application, we will present a geometric description of this space as a (homotopy type of) some Hilbert
scheme. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.
Title: Motivic recognition principle
Abstract: Classically in homotopy theory, infinite loop spaces are recognized as spaces with an additional structure: grouplike
E_{∞}-spaces. The category of such spaces is equivalent to the category of connective spectra. Replacing topological spaces with smooth schemes, we end up in the realm of
motivic homotopy theory, where an analogous statement was sought for since the theory has appeared. In this talk we will discuss the motivic recognition principle, which provides an
equivalence between the category of motivic connective spectra and the category of grouplike motivic spaces with so called framed transfers. This is joint work with Elden Elmanto, Marc
Hoyois, Adeel Khan and Vladimir Sosnilo.
Title: How to encode transfers in motivic homotopy theory
Abstract: Voevodsky’s approach to constructing the (derived) category of motives starts with introducing so called presheaves with transfers. Later Calmès
and Fasel added extra data of quadratic forms to this construction, by considering presheaves with Milnor-Witt transfers. We will show that these transfers are examples of a more general
construction of E-transfers, defined for any motivic ring spectrum E. We will also discuss relation of E-transfers with framed transfers, and some consequences for understanding the category of
E-modules. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.
Title: Motivic stable homotopy groups via framed correspondences
Abstract: A modern approach to the motivic stable homotopy category allows one to express its mapping spaces in terms of geometric data called "framed correspondences". We will explain
this approach and illustrate it by computing Gm-homotopy groups of the special linear algebraic cobordism spectrum MSL.