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Conference Higher Invariants: interactions between arithmetic geometry and global analysis, October 6.-10., 2025 organized by Ulrich Bunke, Denis-Charles Cisinski and Guido Kings
List of Speakers
- Federico Binda
- Shachar Carmeli
- Dustin Clausen
- Hélène Esnault
- Fabian Hebestreit
- Hokuto Konno
- Manuel Krannich
- Akhil Mathew
- Thomas Nikolaus
- Viktoriya Ozornova
- Maxime Ramzi
- Charanya Ravi
- Peter Scholze
- Georg Tamme
- Maria Yakerson
- Inna Zakharevich
Abstracts
Federico Binda - Relative Hyodo-Kato cohomology (joint work in progress with F. Andreatta and A. Vezzani) We introduce a definition of relative Hyodo-Kato cohomology for rigid analytic varieties, based on the relative motivic nearby cycle functor, and a corresponding theory of coefficients. We prove a Hyodo-Katoisomorphism with de Rham cohomology, provide an explicit description of the "constructible" objects, and compare this definition to relative log-rigid cohomology.
Shachar Carmeli - Steenrod Operations in Syntomic Cohomology and Duality for Brauer Groups in Characteristic 2 The Brauer group of a surface over a finite field of characteristic p carries a canonical skew-symmetric pairing, constructed by Artin, Tate, and Milne. Tate conjectured that this pairing is in fact alternating. I will discuss a joint work with Tony Feng addressing this conjecture for surfaces in characteristic 2. The case of odd primes was previously settled by Feng using Steenrod operations on mod 2 étale cohomology. Our extension of Feng's approach to the characteristic 2 case relies on the construction and analysis of Steenrod operations in syntomic cohomology, building on Voevodsky's motivic Steenrod algebra and Lurie's theory of spectral prismatization.
Dustin Clausen - Refining Weil groups In number theory, the absolute Galois group of Q gets refined to the Weil group of Q, a locally compact group whose profinite completion recovers the Galois group. I will explain that this process can and should be continued, by carefully adding higher homotopy groups to get an object (formally, a condensed anima) which, from various perspectives, is better-behaved than both the Weil group and the absolute Galois group.
Hélène Esnault - On the restriction map in p-adically complete de Rham or prismatic cohomology (joint work with Mark Kisin and Alexander Petrov): For $X$ smooth proper over ${\mathbb Z}_p$ and $U $ a smooth affine, we show that the restriction homomorphism $H_{prism}^i(X) \to H^i_{prism}(U)^{sep}$ in the separated quotient of prismatic cohomology $H^i_{prism}(U)$ dies if $H^0(X mod p, \Omega^i)=0$, likewise for $p$-adically complete de Rham cohomology: under the same assumption the restriction homomorphism $H^i_{dR}^i( \hat X) \to H^i_{dR( \hat U)^{sep}$ in the separated quotient of $p$-adically complete de Rham cohomology $H^i_{prism}(U)$ dies. Daniel Caro and Marco D’Addezio lifted the vanishing of $H^i_{dR}^i( \hat X) \to H^i_{dR}( \hat U)^{sep}$ to the vanishing of $H^i_{dR}^i( \hat X) \to H^i_{dR}( \hat U)$ under the extra assumption $H^1(X, \Omega^{i-1})=0$. Indeed, their theorem can in turn be lifted to prismatic cohomology, so far modulo torsion: under the extra assumption $H^1(X, \Omega^{i-1})=0$, the restriction$H _{prism}^i(X) \to H^i_{prism}(U)_{m}$ dies, where $m$ is the maximal ideal of the prism. On the other hand. the algebra structure on prismatic, resp. $p$-adically complete de Rham cohomology kills the square of the non-separated part. On the de Rham side this can be deduced from a more precise formulation of the theorem ofCaro-D’Addezio mentioned above.
Fabian Hebestreit - On the Weiss-Williams index
(based on work in progress with A.Bianchi, K.Hilman, D.Kirstein, C.Kremer, M.Land, T.Nikolaus, W.Steimle):
I will explain a construction of the Weiss-Williams index classes for manifold bundles in the language of Poincaré categories. The particular view this affords allows one to coalesce these classes into a topological field theory with values in cobordism categories of self-dual parametrised spectra. The underlying homotopy types of these categories are Weiss-Williams' LA-spectra, originally custom made as the home for their indices. This observation on the one hand refines and unifies their work with separate constructions of Bökstedt and Madsen (with target Waldhausen's A-spectra) and Basterra, Bobkova, Ponto, Tillmann and Yaekel (with target hermitian K-spectra of the integers) and on the other provides a simple proof of the Weiss-Williams index theorem and a fresh perspective on their map connecting block homeomorphisms and Whitehead spectra.
Manuel Krannich - Pontryagin–Weiss classes Pontryagin classes were originally considered as invariants of real vector bundles, but it was realised in the 60s that they can be defined more generally for Euclidean bundles, that is, fibre bundles whose fibres are homeomorphic to Euclidean space. Thisled to the question whether the well-known vanishing of large-degree Pontryagin classes for small-dimensional vector bundles continues to hold in the setting of Euclidean bundles. Surprisingly, Weiss proved a few years ago that this often fails. I will explaina strengthening of his result resulting from joint work with A. Kupers: For every k>0, there exists a 6-dimensional Euclidean fibre bundle over a sphere whose kth Pontryagin class is nontrivial.
Hokuto Konno - Family gauge theory and diffeomorphism groups: One of the major advances in topology over the past decade has concerned diffeomorphism groups of higher-dimensional manifolds. On the other hand, it has long been known that dimension 4 is special in the classification of manifolds. Such special phenomena are detected using gauge theory. Recent advances in gauge theory for families have revealed that similar phenomena in dimension 4 also appear in diffeomorphism groups, in striking contrast with results in higher dimensions. In this talk, I will survey these new phenomena in dimension 4.
Akhil Mathew - Sheared Witt vectors (after V. Drinfeld, E. Lau, and T. Zink): Motivated by Dieudonné theory, V. Drinfeld and E. Lau introduced a "decompletion" of the ring of Witt vectors W(R) of a derived p-complete ring R such that (R/p)_{red} is perfect, extending a construction of T. Zink. I will explain various characterizations of this decompletion (called the sheared Witt vectors) and some examples. (Joint work in progress with Bhargav Bhatt, Vadim Vologodsky, and Mingjia Zhang.)
Thomas Nikolaus -Delta-Rings in Arithmetic and homotopy theory This talk will be about different instances and applications of the concept of a delta rings (and Witt vectors) in arithmetic andhomotopy theory. In particular we will analyse how generalizations control E_\infty-rings and state some conjectures and results along these lines. If time permits we will also introduce the dual notion and report on recent progress on coalgebras.
Viktoriya Ozornova - What is an (∞,∞)-category? In this talk, we will address the abstract frameworks for inductive and coinductive notions of (∞,∞)-categories. We will examine also the strict case as a blueprint for the actual study, as well as some of the differences between the strict and the weak case. This is joint work in progress with Martina Rovelli.
Maxime Ramzi - K(1)-local K-theory of Azumaya algebras The determinant map provides a way to recover a line bundle from its K-theory class. In this talk, I will discuss a categorified variant of this fact, in an attempt to answer a question of the form : how much of an Azumaya algebra can one recoverfrom its K-theory ? In particular, I will explain how for fields, the p-power torsion Brauer group decategorifies exactly as a p-power torsion strict Picard group.
Charanya Ravi - On the algebraic K-theory of Deligne-Mumford stacks For a finite group G, Artin's induction theorem states that the rational representation ring of G admits a decomposition in terms of representation rings of its cyclic subgroups. This gives an expression for the zeroth algebraic K-group of BG in terms of its cyclic subgroups. Inspired by this, the work of Vistoli gives such a decomposition of the algebraic K-group of coherent sheaves (or G-theory) of quotient stacks for finite group actions. We prove a similar induction theorem for the G-theory of all Deligne-Mumford stacks by looking at the automorphism groups at points of the stack. As an application, we see that the rational G-theory of a Deligne-Mumford stack coincides with the rational étale G-theory of its cyclotomic loop stack. This leads to an orbifold version of the Grothendieck-Riemann-Roch theorem. This is based on an ongoing joint work with Adeel Khan.
Georg Tamme - Localizing invariants of pushouts and non-commutative Hodge theory Singular cohomology of a smooth, proper complex variety carries a pure Hodge structure consisting of a rational lattice given by singular cohomology with rational coefficients and a Hodge filtration on cohomology with complex coefficients coming from thecomparison with de Rham cohomology, these two structures interacting in a specific way. Katzarkov, Kontsevich, and Pantev conjecture that this should generalize to non-commutative smooth, proper varieties, i.e. to smooth, proper ℂ-linear stable ∞-categories.In this case, de Rham cohomology is replaced by periodic cyclic homology, and the rational lattice is conjecturally given by Toën’s and Blanc’s topological K-theory. As Deligne constructed a mixed Hodge structure on the singular cohomology of arbitrary complexvarieties, one may also ask what happens for not necessarily smooth or proper ℂ-linear stable ∞-categories. In this talk, I will indicate how results about localizing invariants on pushouts of rings can be used to prove some cases of the lattice conjecture,for example for certain group rings. Most of these results have also been obtained by different techniques by Konovalov. This is joint work with Markus Land.
Peter Scholze - Geometry and (higher) category theory over the liquid complex numbers
Maria Yakerson - An alternative to spherical Witt vectors: Witt vectors of a ring form a “bridge” between characteristic p and mixed characteristic: for example, Witt vectors of a finite field F_p is the ring of p-adic integers Z_p. Spherical Witt vectors of a ring is a lift of classical Witt vectors to the world of higher algebra, much like sphere spectrum is a lift of the ring of integers.In this talk we will discuss a straightforward construction of spherical Witt vectors of a ring, in the case when the ring is a perfect F_p-algebra. Time permitting, we will further investigate the category of modules over spherical Witt vectors, and explain a universal property of spherical Witt vectors as an E_1-ring. This is joint work with Thomas Nikolaus.
Inna Zakharevich - Towards a trace on the Grothendieck spectrum of varieties In this talk we discuss an alternate model of topological Hochschild homology. This model has the advantage that it is largely combinatorial, and thus exists independently of a choice of enrichment. The hope is that this model can lead to a meaningful trace andmodel of THH for the Grothendieck spectrum of varieties. This is joint work with Jonathan Campbell and Kate Ponto.
Program and Schedule
| Day | Time | Speaker | Topic | Room |
|---|---|---|---|---|
| Mon, 6.10 | 09:00 - 10:00 | Georg Tamme | Localizing invariants of pushouts and non-commutative Hodge theory | H32 |
| 10:30 - 11:30 | Dustin Clausen | Refining Weil groups | H32 | |
| 14:00 - 15:00 | Maria Yakerson | An alternative to spherical Witt vectors | H32 | |
| 15:30 - 16:30 | Maxime Ramzi | K(1)-local K-theory of Azumaya algebras | H32 | |
| Tue, 7.10 | 09:00 - 10:00 | Thomas Nikolaus | Delta-Rings in Arithmetic and homotopy theory | H32 |
| 10:30 - 11:30 | Federico Binda | Relative Hyodo-Kato cohomology | H32 | |
| 13:45 - 14:45 | Hélène Esnault | On the restriction map in p-adically complete de Rham or prismatic cohomology | H32 | |
| 15:15 - 15:25 | Gong show - Georg Lehner | Algebraic K-theory of coherent spaces | H32 | |
| 15:25 - 15:35 | Gong show - Kaixing Cao | Comparison of Hyodo-Kato and de Rham-Fargues-Fontaine Cohomology Theories | H32 | |
| 15:35 - 15:45 | Gong show - Nikita Müller | Higher derivators and universal properties | H32 | |
| 15:50 - 16:00 | Gong show - Noam Nissan | Orthogonal motivic spectra | H32 | |
| 16:00 - 16:10 | Gong show - Jonas Stelzig | Kähler manifolds are not that formal | H32 | |
| 16:10 - 16:20 | Gong show - Bhavna Ashok Joshi | Motivic Filtrations on Localizing Invariants | H32 | |
| 16:25 - 16:35 | Gong show - Johannes Glossner | A Model-Independent Universal Property of the Lax 2-Functor Classifier | H32 | |
| 16:35 - 16:45 | Gong show - Andreas Gieringer | Vorst's conjecture for ample line bundles in characteristic zero | H32 | |
| Wed, 8.10 | 8:45 - 9:45 | Inna Zakharevich | Towards a trace on the Grothendieck spectrum of varieties | H32 |
| 10:15 - 11:15 | Shachar Carmeli | Steenrod Operations in Syntomic Cohomology and Duality for Brauer Groups in Characteristic 2. | H32 | |
| 11:30 - 12:30 | Charanya Ravi | On the algebraic K-theory of Deligne-Mumford stacks | H32 | |
| 14:00 | --- | Hike | Hotel Münchner Hof (Tändlergasse 9) | |
| Thu, 9.10 | 09:00 - 10:00 | Viktoriya Ozornova | What is an (∞,∞)-category? | H32 |
| 10:30 - 11:30 | Hokuto Konno | Family gauge theory and diffeomorphism groups | H32 | |
| 11:30 - 11:45 | Organizers | Conference Photo | The stairs are in front of the Mathematics building. | |
| 14:00 - 15:00 | Peter Scholze | Geometry and (higher) category theory over the liquid complex numbers | H32 | |
| 15:40 - 15:50 | Gong show - Polyxeni Spilioti | The twisted Ruelle zeta function and the Ray-Singer metric | H32 | |
| 15:50 - 16:00 | Gong show - Phil Pützstück | Condensed Anderson Duality | H32 | |
| 16:10 - 16:20 | Gong show - Marcin Lara | The Condensed Homotopy Type of a Scheme | H32 | |
| 16:25 - 16:35 | Gong show - Andrea Panontin | p-adic cohomologies and the cohomology of Monsky--Washnitzer | H32 | |
| 16:35 - 16:45 | Gong show - Christoph Winges | Localisation in connective algebraic K-theory | H32 | |
| Evening (19:00) | --- | Dinner | Bischofshof | |
| Fri, 10.10 | 08:45 - 09:45 | Manuel Krannich | Pontryagin–Weiss classes | H32 |
| 10:15 - 11:15 | Fabian Hebestreit | On the Weiss-Williams index | H32 | |
| 11:30 - 12:30 | Akhil Mathew | Sheared Witt vectors | H32 |
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 and research talks are in the Lecture Hall H32, at the first floor of the Mathematics Department of Regensburg University (Attention: not the department "Mathematik und Informatik" of the OTH).
The registration and coffee breaks are at the third floor of the Mathematics Department, in the SFB seminar room.
One can reach the University of Regensburg by following the instructions here and see here for maps of the campus. Many of the participants will probably arrive by bus at the Central Bus Station of the university which is close to the math building, a bit more towards North (=top of the plan).
There will be no talks on Wednesday afternoon. Instead, we warmly invite all participants to join us for a hike in the hills around Regensburg with a beautiful view on the Danube. We plan to have a break at a beautiful Biergarten (bring somecash). One-way, the hike from the city center to the Biergarten is roughly 7 km. To get back, one can either take the same route, or walk 2.5 km to a bus stop. At the Biergarten, it is possible to sit inside. The weather forecast currently is 15°C. Meeting point & time: We meet in front of the Hotel Münchner Hof (Tändlergasse 9) and leave at 14:00.
List of Participants
Registration and financial support
Registration is closed. Registration was open until 31 August 2025. (The deadline for financial support queries was 22 July 2025.)
If Participants want to apply for a short presentation in the Gong Show slots, please use the following link: https://s2survey.net/1085/
Conference Poster
You can download the conference poster here.
Organizers
Ulrich Bunke, Denis-Charles Cisinski and Guido Kings
Conference Picture
Sponsors of the conference
This conference is funded by SFB 1085 "Higher Invariants"
