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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
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.
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.
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 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 $H^0(X mod p, \Omega^i)=0$, so 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. This already produces classes in algebraic de Rham cohomology of $X$ which die in $p$-adically complete de Rham cohomology $H^i( \hat U )$ of $U$. In particular it shows that no $p$-adic method, even if we take into account almost all $p$, can shed light on Grothendieck’s conjecture. Caro-D’Addezio lifted the vanishing of $H^i_{dR}^i( \hat X) \to H^i_{dR( \hat U)^{sep}$ to the vanishing of $H^i_{dR}^i( \hat X) \to H^i_{dR}( \hat U)_{\mathbb Q}$ under the extra assumption $H^1(X, \Omega^{i-1})=0$. May be by the time of the conference we’ll know the precise condition to lift their result to prismatic cohomology.
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.)
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.
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.
Program and Schedule
The list of talks and abstracts will appear here before the conference.
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
TBA
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
TBA
Sponsors of the conference
This conference is funded by SFB 1085 "Higher Invariants"
