This workshop took place from Monday 3 to Wednesday 5 July 2023 at Radboud University Nijmegen, The Netherlands.

A pdf-copy of the program of this workshop and the companion conference **Homotopy theory, K-theory, and trace methods** is available here.

A youtube-playlist with all recorded talks is available here.

- 09:00 - 09:30: Registration
- 09:30 - 10:30: Hoyois (video)
- 10:30 - 11:00: Coffee break
- 11:00 - 12:00: Feld (video)
- 12:00 - 13:45: Lunch break
- 13:45 - 14:45: Harpaz
- 14:45 - 15:15: Coffee break
- 15:15 - 16:15: Pauli (video)
- 16:30 - 17:30: Azouri (video)

- 09:30 - 10:30: Robalo (video)
- 10:30 - 11:00: Coffee break
- 11:00 - 12:00: Ravi (video)
- 12:00 - 13:45: Lunch break
- 13:45 - 14:45: Moulinos (video)
- 14:45 - 15:15: Coffee break
- 15:15 - 16:15: Jin (video)
- 16:30 - 17:30: Gongshow

This workshop was organized around the themes of using refined invariants of algebraic varieties to study enumerative questions, with ideas coming from motivic homotopy theory, quadratic enumerative geometry, hermitian K-theory and beyond.

In the second half of the same week, Wednesday 5 to Wednesday 7 July 2023, there was a conference on

in honor of Bjørn Ian Dundas' 60th birthday at the same location. The two events had one shared day (Wednesday) with talks that fitted both topics.- Ran Azouri (Paris):
*Computing motivic nearby cycles and their mondromy*

**Abstract:**A major result in Ayoub’s thesis is on the motivic nearby cycles functor of semi-stable schemes; the value of the functor at the unit, on a stratum of the special fibre, is determined by its smooth locus. In the talk I will explain how to make motivic nearby cycles computable by reducing to the case of the theorem. One result is a formula for the quadratic Euler characteristic of nearby cycles around a semi-homogeneous singularity. This allows to extend the quadratic conductor formula for projective hypersurfaces of Levine, Pepin Lehalleur and Srinivas, to more general schemes. In another work in progress, joint with Emil Jacobsen, we compute the motivic monodromy operator around a singularity. By adapting Illusie’s approach to the setting of Voevodsky motives, we obtain a motivic version for the Picard-Lefschetz formula of Deligne and Katz.

- Tom Bachmann (Mainz):
*cdh-local motivic cohomology for singular schemes*

**Abstract:**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)

- Niels Feld (Toulouse):
*From motivic homotopy theory to birational geometry*

**Abstract:**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 Milnor-Witt cycle modules and their associated Chow-Witt 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 0-cycles of degree 1.

- Yonatan Harpaz (Paris):
*A categorical approach to the quadratic conductor formula*

**Abstract:**For a degeneration of a family of varieties, Bloch conjectured a formula describing the Artin conductor in terms of a certain localized intersection number. In the work of Toën and Vezzosi, followed by complementary work of Beraldo and Pippi, this localized intersection number was reinterpreted as the trace of the associated singularity category, leading to a proof of Bloch’s conductor formula in the case of unipotent monodromy. In equal characteristic zero, the Artin conductor amounts to the difference in Euler characteristics between the special and generic fibre. In this setting, quadratic refinements of Bloch’s conductor formula were recently considered by Levine, Pepin-Lehalleur and Srinivas, where the Euler characteristic is replaced with the motivic Euler characteristic. In this talk I will describe work in progress with Tasos Moulinos and Ran Azouri exploring a categorical approach in the spirit of Toën and Vezzosi to the motivic conductor formula, where one considers the trace of the singularity category not as a stable category, but as a Poincaré category.

- Marc Hoyois (Regensburg):
*Non-A^1-invariant motivic spectra*

**Abstract:**Recently, there have been several attempts at developing extensions of motivic homotopy theory that include non-A^1-invariant phenomena, such as the algebraic K-theory of singular schemes or de Rham cohomology in positive characteristic. These are usually based on extensions of the category of schemes itself, such as schemes with modulus or log structures. In joint work with Toni Annala and Ryomei Iwasa, we consider a very naive extension of stable motivic homotopy theory, in which we simply remove the A^1-invariance axiom. We show that a surprising number of basic results in A^1-homotopy theory can be proved in this context, using the invertibility of P^1 in a clever way. We construct in particular a non-A^1-invariant refinement of algebraic cobordism, which is related to algebraic K-theory by a Conner-Floyd isomorphism.

- Fangzhou Jin (Shanghai):
*The quadratic Artin conductor of a motivic spectrum*

**Abstract:**We define the quadratic Artin conductor of a motivic spectrum over a smooth scheme under some assumptions, and use it to prove a quadratic refinement of the Grothendieck-Ogg-Shafarevich formula. This is a joint work with Enlin Yang.

- Hana Jia Kong (Princeton):
*Structures and computations in the motivic stable homotopy categories*

**Abstract:**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.

- Marc Levine (Essen):
*Bott residue theorems in quadratic intersection theory, with applications*

**Abstract:**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.

- Tasos Moulinos (Paris):
*Betti realizations of noncommutative motives*

**Abstract:**In this talk, I will survey some past work of mine on the topological K-theory of dg categories. This is an invariant of complex dg-categories, taking values in the infinity category of KU-module spectra. I will begin the talk with some motivation coming by way of Hodge theoretic mirror symmetry, and will proceed to describe the basic construction, originally due to A. Blanc. I will then describe a variant of this construction from previous work, relative to any base complex scheme, together with applications of such a construction towards computations in twisted K-theory, and towards the theory of variations of Hodge structures. Time permitting, I will describe ongoing work, incorporating Hermitian structures and real topological K-theory into the picture.

- Thomas Nikolaus (Münster):
*(Generalized) Prismatic cohomology and the motivic filtration*

**Abstract:**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.

- Sabrina Pauli (Essen):
*Counting curves over an arbitrary field using tropical geometry*

**Abstract:**Mikhalkin’s celebrated correspondence theorem establishes a correspondence between algebraic curves and tropical curves. Consequently, the count of algebraic curves equals a weighted count of tropical curves. There are several versions of this correspondence including one for the count of complex curves and one for the real count. In the talk, I will present work in progress joint with Andrés Jaramillo Puentes in which we provide a version Mikhalkin’s correspondence theorem over an arbitrary base field k where the tropical weights live in the Grothendieck-Witt ring of k. I will also give a short introduction to counting curves over different fields and tropical geometry.

- Charanya Ravi (Bangalore):
*Equivariant localization theorem*

**Abstract:**The Atiyah-Bott localization theorem says that the equivariant cohomology of a space can be recovered, up to inverting some elements, from the equivariant cohomology of the fixed point subspace. We discuss motivic and categorified versions of this result which allows us to deduce the theorem for all oriented theories (cohomology and Borel-Moore homology). This is based on a recent joint work with Adeel Khan.

- Marco Robalo (Paris):
*Gluing Invariants of Donaldson-Thomas Type*

**Abstract:**We will explain how categorified Donaldson–Thomas invariants of Calabi–Yau 3-folds can be obtained by gluing singularity invariants from local models of a suitable moduli space endowed with a (-1)-shifted symplectic structure. We will show how to recover Brav–Bussi–Dupont–Joyce–Szendroi’s perverse sheaf categorifying the DT-invariants, as well as how to glue more evolved singularity invariants, such as matrix factorizations (thus answering a conjecture of Kontsevich and Soibelman).

This is joint work with B. Hennion and J. Holstein.

- Oliver Röndigs (Osnabrück):
*A motivic version of the Adams conjecture*

**Abstract:**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).

- Simon Pepin Lehalleur (Amsterdam)
- Steffen Sagave (Nijmegen)
- Lenny Taelman (Amsterdam)