Oberseminar SoSe 25
Die Vorträge finden jeweils donnerstags um 16:45 Uhr im Raum WSC-N-U-3.05 (im Mathematikgebäude ) statt. Directions from the train station.
Der Tee findet ab 16:15 statt.
Alle Interessenten sind herzlich eingeladen!
The seminar takes place on Thursday, starting at 4:45pm. The duration of each talk is about 60 minutes.
Everybody who’s interested is welcome to join.
| 10.04.2025 | Luca Marannino (Paris Jussieu) | Anticyclotomic Iwasawa theory of modular forms at inert primes via diagonal classes |
| 24.04.2025 | Alex Ivanov (Bochum) | Meromorphic vector bundles on the Fargues-Fontaine curve |
| 08.05.2025 | Immanuel Halupczok (HHU Düsseldorf) | Poincaré series and stratifications |
| 15.05.2025 | Angelina Zheng (Tübingen) | Rational cohomology of M_{4,1} |
| 22.05.2025 | James Taylor (Padova) | Categories of Equivariant Vector Bundles on Drinfeld Symmetric Spaces |
| 05.06.2025 | Cécile Gachet (RUB) | Finiteness of minimal models for some Calabi—Yau pairs, equivariantly |
| 12.06.2025 | N.N. | t.b.a. |
| 26.06.2025 | – | Symposium Düsseldorf-Essen-Wuppertal |
| 03.07.2025 | ALGANT Students | Thesis Rehearsal |
| 10.07.2025 | David Loeffler (UniDistance Brig) | Euler systems, ultraprimes and local conditions |
| 17.07.2024 | N.N. | t.b.a. |
Abstracts
Luca Marannino: Anticyclotomic Iwasawa theory of modular forms at inert primes via diagonal classes
Abstract: In this talk, we outline an approach to the study of anticyclotomic Iwasawa theory of modular forms when the fixed prime p is inert in the relevant quadratic imaginary field. Following ideas of Castella-Do for the “p split” case, one can envisage a construction of an anticyclotomic Euler system arising from a suitable manipulation of certain Galois cohomology classes known as diagonal classes. We will report on this work in progress, trying to underline the main difficulties arising in the “p inert” setting.
Alexander Ivanov: Meromorphic vector bundles on the Fargues-Fontaine curve
Abstract: Recently, two different constructions of a geometric local Langlands category were given: the analytic construction of Fargues—Scholze and the schematic one of Hemo—Zhu (in progress). Motivated by the goal of finding a relation between these, we construct the stack of meromorphic vector bundles, which interpolates between the geometric objects on analytic and schematic sides. We sketch some of its geometric properties, e.g. how the Fargues—Scholze charts M_b naturally appear in it. This is joint work with Ian Gleason and Felix Zillinger.
Immanuel Halupczok: Poincaré series and stratifications
The Poincaré series $P_V(T)$ associated to a variety $V$ defined over $\mathbb Z$ is a formal power series which creates an interesting connection between arithmetic and geometric properties of $V$. On the one hand, it is defined in a purely arithmetic way: The coefficients of $P_V(T)$ are the numbers $\#V(\mathbb Z/p^r\mathbb Z)$ of $\mathbb Z/p^r\mathbb Z$-rational points of $V$, for some fixed prime $p$ and for $r$ running over $\mathbb N$. On the other hand, the series $P_V(T)$ is closely related to the singularities of $V$. For example, $P_V(T)$ can be computed using a resolution of singularities of $V$.
Instead of using a resolution of singularities, one can also study the singularities of $V$ using a suitable (e.g. “Whitney regular”) stratification of $V$. One would expect that such a stratification yields some precise information about $P_V(T)$, but one result one would hope for turns out to be false for Whitney regular stratifications. Instead, I will present a stronger notion of regularity for which that result becomes true. This is joint work with David Bradley-Williams.
In the talk, I will of course explain all the necessary background about Poincaré series and stratifications.
Angelina Zheng: Rational cohomology of M_{4,1}
The moduli space M_{g,n} of non-singular curves of genus g and n marked points, and its compactification, have been central objects in algebraic geometry for many years.
However, lot is still unkown about their geometry, in particular from the point of view of the rational cohomology: this is completly known only for some values of g,n. We will review what is known and focus mainly on M_{g,n}.
Most of the known cases for which the rational cohomology is completely known are due to Tommasi, via Gorinov-Vassiliev’s method.
We will briefly describe such a method, applied to compute the rational comology of M_{4,1} in a joint work with Yiu Man Wong.
James Taylor: Categories of Equivariant Vector Bundles on Drinfeld Symmetric Spaces
Abstract: In this talk we will describe various categories of equivariant vector bundles and vector bundles with connection on Drinfeld symmetric spaces. These include the category of $G^0$-finite (“torsion”) vector bundles with connection and the category of Lubin-Tate bundles introduced by Kohlhaase.
Cécile Gachét: Finiteness of minimal models for some Calabi—Yau pairs, equivariantly
Abstract: In dimension 3 and higher, it is well-known that minimal resolutions of singularities of a complex projective variety need not be unique. Typically, there arise small birational modifications which toggle back and forth between different minimal models of the same variety. This framework is best understood for Calabi—Yau pairs, whose minimal models are connected by finite sequences of so-called flops. Some finite sequences of flops loop, and thereby define non-trivial birational automorphisms on one minimal model; to that extent, it is not uncommon for a Calabi-Yau pair to have infinitely many marked minimal models. It is however conjectured that any given klt Calabi—Yau pair has finitely many unmarked minimal models.
As the class of klt Calabi—Yau pairs is closed under taking quotients by finite group actions in a very natural way, it is natural to expect birational finiteness properties to descend under finite quotients. This talk presents an equivariant descent result in that spirit for a large subclass of klt Calabi—Yau pairs. We will highlight the role of convex geometry, in particular of self-dual homogeneous cones and hyperbolic reflections, in identifying large subclasses of klt Calabi—Yau pair with well-behaved birational geometry as well as in the proof of the main theorem.
David Loeffler: Euler systems, ultraprimes and local conditions
In order to attack the Bloch–Kato conjecture, one needs to control the size of Selmer groups (subspaces of Galois cohomology defined by local conditions). The basic machinery of Kolyvagin shows that any Euler system for a p-adic representation gives a bound on the “strict” Selmer group, i.e. the cohomology classes whose localization at p is 0. However, in practice one wants to refine this, and show that if the Euler system satisfies some additional local property at p, it bounds a different, larger Selmer group with a less strict local condition; and this is difficult in general, since it is very unclear how Kolyvagin’s “derivative” operator interacts with local conditions. Using a reinterpretation of Kolyvagin’s construction via ultrafilters (due to Naomi Sweeting), we give a simple criterion for which local conditions interact well with the derivative construction, and describe ongoing work to verify this in new cases related to Pottharst’s trianguline Selmer groups.