Motivic homotopy theory 2025

 Conference: Motivic homotopy theory (March 17-21, 2025)

List of Speakers

• Alexey Ananyevskiy (LMU München)
• Toni Annala (IAS)
• Tom Bachmann (Mainz)
• Federico Binda (Milan)
• Tess Bouis (Regensburg)
• Elden Elmanto (Toronto)
• Jean Fasel (Grenoble Alpes)
• Shane Kelly (Tokyo)
• Josefien Kuijper (Utrecht)
• Jinhyun Park (KAIST)
• Sabrina Pauli (TU Darmstadt)
• Oliver Röndigs (Osnabrück)
• Fabio Tanania (TU Darmstadt)
• Longke Tang (Princeton)
• Alexander Vishik (Nottingham)
• Vladimir Sosnilo (Regensburg)
• Maria Yakerson (CNRS)
  

Program and Schedule

Monday March 17	Tuesday March 18	Wednesday March 19	Thursday March 20	Friday March 21
9:30 - 10:30 Alexey Ananyevskiy	9:30 - 10:30 Shane Kelly	9:00 - 10:00 Jean Fasel	9:30 - 10:30 Josefien Kuijper	9:30 - 10:30 Tom Bachmann
Coffee break & Registration	Coffee break	Coffee break	Coffee break	Coffee break
11:00 - 12:00 Fabio Tanania	11:00 - 12:00 Jinhyun Park	10:30 - 11:30 Longke Tang	11:00 - 12:00 Federico Binda	11:00 - 12:00 Alexander Vishik
Lunch	Lunch	11:45 - 12:45 Toni Annala	Lunch	Lunch
14:00 - 15:00 Sabrina Pauli	14:00 - 15:00 Tess Bouis	Free afternoon	14:00 - 15:00 Vladimir Sosnilo
Coffee break	Coffee break		Coffee break
15:30 - 16:30 Maria Yakerson	15:30 - 16:30 Elden Elmanto		15:30 - 16:30 Oliver Röndigs
			19:00 Conference dinner

Alexey Ananyevskiy: Nowhere vanishing sections of vector bundles revisited

Consider a vector bundle over a smooth affine variety and suppose that the rank of the bundle equals the dimension of the variety. It is a classical result of Murthy that if the base field is algebraically closed, then triviality of the top Chern class in Chow group yields the existence of a nowhere vanishing section. Morel generalized this to arbitrary fields, enhancing top Chern classes to Euler classes valued in Chow-Witt groups and showing that these are universal obstruction classes. Asok and Fasel further developed this approach, in particular, they showed that Murthy's result holds also over the fields of 2-cohomological dimension at most one. In a complementary direction, Bhatwadekar and Sridharan, and Bhatwadekar, Das and Mandal using some delicate commutative algebra showed that Murthy's result holds over the field of real numbers provided that either the rank of the bundle is odd, or the manifold of real points X(R) satisfies some additional assumption. In the talk I will discuss how one can generalize these results to fields of virtual 2-cohomological dimension at most one.

Toni Annala: Atiyah Duality and Logarithmic Cohomology Theories

In recent work with Marc Hoyois and Ryomei Iwasa, we establish a non-𝐀¹-invariant refinement of Atiyah duality, identifying the dual of a smooth projective scheme with the Thom space of its negative cotangent bundle. Interestingly, this result provides a new conceptual framework for logarithmic cohomology theories, that is complementary to that of logarithmic motives of Binda–Park–Østvær, and in particular allowed us to show that the log-crystalline cohomology of a projective SNCD pair (X,D) is an invariant of the open complement U = X - D. More generally, we expect analogous invariance results for all cohomology theories that admit a logarithmic refinement. Additionally, there is a weight filtration on these cohomology groups in broad generality, which likewise depends only on U.

This talk is based on joint work with Hoyois–Iwasa and Pstrągowski, and relies crucially on forthcoming results of Longke Tang.

Tom Bachmann: Motivic stable stems

I will report on joint work with Robert Burklund and Zhouli Xu, where we determine the bigraded homotopy groups of the p-adically completed motivic sphere spectrum, over "most" fields of characteristic not p. I will begin by explaining an easy special case, namely fields containing an algebraically closed field, in which case our results are straightforward consequences of the motivic Adams-Novikov spectral sequence. I will then explain how we relaxed this assumption (containing an algebraically closed field) by touching on a dizzying amount of classical homotopy theory -- including equivariant orientations, the real Adams conjecture, a question of Sullivan, and more.

Federico Binda: Infinite root stacks and the Beilinson fiber square

Following a suggestion by Mathew, we will explain how to identify some arithmetic invariants of logarithmic schemes using the infinite root stack machinery developed by Talpo and Vistoli. A key ingredient is a form of flat descent for logarithmic invariants that we call "saturated descent". Using this, we can identify the homotopy-theoretic version of logarithmic prismatic and syntomic cohomology with the site-theoretic version introduced by Koshikawa and Koshikawa-Yao. As a sample application, we will explain how to get a log variant of the (graded version of the) Beilinson fiber square of Antieau-Mathew-Morrow-Nikolaus. (Joint work with T. Lundemo, A. Merici, and D. Park).

Tess Bouis: From p-adic Hodge theory to motivic cohomology and back

Abstract: The initial goal of p-adic Hodge theory, as formulated by the foundational conjectures of Fontaine in the 1980s, is to compare the different p-adic cohomology theories one can associate to schemes of mixed characteristic (0,p). If Fontaine's conjectures have now been solved by the work of many people, the recent development of prismatic cohomology has shed new light on integral aspects of this theory. In this talk, I want to explain how one can use these recent advances in p-adic Hodge theory to construct a new theory of motivic cohomology for general (qcqs) schemes. This theory generalises the recent construction of Elmanto-Morrow over a field to mixed characteristic, and allows us to give a simplified motivic approach to certain classical results in p-adic Hodge theory. This is part of joint works in progress with Quentin Gazda and Arnab Kundu.

Elden Elmanto: Symmetric bilinear forms and mod-2 syntomic cohomology

The first successful deployment of motivic homotopy theory was Voevodsky and Orlov-Visihik-Voevodsky's resolution of Milnor's conjecture relating the graded pieces of the Witt ring, mod-2 Milnor K-theory and mod-2 étale cohomology for fields of odd characteristics. Earlier, Kato had discovered the same story for characteristic two fields, with logarithmic de Rham forms in place of étale cohomology. I will report on joint work in progress with Arnab Kundu on this story with mixed characteristics, crucially in the (2,0) case. I will explain the relationship between the graded pieces of a filtration on symmetric L-theory (in the sense of the 9 authors) of a 2-henselian local ring, with mod-2 syntomic cohomology (in the sense of Bhatt-Morrow-Scholze). Time permitting, I will explain how this relates to an extension of Milnor-Witt motivic cohomology to qcqs schemes following work of myself with Morrow and Bouis.

Jean Fasel: The quadratic Riemann-Roch theorem

In this talk, I will explain the quadratic Riemann-Roch theorem, in which the Chern character linking K-theory and rational Chow groups is replaced by the Borel character linking Hermitian K-theory (aka higher Grothendieck-Witt groups) with rational Chow-Witt groups. I will also explain how to compute the relevant Todd classes, in link with the formal ternary laws.

Shane Kelly: Procdh topologies

We will discuss procdh topologies in the classical setting, and the formal schemes setting (à la EGA), in particular, focussing on applications to algebraic K-theory and making remarks on cohomological dimension. If there is time we may make some comments about the derived schemes setting. This is mostly joint work with Shuji Saito.

Josefien Kuijper: Spherical scissors congruence as spectral Hopf algebra

The Dehn invariant is known to many as the satisfying solution to Hilbert’s 3rd problem: a polyhedron P can be cut into pieces and reassembled into a polyhedron Q if and only if Q and P have not only the same volume, but also the same Dehn invariant. Generalized versions of Hilbert’s 3rd problem concern the so-called scissors congruence groups of euclidean, hyperbolic and spherical geometry in varying dimensions, and in these contexts one can define a generalized Dehn invariant. In the spherical case, Sah showed that the Dehn invariant makes the scissors congruence groups into a graded Hopf algebra. Zakharevich has shown that one can lift the scissors congruence group to a K-theory spectrum. In this talk I will discuss the spectral Dehn invariant for spherical geometry, and we will see how it gives rise to a spectral version of Sah’s Hopf algebra. This talk is based on joint work in progress with Inbar Klang, Cary Malkiewich, David Mehrle and Thor Wittich.

Jinhyun Park: On the relative Milnor K-theory of some Artin local algebras over a field of any characteristic

When A is a local ring with an ideal I, the n-th relative Milnor K-theory of (A,I) has a good set of generators described by Kato-Saito.

For a k-algebra A with a nilpotent ideal I over a field k with char(k) = 0, using the above, we can describe a map called the Bloch map (some call it even Bloch-Artin-Hasse map) from the relative Milnor K-group to a group originating from the absolute Kähler differentials, which is often an isomorphism as observed by several people.

For char(k) = p > 0, under the mild assumption “weakly 5-stable” (due to M. Morrow improving an older notion of W. van der Kallen), together with a rather strong assumption “N! Is invertible in k” where I^N = 0, a similar result was obtained, e.g. by Gorchinskiy-Tyurin. However, if we want to consider the pro-system over N ≥ 1, this result might be useful effectively when char(k) = 0.

In this talk, in a response to a question posed by M. Morrow years ago, I would like to talk about a new approach to study the relative Milnor K-theory from the geometric perspective of algebraic cycles, and discuss how we may improve the above known results to a field of any characteristic, without imposing an exorbitant invertibility assumption.

Sabrina Pauli: A Tropical Correspondence Theorem for Quadratic Plane Curve Counts

The number of rational plane curves of a given degree satisfying point conditions is a classical enumerative invariant. In the real setting, restoring invariance requires a signed count, while more generally, curves should be counted with a weight in the Grothendieck-Witt ring of quadratic forms.

To compute such counts, one can translate the problem into tropical geometry, where it reduces to a weighted count of certain graphs with point conditions. This significantly simplifies the computation. In this talk, I will present a tropical correspondence theorem for quadratic plane curve counts.

Oliver Röndigs: Low integral Milnor-Witt stems over the integers

Work of Calmes, Harpaz, and Nardin, based on their collaboration with Dotto, Hebestreit, Land, Moi, Nikolaus, and Steimle, provides a reasonable motivic ring spectrum KQ representing hermitian K-theory in the motivic stable homotopy category of S. Here S can be any regular Noetherian base scheme of finite Krull dimension; the inconvenient restriction that 2 be invertible on S is not required anymore. Recent joint work with K. Arun Kumar shows that this motivic spectrum coincides with the one constructed in Kumar's thesis, and in particular is cellular. One consequence, derived in joint work with Håkon Kolderup and Paul Arne Østvær, is a determination of various filtrations on hermitian K-theory over Dedekind rings. As another consequence, effectivity and connectivity properties of the unit map from the motivic sphere spectrum to KQ induce information on motivic stable homotopy groups of spheres over the integers.

Fabio Tanania: Real isotropic cellular spectra

In this talk, I will introduce the category of isotropic motivic spectra over the real numbers. In brief, this is obtained from SH(R) by annihilating anisotropic quadrics. I will then discuss in more detail the structure of the subcategory of isotropic cellular spectra. This can be described as a one-parameter deformation, whose generic fiber is the classical stable homotopy category, while the special fiber is the derived category of comodules over the topological dual Steenrod algebra. This reveals a striking similarity between real isotropic cellular spectra and F2-synthetic spectra.

Longke Tang: The 𝐏¹-motivic Gysin map

Recently, Annala, Hoyois, and Iwasa have defined and studied the 𝐏¹-homotopy theory, a generalization of 𝐀¹-homotopy theory that does not require 𝐀¹ to be contractible, but only requires pointed 𝐏¹ to be invertible. This makes it applicable to non-𝐀¹-invariant cohomology theories such as Hodge, de Rham, and prismatic. I will recall basic facts in their theory, and construct the 𝐏¹-motivic Gysin map, thus giving a uniform construction for the Gysin maps of the above cohomology theories that are automatically functorial. If time permits, I will also explain and possibly prove basic properties of the Gysin map.

Alexander Vishik: Balmer spectrum of Voevodsky motives and pure symbols

The difference in complexity between algebraic geometry and topology is reflected in the structure of the Balmer spectra of the respective “motivic” categories. While the Balmer spectrum of the topological version is just the Zariski spectrum of integers, its algebro-geometric counterpart is much wilder. The vast majority of known points there are given by isotropic realisations. These are parametrised by a choice of a prime p and a p-equivalence class of field extensions of the base field. Points of characteristic 2 are understood better, since, in this case, it is known that the isotropy of algebraic varieties is controlled by pure symbols over flexible closure (so, every variety is a “norm-variety”, in a sense). This permits to impose certain “coordinates” on the Balmer spectrum coming from pure symbols. In particular, it shows that isotropic points are closed.

Kirsten Wickelgren: Equivariantly enriched Gromov–Witten invariants

Abstract: We compute the equivariant Euler characteristic as an Euler number and give several related results on degrees in equivariant homotopy theory. As an application, we consider certain enriched Gromov–Witten invariants. For example, we compute an equivariantly enriched count of rational plane cubics through a G-invariant set of 8 general points in CP^2. When Z/2 acts by pointwise complex conjugation, this recovers a signed count of real rational cubics. This is joint work with Candace Bethea.

Maria Yakerson: Fun facts about p-perfection

In this talk we will discuss the structure of E-infinity-monoids on which a prime p acts invertibly, which we call p-perfect. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the p-perfection functor, and describe it in terms of Quillen's +-construction, similarly to group completion. This is joint work with Maxime Ramzi.

Practical Information

City of Regensburg: Regensburg is a Unesco World Heritage site that is famous for its well-preserved medieval city center and its beautiful Gothic cathedral.
Further information about Regensburg can be found here.

Internet: Access to eduroam and BayernWLAN is available throughout the Mathematics Building.

Accomodation: Regensburg is a tourist destination and we encourage guests to book their rooms as early as possible. Here is a list of hotels in the area:

- Hotel Münchner Hof (In the city center, one must take a bus to the University.)

- Hotel Kaiserhof am Dom (In the city center, one must take a bus to the University.)

- Hotel Jakob (In the city center, one must take a bus to the University.)

- Hotel Wiendl (Between the city center and the University.)

- Hotel Central (Between the city center and the University, one must take a bus to the University.)

Family Friendly Campus: Our UR family service offers various rooms for families and services. If you need further information look here.

Venue

All lectures are in the lecture hall H51 at the University of Regensburg University. This is not in the math building, but the easiest way to reach it is to enter the math building and walk south for about 200m (see the campus plan). There is currently a large yellow crane next to the math building due to some construction on the roof.

One can reach the University of Regensburg by following the instructions here. Many of the participants will probably arrive by bus at the Central Bus Station of the university which is close to the math building (see the campus plan above).

Conference Dinner

A conference dinner will take place Thursday at 19:00 at the restaurant Brauhaus am Schloss in the town centre.

There will be a registration sheet for the dinner on Monday, where you can also choose from three menu options. For non-speakers, we ask for a contribution of 35 euros.

Conference Poster

You can download the conference poster here.

Organizers

Denis-Charles Cisinski
Marc Hoyois

Conference Picture

Sponsors of the conference

This conference is funded by SFB 1085 "Higher Invariants"