This conference took place from Wednesday 5 to Friday 7 July 2023 at Radboud University Nijmegen, The Netherlands.
A pdf-copy of the program of this conference and the companion workshop Motivic and non-commutative aspects of enumerative geometry is available here.
A youtube-playlist with all recorded talks is available here.5 July
This conference was organized around the themes of K-theory, (motivic) homotopy theory, topological Hochschild homology, trace methods, and related topics. It is dedicated to Bjørn Ian Dundas on the occasion of his 60th birthday.
Given a ring spectrum A and a class a in the homotopy groups of A, we define under suitable assumptions an extension ring spectra realizing the adjunction of a root of a. This provides examples of tamely ramified extensions of rings spectra, with spliting results in topological Hochschild homology and algebraic K-theory of the extension.
This is joint work with Haldun Özgür Bayındır and Tasos Moulinos.
The formalism of the motivic stable categories allows us to define motivic cohomology of arbitrary schemes: just pull back the motivic spectrum HZ constructed by Spitzweck over the integers to any scheme, and evaluate. In this talk I will explain how the above very abstract definition can be identified with a more concrete one, and also reconciled with other recent approaches to motivic cohomology of general schemes via syntomic cohomology. After that I will mention two applications: the resolution of Voevodsky’s zero-slice conjecture as well as the construction of the motivic filtration on homotopy K-theory, both over arbitrary base schemes. (jt. work with Elden Elmanto and Matthew Morrow)
Hesselholt and Madsen establish an isomorphism between pi_0 of the cyclic fixed-points of topological Hochschild homology and the ring of Witt vectors. Under this isomorphism, the effect of the trace map on the components of cyclic K-theory identifies with the characteristic polynomial. In this talk, I will discuss how the components of the dihedral fixed-points of real topological Hochschild homology exhibit an algebraic structure reminiscent of the Witt vectors and use this structure to refine the characteristic polynomial of a self-adjoint endomorphism.
Hochschild homology, and its analogue for ring spectra, topological Hochschild homology (THH), play an essential role in the trace method approach to algebraic K-theory. For ring spectra with an anti-involution there is a theory of Real topological Hochschild homology (THR), which is an O(2) -equivariant spectrum that receives a trace map from Real algebraic K-theory. In this talk, I will discuss how Real topological Hochschild homology can be characterized as an equivariant norm, and how this leads to a definition of an algebraic analogue of THR, Real Hochschild homology. This is joint work with Gabriel Angelini-Knoll and Mike Hill.
The theory of shadows, first introduced by Ponto, is an axiomatic, bicategorical framework that generalizes (topological) Hochschild homology and satisfies analogous important properties, such as Morita invariance and agreement. I’ll describe a general model category theoretic framework in which to define Hochschild homology of “ring objects” and to show that it is a shadow and explain how it applies to motivic spaces and spectra.
(based on work with K. Adamyk, T. Gerhardt, I. Klang, and H. Kong and with A. Beaudry, M. Kedziorek, M. Merling, and V. Stojanoska)
The equivariant slice filtration is really more of a recipe that takes a series of collection of representations for various subgroups and produces a tower of approximations in G-spectra. From this lens, the ordinary Postnikov tower and the slice and regular slice filtrations are seen as avatars of the same construction. I will talk about how to build a version of the slice filtration adapted to studying multiplicative questions more generally, building universal filtrations for various N_\infty operads. Time permitting, I will connect this to the motivic slice filtration and to motivic versions of the intermediaries.
A fundamental question in classical stable homotopy theory is to understand the stable homotopy groups of the spheres. A relatively new method is via the motivic approach. Motivic stable homotopy theory has an algebro-geometric root and closely connects to questions in number theory. Besides, it relates to the classical and the equivariant theories. The motivic category has good properties and allows different computational tools. I will talk about some of these properties and computations, and will show how it relates to the classical and equivariant categories.
This is joint work with Tom Bachmann, Guozhen Wang, and Zhouli Xu.
Torus localization has been a useful tool for computing characteristic classes in many settings. We will describe some background for “quadratic intersection theory”, which is an intersection theory that replaces the role of the integers, as coefficients, intersection multiplicities or degrees, with quadratic forms. In this context, we have developed quadratic versions of the Atiyah-Bott localization theorem, the Bott residue formula and Graber-Pandharipande localization of virtual fundamental classes; these have been extended in scope by Alessandro D’Angelo. We describe these results and applications to some counting problems, such as twisted cubics on hypersurfaces, and computations of some quadratic Donaldson-Thomas invariants. The work on twisted cubics is joint with Sabrina Pauli and computations of quadratic DT invariants are due to Anneloes Viergever.
Given a commutative ring or ring spectrum, there is a functor from finite sets to commutative rings sending a set to the tensor product of copies of the ring (or ring spectrum) indexed by that set. The Loday construction extends this to simplicial sets. The simplest nontrivial example is the Loday construction over a circle, which gives Hochschild homology (or, respectively, topological Hochschild homology).
I will discuss Loday constructions on higher spheres and on higher tori. The calculations for spheres are inductive, using the splitting of an n-sphere into two hemispheres, joined along the equator (n-1)-sphere. The calculations for tori which we know are for cases where the Loday construction on the torus is equivalent to the Loday construction on the wedge of spheres whose suspension is homotopy equivalent to the suspension of the torus. This kind of stability is unexpected, definitely not generally true, and yet there are surprisingly many cases where it holds.
(All joint work with Birgit Richter; some parts are also joint with Bjørn and with others)
We will explain how the motivic filtration on topoloical periodic homology (due to Bhatt-Morrow-Scholze and drastically generalized by Hahn-Raksit-Wilson) is constructed. The associated graded is given by prismatic cohomology. The main goal of the talk is to explain prismatic cohomology without assuming any preknowledge. Then we discuss a recent generalization, which is joint with Antieau and Krause. This generalization is one of the keys for the computation of K(Z/p^n). The other ingredient is the Dundas–McCarthy theorem. If time permits we will explain some conjectures about topological refinements of prismatic cohomology.
Brun showed that pi_0 of every genuine commutative G ring spectrum is a G-Tambara functor. We define a Loday construction for G-Tambara functors for any finite group G. This definition builds on the Hill-Hopkins notion of a G-symmetric monoidal category and the work of Mazur, Hill-Mazur and Hoyer who prove that for any finite group and any G-Tambara functor R there is a compatible definition of tensoring a finite G-set X with R. We extend this to a tensor product of a G-Tambara functor with a finite simplicial G-set, defining the Loday construction this way. We investigate some of its properties and describe it in examples.
In joint work in progress with Alexey Ananyevskiy, Elden Elmanto, and Maria Yakerson, the following version of the Adams conjecture is obtained: Given a vector bundle E over a smooth scheme X over a field F and an integer b invertible in F, there exists a natural number N such that the Thom spectrum of the b^N-fold direct sum of E is equivalent to the Thom spectrum of the b^N-fold direct sum of the b-th Adams operation of E. This equivalence exists in the motivic stable homotopy category SH(X).