# Trimester Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45, Bonn

Organizers: Guillermo Cortiñas, Hélène Esnault, Christian Haesemeyer, Holger Reich, Jonathan Rosenberg

## Tuesday, May 9

15:00 - 16:00 Welcome Meeting

Abstract: Welcome meeting with the participants and organizers of the Trimester Program.

## Wednesday, May 24

14:00 Organizing Meeting

Abstract: This is an organizing meeting for the trimester seminar. There is also the opportunity to suggest and discuss informal working seminars and reading groups.

## Wednesday, May 31

14:00 Manh Toan Nguyen: Equivariant motivic cohomology

Abstract: Motivic cohomology for smooth schemes over a field is now a well-established theory. This led to the solution to the Milnor-Bloch-Kato conjecture by Voevodsky and Rost. At present, the corresponding equivariant theory is developing and attracting many attentions.

In this talk, I will discuss about two constructions of motivic cohomology for algebraic varieties over a field equipped with an action of a finite group: one is given by Levine-Serpé in terms of equivariant cycles and the other, follows an idea of Grayson, is in terms of the K-theory of equivariant automorphisms. I will also explain how these objects relate to each other as well as to other interesting theories such as orbifold Chow ring and equivariant algebraic K-theory.

## Wednesday, June 7

14:00 Vivek Sadhu: A relative version of Weibel's vanishing conjecture

Abstract: Weibel conjecture says that for a d-dimensional noetherian scheme X, K_{n}(X)=0 for n<-d and X is K_{-d}-regular. Very recently, a complete solution of this conjecture has been given by Kerz-Strunk-Tamme (KST). In the first part of this talk, I will discuss the progress history of this conjecture and the technique of KST. In the second part, I will present a relative version of this conjecture.

## Wednesday, July 5

14:00 Rufus Willett: Classification of C*-algebras and computing K-theory

Abstract: Over the past several years huge progress has been made in the program to classify simple C*-algebras by K-theoretic invariants (essentially, topological K-theory, plus the pairing with traces on the algebras) due to Elliott and others. The remaining steps needed to complete the program are K-theoretic, revolving around the so-called Universal Coefficient Theorem (UCT). I’ll survey what’s known here, including an example of Skandalis showing the UCT can fail for reduced group C*-algebras and connections to the so-called Baum-Connes conjecture, and what remains to be done. I’ll then outline a joint (large) project formulated by Nate Brown, Guoliang Yu, and myself to tackle some of the remaining questions. I’ll try not to assume any background in C*-algebra classification or Baum-Connes theory.

## Wednesday, July 12

14:00 Lars Hesselholt: K-theory of division algebras over local fields

Abstract: This is a report on joint work with Michael Larsen and Ayelet Lindenstrauss. Let K be complete discrete valuation field with finite residue field of characteristic p > 0, and let D be a central division algebra of finite dimension d over K. We show that for every positive integer i, there exists an isomorphism of p-adic K-groups Nrd_{D/K}: K_{i}(D,ℤ_{p}) → K_{i}(K,ℤ_{p}) such that d Nrd_{D/K} = N_{D/K} is the norm. The case where p does not divide d was proved thirty years ago by Suslin and Yufryakov, and the new ingredient that makes it possible now to prove the general case is the cyclotomic trace map.

## Thursday, July 13

16:30 Alexander Berglund: A dg Lie algebra model for the block diffeomorphism group

Abstract: I will describe a differential graded Lie algebra model, in the sense of Quillen's rational homotopy theory, for the block diffeomorphism group of a simply connected manifold with boundary a sphere. The model yields computations of the rational homotopy groups and, in favorable situations, it can be used to compute the rational cohomology (in a stable range) in terms of certain graph complexes. This is joint work with Ib Madsen.

## Tuesday, July 18

15:00 Thomas Nikolaus: On topological cyclic homology I

Abstract: We review the notion of topological cyclic homology and the recent progress made in joint work with P. Scholze. Our main result is a drastic simplification of the Definition. This also also to better understand the structural meaning. We will discuss some example and application and if time allows outline some future directions.

## Wednesday, July 19

15:00 Alexander Varchenko: Elliptic dynamical quantum group E_{τ,h}(gl_{2}) and elliptic equivariant cohomology of the cotangent bundles of Grassmannians

Abstract: The torus T equivariant elliptic cohomology defines a functor Ell_{T}: {T - spaces X} → {schemes}. For example, for the cotangent bundle of a Grassmannian the scheme Ell_{T}(T*Gr(k,n)) is some explicitly given sub-scheme of S^{k}E × S^{n-k}E × E^{n} × E^{2} with coordinates t_{1}, …, t_{k}, s_{1}, …, s_{n-k}, z_{1}, …, z_{n}, h, λ, where t_{i}, s_{j} correspond to the Chern roots of the two standard vector bundles over the Grassmannian, z, y correspond to the torus parameters, λ is the dynamical parameter also called the Kähler parameter, and E is an elliptic curve.

I will define a class of line bundles on the scheme ∪_{k=0,…,n} Ell_{T}(T*Gr(k,n)) such that the operator algebra of the elliptic dynamical quantum group E_{τ,h}(gl_{2}) will act on sections of those line bundles (a generator of the operator algebra will send a section of such a line bundle to a section of possibly another line bundle). That construction is an analog of the Yangian Y(gl_{2}) action on the direct sum ⊕_{k=0,…,n} H*_{T}(T*Gr(k,n)) of equivariant cohomology.

This is a joint work with G. Felder and R. Rimanyi.

## Thursday, July 20

10:00 Cary Malkiewich: Periodic orbits and equivariant traces

Abstract: This informal talk is about an emerging connection between transfers on K-theory and THH on the one hand, and periodic orbits of continuous dynamical systems on the other. The centerpiece is a model for the C_{n} equivariant transfer map on THH, which we are calling the "n-th power trace". We began studying this construction in order to calculate transfers on algebraic K-theory. Happily, it turned out to be the key ingredient needed to resolve a question of Klein and Williams, which gives us an effective way to connect the theory of periodic points to the equivariant theory of fixed points. We suspect that the nth power trace will be useful for further questions that interpolate between geometric phenomena in dynamics and algebraic invariants in K-theory and THH. I will describe one such question that we plan to attack next. This is joint work with Kate Ponto.

## Wednesday, July 26

16:30 Thomas Nikolaus: On topological cyclic homology II

## Wednesday, August 2

15:00 Adeel Khan: Motivic infinite loop spaces

Abstract: Given a topological space X, May’s recognition principle says that a structure of infinite loop space on X is equivalent to a group-like E_{∞}-monoid structure. We will discuss an analogous result in motivic homotopy theory which says that a structure of infinite P^{1}-loop space on a motivic space X is equivalent to a homotopy coherent system of finite lci transfer maps. This is based on an analogue of the Pontrjagin-Thom construction which identifies stable homotopy groups of spheres with groups of framed bordisms. We will also discuss some applications, including a derived version of the Chow group which computes rational motivic cohomology. This is joint work with E. Elmanto, M. Hoyois, V. Sosnilo and M. Yakerson, and the applications also rely on joint projects with D. Rydh and with F. Déglise.

## Thursday, August 3

15:00 Shantanu Dave: Geometric hypoellipticity and topological invariants

Abstract: Many common geometric structures on a manifold produce natural differential operators. These operators are easily seen to be elliptic in usual Riemannian, spin, complex setup and include the Dirac and Laplace operators. Their analysis then leads to various topological and geometric invariants such as K-theory classes and torsion. However many of the other operators, such as the curved BGG operators are not elliptic in the usual sense. In this talk we shall describe how these operators can be constructed. We will also explain why, a more exotic tangent groupoid construction and harmonic analysis on nilpotent Lie groups, guarantee that these operators have the same analytic properties as elliptic operators. We shall then describe the invariants that can be constructed from these operators. This is joint work with Stefan Haller in Vienna.

## Tuesday, August 8

15:00 Irakli Patchkoria: THH of differential graded algebras and exotic equivalences

Abstract: We will compute topological Hochschild homology of certain universal differential graded algebras obtained by killing primes. For this we will use THH of Thom spectra and Bökstedt's theorem. The calculation reduces to computing homology of free loop spaces with twisted coefficients. Our main examples of DGAs are constructed from specific number rings. The computation shows that these DGAs are not formal over the sphere spectrum. As an application we get new examples of exotic equivalences. At the end we will construct a new family of DGAs which are not formal over the integers and which generalize the above mentioned examples. Computing THH of the these remains open.

## Wednesday, August 16

15:00 Heng Xie: Algebraic KR-theory of algebraic varieties

Abstract: In topology, KR-theory was invented by Atiyah in the 1960's. In this talk, I will define a version of algebraic KR-theory of algebraic varieties. I will explain recent joint work with Marco Schlichting and Girja Tripathi on a geometric model of algebraic KR-theory in the C_{2}-equivariant motivic homotopy category of Heller, Krishna and Østvær. I will also give a version of the dévissage theorem for algebraic KR-theory. This result generalizes Gille's dévissage theorem for Balmer's Witt groups and the dévissage theorem of Hu, Kriz and Ormsby. It provides a useful tool for us to compute algebraic KR-theory of a smooth scheme (with C_{2}-action) supported in an invariant closed subscheme in terms of algebraic KR-theory of that invariant closed subscheme. If time permits, I will use this dévissage theorem to prove the Real projective space formula over regular bases.

## Thursday, August 17

15:00 Bram Mesland: A Hecke module structure on the KK-theory of arithmetic groups

Abstract: Let G be a locally compact group, Γ a discrete subgroup and C_{G}(Γ) the commensurator of Γ in G. The cohomology of Γ is a module over the Shimura Hecke ring of the pair (Γ,C_{G}(Γ)). This construction recovers the action of the Hecke operators on modular forms for SL(2,Z) as a particular case. In this talk I will discuss how the Shimura Hecke ring of a pair (Γ, C_{G}(Γ)) maps into the KK-ring associated to an arbitrary Γ-C*-algebra. From this we obtain a variety of K-theoretic Hecke modules. In the case of manifolds the Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions. We discuss Hecke equivariant exact sequences arising from possibly noncommutative compactifications of Γ-spaces. Examples include the Borel-Serre and geodesic compactifications of the universal cover of an arithmetic manifold, and the totally disconnected boundary of the Bruhat-Tits tree of SL(2,Z). This is joint work with M.H. Sengun (Sheffield).