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

Past talks

May 6, 12:30-14:30

Katharina Hübner, Einstein Institute of Mathematics

TBA

 

 

April 22, 12:30-14:30

Frédéric Déglise, ENS de Lyon

TBA

 

 

April 15, 12:30-14:30

Ran Azouri, Universität Duisburg-Essen

TBA

 

 

April 1, 13:00-15:00

Mikhail Bondarko, SPbU

On the motivic t-structure on DM(k,Q)

I will describe my recent observations on certain candidates for the (conjectural) motivic t-structure and on objects that represent “classical” cohomology theories.

General (and recent) abstract nonsense easily implies that any homology theory H on the “big” category DM(k,Q) of motives over a field defines a t-structure t_H (that is, the “cohomological left hand side” of  t_H is characterized by the vanishing of H(M[i]) for i>0). In particular, this is the case when H is the Ql-etale homology (clearly, one can also take etale cohomology here).

t_H restricts to compact motives whenever the weight structure w right adjacent to it “respects coproducts”, and the converse implication follows from standard conjectures. Moreover, the “right hand side” of this t_H can be described “almost explicitly” if the object R that represents etale cohomology is connective (that is, there are no non-zero morphisms RR[i] for i>0).

I have also proved that the so-called Chow t-structure splits this object R. Consequently, the weight filtration on the Q_l-etale cohomology of motives splits functorially; this is also true for any other cohomology that possesses similar properties. Moreover, I have some ideas how to prove that this splitting respects cup products and to verify the aforementioned connectivity.

 

March 25, 12:30-14:30

Sabrina Pauli, University of Duisburg-Essen

Formulas for the A1-degree

Morel´s A1-degree is a very important tool in A1-enumerative geometry and one wishes to have nice algebraic formulas for it. One way to compute it is to express it as a sum of local degrees for which there exist formulas whenever the residue fields are separable field extensions over the ground field. Cazanave provides a global formula for the A1-degree of an endomorphism of P1 given by the so called Bezoutian. In my talk I will show how a multivariate version of the Bezoutian can be used to compute the A1-degree generalising Cazanave´s result and removing the assumptions on the residue fields.

This is joint work with Thomas Brazelton and Stephen McKean.

 

March 18, 12:30-14:30

Heng Xie, University of Wuppertal

Witt groups of spinor varieties

In this talk, I will discuss the computation of Witt groups of maximal isotropic Grassmannians, aka. spinor varieties. They are examples of type D homogeneous varieties. The method relies on the Blow-up setup of Balmer-Calmès, and we investigate the connecting homomorphism via the projective bundle formula of Walter-Nenashev, the projection formula of Calmès-Hornbostel and the excess intersection formula of Fasel. The computation in the Type D case can be presented by so called “even shifted young diagrams”.

This is joint work with Thomas Hudson and Arthur Martirosian.

 

March 11, 12:30-14:30

David Kumallagov, SPbU

Smooth weight structures and weakly birational objects

 

In this talk, we will introduce a family of aisles on the various motivic categories, defined in terms of smooth varieties. These aisles vastly generalize Chow weight structures as well as slice filtrations. Then we will prove that the filtration on Ht_hom defined by these aisles can be described in some ”classical” terms. Next, we will discuss weight structures w^s_Smooth defined by these aisles. This allows us to construct a filtration on the Ht^eff_hom with the help of which to prove some results that widely generalize some results of Kahn-Sujatha.

This is a joint work with M. Bondarko.

March 4, 12:30-14:30

Chetan Balwe, IISER Mohali

Geometric criteria for A^1-connectedness

 

We prove that for a large class of varieties, A^1-connectedness can be characterized as the condition that the generic point of the variety can be connected to a k-rational point by an A^1-homotopy. As an application, we will show that over an algebraically closed field of characteristic zero, given any smooth, proper A^1-connected variety, the smooth models of its symmetric powers are also A^1-connected. This can be applied to deduce that the norm varieties constructed by Rost are A^1-connected (over an algebraically closed field of characteristic zero). This talk is based on joint work with Amit Hogadi and Anand Sawant. 

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 η-periodic motivic sphere over fields has attracted considerable interest in the past: Ananyevskiy–Levine–Panin have proved that its rationalization is Eilenberg–MacLane, and various authors have studied its homotopy groups over specific fields. In this talk I will explain recent results obtained in joint work with Mike Hopkins, in which we establish structural properties of the η-periodic category, over in principle general bases (containing 1/2). In particular we compute over Z[1/2] the η-periodic stable stems, as well as the η-periodic algebraic SL-cobordism groups.

 

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 A^1-invariance and lack of a Kunneth formula. The overall goal of this project is to describe the K-theory of the affine spaces over a scheme over Z_p in terms of the K-theory of the scheme itself and information controlled by “Hochschild-type” information. I aim to explain the approach of Cortinas-Haesemeyer-Weibel-Walker in characteristic zero and some progress made in mixed characteristics with Martin Spers.

 

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 A1-homology of a smooth scheme admitting a nice stratification.  We will discuss how to orient the cellular A1-chain complex (whose homology is the cellular A1-homology) and make computations using it, with focus on the Bruhat decomposition of a split reductive group and the associated flag variety.  The talk is based on joint work with Fabien Morel.

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 -categories and how this is related with classical notions of Grothendieck-Witt theory and L-theory. We will use the properties of these invariants to compute the hermitian K-theory groups for Dedekind domains in term of other invariants like the K-groups and the Picard group. This is joint work with B. Calmès, E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, T. Nikolaus and W. Steimle.

October 22, 12:30-14:30

Grigory Garkusha, Swansea University

Semilocal Milnor K-theory

In this talk I will tell about semilocal Milnor K-theory of fields. A strongly convergent spectral sequence relating semilocal Milnor K-theory to semilocal motivic cohomology is constructed. In weight 2, the motivic cohomology groups  are computed as semilocal Milnor K-theory groups  . The following applications are given: (i) several criteria for the Beilinson-Soulé Vanishing Conjecture; (ii) computation of   of a field; (iii) the Beilinson conjecture for rational K-theory of fields of prime characteristic is shown to be equivalent to vanishing of rational semilocal Milnor K-theory.

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 -rings.

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 -rings.