| Monday | Tuesday | Wednesday | Thursday | Friday | |
|---|---|---|---|---|---|
| 9:30 | Leonhardt | Anschütz | Maire | Tamagawa | Levine |
| 11:00 | Stix | Esnault | Nakamura | Huber-Klawitter | Geißer |
| 12:00 | Lunch | Lunch | Lunch | Lunch | Snack(*) |
| 13:30 | Spiess | Szamuely | Harari | Kerz | Puschnigg(**) |
| 15:00 | Morin | Tschinkel | Pop | Hike | |
| Evening | Dinner |
(*): We will provide a savory snack in the Common Room before the last talk.
(**): This talk starts at 12:30 pm.
In diesem Votrag möchte ich verschiedene Modulräume von Grad 1 Divisoren auf der Fargues--Fontaine-Kurve diskutieren, und erklären wie diese eine Geometrisierung von Fontaines $(\phi,\Gamma)$- bzw. $(\phi,N,\mathit{Gal}_{\mathbf{Q}_p})$-Moduln erlauben. Desweiteren möchte ich einen Ausblick geben, wie diese Theorie eine analytische Prismatisierung und eine Geometrisierung einer möglichen lokal-analytischen $p$-adischen Langlandskorrespondenz erlaubt. Der Vortrag beruht auf gemeinsamen Projekten mit Bosco, Le Bras, Scholze und Rodriguez-Camargo.
We study the restriction map in prismatic cohomology of a smooth proper variety to the one of an affine, showing vanishing under a Hodge type cohomological condition.
We discuss independence of the algebraically closed base field and finiteness of motivic and étale motivic cohomology with finite coefficients of smooth and proper varieties. If the variety is defined over a finite field, we also discuss the weaker question of independence of the fiber of the Frobenius on the algebraically closed field.
We present various arithmetic duality theorems related to the étale or fppf cohomology of commutative group schemes. We provide applications to the arithmetic of linear algebraic groups over global fields of characteristic $p$ and other function fields.
Periods are numbers defined as values of integrals of arithmetic nature. They are a classical object of transcendence theory. The Period Conjecture gives a conceptual explanation for all relations between them in terms of two fibre functors on the category of motives over $\mathbb{Q}$. On the other hand, they have a very explicit description in terms of volumes of $\mathbb{Q}$-semialgebraic sets. More recently, exponential periods have come into focus. By joint work with Commelin and Habegger, they can be described as volumes of definable sets in a certain $o$-minimal structure. In the talk, we are going to argue that tame ($o$-minimal) geometry is the natural setting for the Period Conjecture and its generalisations. We will explain the state of the art, in particular de Rham cohomology in the $o$-minimal setting. No knowledge of $o$-minimality is assumed.
In joint work with H. Esnault, we investigate the definability of flat sections of algebraic flat connections within the framework of $o$-minimal theory. We demonstrate that the connection is regular-singular with unitary monodromy eigenvalues at infinity if and only if a specific definability property is satisfied, thereby refining previous results established by Bakker and Mullane.
Given a hyperbolic curve $Y$ defined over the integers and a finite set of primes $S$, the set of $S$-integral points $Y(\mathbb{Z}_S)$ is finite by theorems of Siegel, Mahler, and Faltings. Determining this set in practice is a difficult problem for which no general method is known. In this talk I report on joint work with Martin Lüdtke in which we develop a Chabauty--Coleman method for finding $S$-integral points on affine curves. We achieve this by bounding the image of $Y(\mathbb{Z}_S)$ in the Mordell--Weil group of the generalised Jacobian using arithmetic intersection theory on a regular model.
We give a report on versions of classical Atiyah-Bott localization in the setting of generalized motivic cohomology theories living in the minus part of the motivic stable homotopy category, with the main examples being Witt sheaf cohomology and Witt theory. The main thrust is to replace the classical approach relying on a $\mathbb{G}_m$-action with one that uses instead an action by $\mathit{SL}_2$ or the normalizer of the torus in $\mathit{SL}_2$. We will also discuss recent work involving other groups. The new input involves a description of the real points of $\mathit{BG}$, from work of Ambrosi-de Gaay Fortnam and Mantovani-Matszangosz-Wendt, together with Bachmann’s description of the real realisation as $\rho$ localisation.
In this talk, we recall Alexander Schmidt’s theorem on the cohomological dimension of the pro-$p$ groups $G_S$. We emphasize the importance of a governing field, which already appears in the proof of the Scholz--Reichardt theorem, providing a positive answer to the Inverse Galois Problem for $p$-groups. We then present several variants and related topics: the Galois problem for the $p$-Hilbert tower; constructions of Galois representations with only tame ramification; and the Strong Massey property for number fields. This is joint work with Farshid Hajir, Michael Larsen, Jan Minac, Ravi Ramakrishna, and Nguyen Duy Tan.
We give a conjectural description of Zeta-values of arithmetic schemes in terms of two perfect complexes of abelian groups and a canonical trivialization. Then we state a (proven) archimedean analogue of this conjecture. In order to show compatibility with the classical Bloch-Kato conjecture, we study the Beilinson fiber square from a prismatic viewpoint, which provides precise integral information. This implies new cases of the Bloch-Kato conjecture. This is joint work with Matthias Flach and Achim Krause.
The monodromy action on the meta-abelian fundamental group of a once-punctured elliptic curve provides a certain family of quantities ``Eisenstein invariants''. In this talk, we discuss their extensions to those quantities on what B. Enriquez (2014) introduced as the elliptic Grothendieck--Teichmüller group $GT_{ell}$. We also provide a snapshot of $GT_{ell}$ with the profinite braid tower in the background from a viewpoint of my collaboration with A. Minamide (2022).
After briefly recalling the $t$-birational Section Conjecture ($t$-BSC) in the larger context of the SC, I will mention old and new results concerning the $t$-BSC. A main theme of my talk is giving generalizations and ``minimalistic'' refinements of the classical $t$-BSC (over quite general base fields, beyond the context of Grothendieck's SC). I will also give a list of open problems.
Die Operator-$K$-Theorie hat sich als die bedeutendste homologische Invariante von Banach- und $C^*$-Algebren erwiesen. Trotz der völlig elementaren Definition ist ihre Berechnung aber ausgesprochen schwierig. Mit der (lokalen) zyklischen Homologie wurde eine Homologie-Theorie für Banach-Algebren konstruiert, die einerseits eine gute Approximation der K-Theorie liefert, andererseits aber mit Mitteln der homologischen Algebra relativ gut berechenbar ist. Wir erklären im Vortrag die Grundzüge dieser Theorie und studieren an diversen Beispielen, wie viel Information beim Übergang von der K-Theorie zur zyklischen Homologie verloren geht. Dabei spielen die (reduzierten) $C^*$-Algebren Gromov-hyperbolischer Gruppen mit Kazhdans Eigenschaft $(T)$ eine zentrale Rolle.
I will present recent applications of Eisenstein classes to regulators and special values of $L$-functions for totally real number fields (joint work with S. Dasgupta, A. Galanakis and M. L. Honnor).
Anabelian geometry has been understood predominantly as the geometry encoded in the arithmetic of fundamental groups. An early approach in the 1990ies by Voevodsky, that is also based on Grothendieck's famous letter to Faltings, extracts geometry from the étale topos of a variety. `Voevodsky anabelian geometry' has been revisited and expanded recently by Carlson, Haine and Wolf. In this talk we will report on work together with Magnus Carlson that covers varieties over sub-$p$-adic fields.
Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. I shall be reporting on recent joint work with Davide Lombardo where we prove that the geometric étale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the Galois group of the maximal Kummer extension of $K$ in odd degrees. In particular, every abelian variety has finite torsion over the maximal Kummer extension. This improves results by Rössler and the speaker as well as Murotani and Ozeki.
A local system on a variety is interpreted as a linear representation of the fundamental group of the variety. In the algebro-geometric setting (over an algebraically closed field), the restriction of a local system to a (closed) point is clearly trivial, while in the arithmetico-geometric setting (over an ``arithmetic'' field), it is usually highly nontrivial and sometimes even strongly controls the behavior of the original local system. In this talk, I will treat some topics from my long-time collaboration with Anna Cadoret on this issue.
I will discuss recent results and constructions in equivariant birational geometry (joint with I. Cheltsov, A. Kresch, B. Hassett, and Zh. Zhang).