• K-Theory and Related Fields
• Workshop: K-theory and related fields

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• Participants

Federico Binda: Towards a motivic (homotopy) theory without A¹-invariance

Motivic homotopy theory as conceived by Morel and Voevodsky is based on the crucial observation that the affine line A¹ plays in algebraic geometry the role of the unit interval in algebraic topology. Following the work of Kahn-Saito-Yamazaki, we constructed an unstable motivic homotopy category "with modulus", where the affine line is no longer contractible. In the talk, we will sketch this construction and we will explain why this category can be seen as a candidate environment for studying representability problems for non A¹-invariant generalized cohomology theories.

Recorded Talk

Real topological Hochschild homology (THR) is a Z/2-equivariant spectrum introduced by Hesselholt and Madsen as the recipient of a trace map from real algebraic K-theory of discrete rings with anti-involution.

In joint work with Moi and Patchkoria we interpret THR as a derived smash product of modules over the Hill-Hopkins-Ravenel norm, and carry out calculations for F_p, group-algebras and in π₀.

In joint work with Ogle we extend the construction of real K-theory to ring spectra, and use the trace to THR to show that the restricted assembly map of the spherical group-ring splits. One can then reformulate the Novikov conjecture in terms of the vanishing of the trace on the kernel of a certain linearization map in rational Hermitian K-theory.

Recorded Talk

In this talk we discuss some first steps towards describing the tensor triangulated spectrum (in Balmer's sense) of the motivic stable homotopy category over a finite field, and some consequences.

The Bass-Quillen conjecture says that vector bundles over polynomial rings come from the base ring if the latter is regular. We study an analogous problem in non-archimedean analytic geometry, where one replaces polynomials by convergent power series. Using a new descent result for algebraic K-theory we prove a stable version of this non-archimedean problem.

Motivic Eisenstein classes have been defined in various situations, for example for G = S be a smooth commutative group scheme of relative dimension d and t ∈ G(S) a torsion section (with A. Huber) where one gets classes Eis_mot^k(t) ∈ H_mot^2d-1(S,Sym^k H_Q(d)) with H_Q the first relative motivic homology of G = S. We discuss p-adic interpolation results for the images of these classes under the étale regulator and explain some arithmetic applications, in particular the explicit reciprocity law for Rankin-convolutions obtained together with Loeffler and Zerbes.

Recorded Talk

Let R → A be a map of rings (or ring spectra) and suppose that A is finitely generated projective (or perfect) as an R-module. Then in addition to the usual map on algebraic K-theory K(R) → K(A), there is a wrong-way "transfer" map K(A) → K(R). In particular, when E → B is a map of spaces whose homotopy fiber F is finitely dominated, this gives a wrong-way map on Waldhausen's functor A(B) → A(E). We will ask a few fundamental questions about this transfer, and present the beginning of a program to answer these questions using trace methods. Our main results concern the corresponding transfer on THH, which in the A-theory case is a stable map of free loop spaces LB₊ → LE₊. If there is time, we will also describe how our techniques are related to the study of fixed points of dynamical systems.

Recorded Talk

Hesselholt has recently been advertising "topological periodic cyclic homology" (TP) as potentially filling some of the same roles for finite primes as periodic cyclic homology plays rationally. It is constructed from topological Hochschild homology (THH) analogously to the way periodic cyclic homology is constructed from Hochschild homology. In joint work with Andrew Blumberg, we prove a strong Kunneth theorem for the periodic topological cyclic homology of smooth and proper dg categories over a finite field k, namely, the derived smash product TP(X) ∧_TP(k) TP(Y) is weakly equivalent to TP(X ⊗_k Y).

Slides

Recorded Talk

One of the fundamental results of Thomason states that the algebraic K-theory of discrete commutative rings satisfies Galois descent after a type of localization. This allows one to calculate new K-groups from known simpler calculations. Based on this Ausoni and Rognes have conjectured that a similar descent result holds for structured ring spectra. During this talk I will discuss recent progress on this conjecture as well as a closely related red-shift conjecture. This is joint work with Clausen, Mathew, and Naumann.

Recorded Talk

Joint work with Amalendu Krishna.

The Bass-Quillen conjecture states that every vector bundle over Aⁿ_R is extended from Spec(R) for a regular noetherian ring R. Lindel proved that this conjecture has an affirmative solution when R is essentially of finite type over a field. We will discuss an equivariant version of this conjecture for the action of a diagonalizable group scheme. We prove a general result about G-equivariant vector bundles over affine G-toric schemes and derive the equivariant analogue of the Bass-Quillen conjecture as a corollary.

Recorded Talk

Topological Hochschild homology (THH) is the target of the trace map from algebraic K-theory. Similarly, iterated algebraic K-theory maps to an iterated version of THH, the so-called torus homology. In joint work with Halliwell, Hoening, Lindenstrauss and Zakharevich and further joint work with Lindenstrauss we develop juggling formulae for higher order THH, and more generally, relating X-homology of a commutative ring spectrum to its ΣX-homology for a simplicial set X. We apply these results in concrete examples, gaining for instance a relationship between higher order Hochschild and higher order Shukla homology.

Recorded Talk

The G-theory of a noetherian ring R is defined as Quillen's K-theory of the category of finitely generated R-modules. In this talk we will discuss the Hambleton-Taylor-Williams conjecture for the decomposition of G_n(ZG) as a direct sum of the G-theory of the factors of a maximal order with suitably chosen rational integers inverted in the factors, where G is any finite group. We will explain the connection between the inverted numbers and the Brauer-Nesbitt theorem on modular representations, and show that the solvable group SL(2, F₃) is a counter-example to the conjectured decomposition.

Recorded Talk

This talk is a report on joint work with Hélène Esnault. We show that if a morphism of smooth projective varieties in char. p induces the trivial map on étale fundamental groups, then the pullback of any stratified vector bundle is trivial, as a stratified bundle.

Recorded Talk

We extend Waldhausen's additivity theorem from algebraic K-theory to the setting of cobordism categories. As sample applications, we re-prove and generalize Genauer's fibration sequence for cobordism categories of manifolds with boundary, and the Bökstedt-Madsen delooping of the cobordism category.

I will present a new and direct proof of a result of Suslin saying that any Tor-unital ring satisfies excision in algebraic K-theory. In fact, I will prove a stronger and more general result which applies to connective ring spectra and implies excision for any localizing invariant. Examples are K-theory, (topological) Hochschild, or (topological) cyclic homology.

Recorded Talk

Joint work with Haesemeyer.

We give a K-theoretic criterion for a quasi-projective variety to be smooth, generalizing the proof of Vorst's conjecture for affine varieties. If L is a line bundle corresponding to an ample sheaf on X, it suffices that K_q(L) = K_q(X) for all q at most d+1, d the dimension of X. Our proof is in characteristic zero, using sheaf cohomology.

Recorded Talk

Building on previous work of Bartels, Lück, Reich and others studying the algebraic K-theory and L-theory of discrete group rings, the validity of the Farrell-Jones Conjecture has been recently extended to the setting of Waldhausen's algebraic K-theory of spaces in numerous cases. I will discuss these generalisations and how they contribute to our understanding of the topology of high-dimensional, closed, aspherical manifolds, in particular of their automorphism groups.

This talk covers joint work with Enkelmann, Kasprowski, Lück, Pieper, Ullmann and Wegner.

Recorded Talk