
Conference on Motivic and Equivariant Topology Mathematics Department, Swansea University, 1517 May 2023 
All talks will be in Computational Foundry Building Lecture Theatre 002
Isotropic and numerical equivalence for cycles and Morava Ktheories and Balmer spectrum of Voevodsky and MorelVoevodsky categories (video) Isotropic realisations assign to a Voevodsky motive its "local" versions residing in much handier categories and parametrized by a prime p and a choice of an equivalence class of a (finitely generated) field extension over k. I will discuss the recent proof of the Conjecture claiming that the category of isotropic Chow motives is equivalent to the category of numerical Chow motives with F_pcoefficients. This implies that isotropic realisations provide points of the Balmer spectrum of the Voevodsky category complementing the classical topological points. It also shows that the "size" of the isotropic category is similar to that of its topological counterpart. Moreover, a generalisation of the mentioned Conjecture is proven: the category of isotropic ChowMorava motives is equivalent to the category of numerical ChowMorava motives. This permits to construct plenty of new "isotropic" points of the Balmer spectrum of the MorelVoevodsky category SH(k). Such a point is parametrised by a prime p, a number n between 1 and infinity, and a choice of a K(p,n)equivalence class of finitely generated extensions of k. The latter construction is a joint work with Peng Du. Heights for equivariant formal groups (video) The stratification by height of formal groups gives rise to the chromatic filtration on the stable homotopy category. In my talk I will discuss joint work with Lennart Meier on the notion of height for Aequivariant formal groups for an abelian compact Lie group A, and how this notion relates to the Balmer spectrum of finite Aspectra. Real projective groups are formal (video) A differential graded algebra is called formal if there is a zigzag of quasiisomorphisms to its cohomology algebra. Formality is a rather strong property and implies, for example, the vanishing of all Massey products. After showing that all triple Massey products of degree one classes in the mod 2 Galois cohomology of global fields of characteristic different from 2 vanish whenever they are defined, Hopkins and Wickelgren asked whether the mod 2 cohomology algebra of such fields is formal. Minac and Tan then formulated the Massey vanishing conjecture that Massey products of degree one classes in Galois cohomology of any field should vanish whenever they are defined. In this talk I will report on joint work with Ambrus Pal in which we prove that the mod 2 cohomology algebras of real projective groups are formal. This implies the HopkinsWickelgren formality and the MinacTan Massey vanishing conjecture for all fields of virtual cohomological dimension 1. Equivariant spectra in the profinite setting (video) Profinite groups occur in many places, as the limit of diagrams of finite groups, as Galois groups or as Etale fundamental groups. They are compact Hausdorff topological groups that can be thought as a generalisation of a finite group. The aim of this talk is to introduce the homotopy theory of (rational) Gspectra, for G a profinite group G and demonstrate its rich structure. We take three viewpoints. Firstly, that Gspectra are the limit of G/Nspectra as N runs over the open normal subgroups of G (at the level of homotopy or infinity categories). Secondly, that there is an algebraic model for the homotopy theory in terms of sheaves over the space of closed subgroups of G. Thirdly, that the tensortriangulated geometry of rational Gspectra has a wellbehaved support theory, at least when the space of subgroups of G is scattered in the sense of CantorBendixson. From motivic homotopy theory to birational geometry: the study of ChowWitt groups (video) In this talk, I will start with some general background in motivic homotopy theory, and study in particular the heart of the stable motivic homotopy category of Morel and Voevodsky. Then I will present some consequences of the theory of MilnorWitt cycle modules and their associated ChowWitt groups in birational geometry. In the end, I will discuss a motivic formulation concerning the existence of rational points in a given variety and its link to the existence of 0cycles of degree 1. Correspondences and stable homotopy theory (video) A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra SH is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory SH(k) from spectral modules over associated spectral categories.
Morava Ktheory of infinite groups and Euler characteristic (video) Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava Ktheory of G and associated Euler characteristic, and give a character formula for the LubinTate theory of G. This will generalise the results of HopkinsKuhnRavenel from finite to infinite groups and the Ktheoretic results of Adem, Lueck and Oliver from chromatic level one to higher chromatic levels. Along the way we will give explicit computations for amalgamated products of finite groups, right angled Coxeter groups and certain special linear groups. We will also discuss connections with special values of zeta functions. This talk is mostly based on a joint work with Lueck and Schwede. On the relation of MilnorWitt Ktheory and Hermitian Ktheory (video) Let R be a commutative local ring with 2 a unit. There is a canonical multiplicative homomorphism K_n^{WM}(R) > GW^n_n(R) from MilnorWitt Ktheory of R to its Hermitian Ktheory. I will explain recent improvements on homology stability of symplectic groups over local rings with infinite residue fields and use this to construct a map GW^n_n(R) > K_n^{MW}_n(R) when n=2,3 mod 4. Finally, we compute the composition K_n^{WM}(R) > GW^n_n(R) > K_n^{MW}_n(R). Using A1homotopy theory, Asok, Fasel and Deglise have obtained similar results when R is in addition regular. Characteristic polynomials of selfadjoint endomorphisms: A tale of Witt vectors (video) I will discuss several variations of the ring of Witt vectors, defined from real topological Hochschild homology. We will explore their algebraic structure and use it to refine the characteristic polynomial of a selfadjoint endomorphism, in particular constructing invariants for symmetric forms. Grothendieck classes of quadric hypersurfaces and involution varieties (video) The Grothendieck ring of varieties, introduced in a letter from Alexander Grothendieck to JeanPierre Serre (August 16th 1964), plays an important role in algebraic geometry. However, despite the efforts of several mathematicians, the structure of this ring still remains poorly understood. In this talk, in order to better understand the Grothendieck ring of varieties, I will describe some new structural properties of the Grothendieck classes of quadric hypersurfaces and involution varieties. More specifically, by combining the recent theory of noncommutative motives with the classical theory of motives, I will show that if two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class, then they have the same even Clifford algebra and the same signature. As an application, this implies in numerous cases (e.g., when the base field is a local or global field) that two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class if and only if they are isomorphic. Model categories and invariants of second kind (video) Koszul duality is a phenomenon occurring widely in mathematics, classical examples being the BernsteinGelfandGelfand correspondence in representation theory or the correspondence between differential graded (dg) Lie algebras and formal moduli problems in deformation theory. For dg algebras, its modern formulation is as a Quillen equivalence between certain model categories of dg algebras and conilpotent dg coalgebras, and their corresponding dg modules and comodules. In particular, the model structures on the coalgebra/comodule side are required to be "of second kind", that is, weak equivalences are finer than quasiisomorphisms. We will survey these ideas and then show how further generalising Koszul duality leads to model structures of second kind on the algebra/module side too, and then speak about resulting invariants such as Hochschild cohomology of second kind.
Integral representations of general linear groups and etale abelian motives (video) After reviewing the envisioned Tannakian viewpoint on mixed motives and giving various definitions of motivic categories I will explain how one can model a subcategory of the category of etale motives generated by a 1motive by modules over a commutative algebra in a stable category of representations of a general linear group over the integers. This builds upon rational results by Iwanari. ChowWitt groups of motivic classifying spaces (video) The ChowWitt groups of a scheme are a quadratic refinement of the more wellknown Chow groups and play an important role in intersection theory. For any linear algebraic group G, one can define the ChowWitt ring of its motivic classifying space, which is the ring of characteristic classes for principal Gbundles over smooth varieties with values in the ChowWitt ring. In this talk we will focus on a geometric model for motivic classifying spaces called an admissible gadget. This is a class of algebraic varieties approximating BG in the sense that their ChowWitt rings coincide in a suitable range of dimensions. We will then discuss some handson computations for some classical algebraic groups and finite groups using this machinery. Rational enriched motivic spaces In this talk a model for connective and very effective motivic spectra with rational coefficients will be given. It uses methods of homological algebra of enriched Grothendieck categories associated with Nisnevich sheaves with specific transfers as well as motivic Gammaspaces introduced and studied by Garkusha, Panin and Ostvaer.  