"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: 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.