# Seminar on A1-topology, motives and K-theory

An online research seminar focusing on motivic homotopy theory and K-theory as well as related subjects from algebraic geometry and topology. Talks are up to 120 minutes uninterrupted. Some talks are recorded. Most talks are in Russian. We usually meet on Thursdays at 12:30 St. Petersburg time (GMT+3) at Zoom channel 818-1526-4739 (passcode required). If you want to get the passcode, (un)subscribe to announcements or give a talk, please contact Alexey Ananyevskiy at alseang@gmail.com.

See also https://indico.eimi.ru/category/12/

**Forcoming talks**

### March 4, 12:30-14:30

Chetan Balwe, IISER Mohali

### TBA

### March 18, 12:30-14:30

Heng Xie, University of Wuppertal

### TBA

**Past talks**

### February 25, 12:30-14:30

Fabio Tanania, LMU Munich

### Stable motivic homotopy groups of the isotropic sphere specrum

In this talk, I will introduce the isotropic stable motivic homotopy category, constructed, following the work of Vishik on isotropic motives, by killing anisotropic varieties. Then, I will discuss the structure of the isotropic Steenrod algebra and properties of the isotropic Adams spectral sequence. At the end, I will present the computation of the stable motivic homotopy groups of the isotropic sphere spectrum. If time allows, I will also discuss some structural results about the category of isotropic cellular spectra.

### February 18, 12:30-14:30

Tariq Syed, University of Duisburg-Essen

### The generalized Vaserstein symbol

I will begin with a brief discussion of the cancellation problem of projective modules over commutative rings (i.e. algebraic vector bundles on affine schemes). Motivated by this, I will introduce the generalized Vaserstein symbol and explain its applications to the cancellation problem and to the generalized Serre question on algebraic vector bundles.

### February 11, 12:30-14:30

Niels Feld, Institut Fourier

### Milnor-Witt sheaves and modules

We present a generalization of Rost’s theory of cycle modules where we use Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a setting to study general cycle complexes and their (co)homology groups. We link this theory with Morel-Voevodsky stable homotopy category and we study homotopy sheaves with generalized transfers. As applications, we discuss a conjecture of Morel about Bass-Tate transfers and a conservativity conjecture due to Bachmann and Yakerson.

### February 4, 12:30-14:30

Fangzhou Jin, Tongji University

### A Gersten complex on real schemes

# We discuss a connection between coherent duality and the Verdier-type duality on real schemes via a Gersten-type complex.

# This is a joint work with H. Xie.

### January 28, 12:30-14:30

Daniil Rudenko, University of Chicago

Goncharov depth conjecture and volumes of orthoschemes

Goncharov conjectured that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. In the first part of the talk I will explain how this conjecture fits into the general scheme of conjectures about mixed Tate motives. In the second part of the talk I will explain an idea behind the proof of the Goncharov conjecture. The proof is based on an explicit formula, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Surprisingly, almost the same formula gives a volume of a hyperbolic orthoscheme generalising the formula of Lobachevsky in dimension 3 to an arbitrary dimension.

### January 21, 12:30-14:30

Tom Bachmann, LMU Munich

η -periodic motivic stable homotopy theory

The

### December 3, 16:00-18:00

Elden Elmanto, Harvard

### On Bass’ NK groups of schemes in mixed characteristics

Two obstacles exist when trying to compute K-theory of schemes: lack of

### November 26, 12:30-14:30

Maria Yakerson, ETH Zürich

### Universality of hermitian K-theory

In this talk, we will give a new model of the motivic hermitian K-theory spectrum, which can be interpreted as a new universality property of hermitian K-theory in terms of its structure of transfers. In addition, we will explain the connection with the analogous property of algebraic K-theory. This is joint work in progress with Marc Hoyois, Joachim Jelisiejew, and Denis Nardin.

### November 19, 12:30-14:30

Adeel Khan, IHES

### Sheaf-theoretic Fourier transforms

In the 70’s, Sato and Deligne introduced sheaf-theoretic versions of the Fourier transform, which interchange sheaves on a vector bundle with sheaves on its dual. The Fourier-Sato transform has proven to be a vital tool in microlocal analysis on manifolds, and the Fourier-Deligne transform is similarly important in the theory of l-adic sheaves. I will talk about a version for motivic sheaves, developed jointly with Cisinski and Zargar, which unifies the Fourier-Sato transform and Laumon’s homogeneous variant of the Fourier-Deligne transform. This leads to a motivic theory of microlocalization which is currently under development. In another direction, I will describe an extension of the Fourier-Sato transform to perfect complexes and, time-permitting, applications of this in Donaldson-Thomas theory.

### November 12, 12:30-14:30

Pavel Sechin, University of Duisburg-Essen

### Landweber-equivariant Grothendieck motives: a toy example of non-oriented phenomena

Landweber-equivariant motives are constructed as a limit of a diagram of categories of Grothendieck motives associated to oriented cohomology theories (such as algebraic cobordism, Chow groups and K-theory) with functors of Riemann-Roch type between them. Thus constructed category is computable to the same degree as algebraic cobordism, but it possesses some non-oriented properties, e.g. a motive of a projective space does not decompose anymore as a direct sum of ‘Tate motives’ and the dual of a motive of a smooth projective variety involves a non-trivial twist along the class of the tangent bundle in K-theory. This category has an exact structure, and many (or maybe all, to an extent) direct sum decompositions in Chow motives (e.g. projective bundle decomposition or a blow-up along a smooth subvariety decomposition) lift to non-trivial extensions in this category.

I will explain the construction and the properties of Landweber-equivariant motives, but at the moment there are no applications.

### November 5, 12:30-14:30

Anand Sawant, TIFR

### Cellular A1-homology of schemes associated with the Bruhat decomposition

We will introduce the notion of cellular

### October 29, 12:30-14:30

Denis Nardin, Universität Regensburg

### Hermitian K-theory of Dedekind domains

In this talk I will explain how to define a notion of Hermitian K-theory for stable

### October 22, 12:30-14:30

Grigory Garkusha, Swansea University

### Semilocal Milnor K-theory

In this talk I will tell about semilocal Milnor

### October 15, 12:30-14:30

Alexey Ananyevskiy, PDMI RAS

### Motivic second Hopf map as an obstruction to symplectic orientation (2/2)

I will give the construction of the motivic second Hopf map ν in terms of framed correspondences and show that the nonvanishing of the corresponding element in generalized motivic cohomology gives an obstruction to the existence of symplectic Thom isomorphisms. As a corollary we will see that the stable A1-derived category does not admit Thom isomorphisms for oriented (and for symplectic) bundles.

### October 8, 12:30-14:30

Pablo Pelaez, UNAM

Some applications of the motivic Becker-Gottlieb transfer

We will discuss the construction (due to Carlsson and Joshua) of the motivic Becker-Gottlieb transfer as well as some of its basic properties (due to Joshua and the speaker) and a few applications.

### October 1, 12:30-14:30

Alexey Ananyevskiy, PDMI RAS

### Motivic second Hopf map as an obstruction to symplectic orientation (1/2)

I will give the construction of the motivic second Hopf map ν in terms of framed correspondences and show that the nonvanishing of the corresponding element in generalized motivic cohomology gives an obstruction to the existence of symplectic Thom isomorphisms. As a corollary we will see that the stable A1-derived category does not admit Thom isomorphisms for oriented (and for symplectic) bundles.

### September 24, 12:30-14:30

Vladimir Sosnilo, PDMI RAS, SPbU

### Excision for algebraic K-theory with respect to categorical Milnor squares (2/2)

In this talk we will define a Milnor square of stable infinity-categories. We will prove that the nonconnective K-theory sends such a square to a homotopy pullback square of spectra. In particular, there is associated long exact sequence of K-groups. We will explain the relation between the new notion and Milnor squares of rings. The classical Milnor excision result of Suslin and Wodzicki will follow from this result. More generally, the theorem generalizes a recent result of Land and Tamme about the K-theory of pullbacks of

### September 17, 12:30-14:30

Vladimir Sosnilo, PDMI RAS, SPbU

### Excision for algebraic K-theory with respect to categorical Milnor squares (1/2)

In this talk we will define a Milnor square of stable infinity-categories. We will prove that the nonconnective K-theory sends such a square to a homotopy pullback square of spectra. In particular, there is associated long exact sequence of K-groups. We will explain the relation between the new notion and Milnor squares of rings. The classical Milnor excision result of Suslin and Wodzicki will follow from this result. More generally, the theorem generalizes a recent result of Land and Tamme about the K-theory of pullbacks of