Schedule

Time: Friday June 16th, 2023, 13:30 - 17:30 (CET)  

Location: Amsterdam, University of Amsterdam

13:30 - 14:30 Francesca Leonardi (Leiden University): FORMALITY OF LOGARITHMIC HOCHSCHILD (CO)HOMOLOGY

Formality is the property for a cochain complex to be quasi-isomorphic to its cohomology. Arinkin and C ̆ald ̆araru use self-intersection techniques to study the formality of the Hochschild complex (generalising the HKR isomorphism). Following Olsson’s definition of Hochschild cohomology for log schemes, we investigate its formality. As an application, we decompose the log Hochschild cohomology for orbifolds in terms of the contributions from every element of the group acting on the log scheme. Joint work (in progress) with M ́arton Hablicsek and Leo Herr.

15:00 - 16:00 Kees Kok (University of Amsterdam): On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces

The failure of the integral Hodge conjecture was first established by Atiyah and Hirzebruch in the 60's. Throughout the years, many techniques have been developed to find even more counter examples. More recently (2012) Colliot-Thélène and Voisin gave an interpretation of the integral Hodge conjecture in degree 2 in terms of unramified cohomology and in 2018, Colliot-Thélène used this, together with a specialization technique to generalize a counter example found by Benoist and Ottem. In this talk, I will explain this specialization technique and show that it can be generalized to recover a counter example found by Shen in 2019. Moreover, by the nature of the argument, there is no need to work over the complex numbers and in turn we obtain a new counter example to the integral Tate conjecture.

16:30 - 17:30 Noah Olander (University of Amsterdam): Fully faithful functors and dimension

Can one embed the derived category of a higher dimensional variety into the derived category of a lower dimensional variety? The expected answer was no. We give a simple proof and prove new cases of a conjecture of Orlov along the way.

Time: Friday January 20th, 2023, 13:00 - 17:00 (CET)  

Location: Nijmegen, Radboud University Nijmegen

13:00 - 14:00 Przemek Grabowski (University of Amsterdam): Foliations and Galois theory in char p>0

Differential operators can explain purely inseparable extensions, and vice versa. We will explain what it means, and how it relates to relative tangent bundles, Galois problems, and fibrations. In particular, we will recover a basic fact from characteristic zero that relative tangent bundles determine fibrations in positive characteristic.

14:15 - 15:15 Art Waeterschoot (KU Leuven): Resolving wild quotient singularities with differents

Given a nice curve C over a p-adic field, ideally one would be able to tell whether it has potentially good reduction from geometric information– for instance its minimal regular model. This is already complicated for elliptic curves, especially if p=2. A decade ago, Lorenzini laid down a framework for studying these questions via resolving wild quotient singularities of arithmetic surfaces. Recently the case of potential good ordinary reduction was completed by Obus and Wewers, using deformation theory and explicit valuation theory – I’ll explain another method using Temkin’s nonarchimedean analytic differents on metric graphs.

15:45 - 16:45 Wim Nijgh (Leiden): On the Galois invariant part of the Weyl group of the geometric Picard lattice of a K3 surface

Let X denote a K3 surface and let O(Pic X) denote the group of isometries of Pic X. Let RX denote the Galois invariant part of the Weyl group of the geometric Picard lattice of X. One can show that each element in RX can be restricted to an element of O(Pic X). The question raised by Bright e.a, (2019), is the following: Is for every K3 surface X, the image of this restriction map RX -> O(Pic X) a normal subgroup? In this talk, we will discuss and answer this question. For those not familiar with lattices and K3 surfaces, a short introduction to these concepts will be given.

Time: Friday October 14th, 2022, 13:30 - 17:30 (CET)  

Location: Leiden, Leiden University, room DM0.17 of the Gorlaeus building

13:30 - 14:30 Corinne Bedussa (Leuven): Berkovich geometry methods in algebraic geometry

The SYZ conjecture is a classical conjecture in Mirror Symmetry predicting the existence of Lagrangian fibrations on Calabi-Yau manifolds with maximal degeneration. Kontsevich and Soibelman suggested a non-Archimedean formulation of this problem (over the non-Archimedean field C((t))) which revealed to be a successful approach as allows to exploit the methods of Berkovich geometry and its interplay with birational geometry. Motivated by this conjecture, I will give a self-contained introduction to Berkovich geometry from the perspective of an algebraic geometer interested in this problem and describe the non-Archimedean  analogue of the SYZ fibration. If time permits, I will overview the main expectations and results in this conjectural framework, with a special interest in compact hyperkahler manifolds (irreducible holomorphic symplectic varieties).

14:45 - 15:45 Alexandra Viktorova (Leuven): On singular cubic threefolds

 In this talk, we will study singular complex cubic threefolds and possible combinations of isolated singularities appearing on them. To get a complete classification of such combinations, one can use deformation theory results together with a geometric method called the projection method. We will mainly focus on the projection method which can be applied in any dimension and in particular is enough to get a classification of singularities on cubic surfaces (the surface case was originally due to Schläfli and later reworked by Bruce and Wall).

16:15 - 17:15 Martin Lüdtke (Groningen): Non-abelian Chabauty for the thrice-punctured line

Let X be a hyperbolic curve defined over the rational numbers. The set of rational points X(Q) is known to be finite by Faltings' Theorem, but determining it is very difficult. The non-abelian Chabauty method, as developed by Minhyong Kim, produces explicit p-adic analytic functions whose vanishing set in X(Qp) contains and conjecturally equals the set of rational points. I will explain the method and show how it applies to the example of the thrice-punctured line. 

Time: Friday April 29th, 2022, 13:00 - 17:00 (CET)  

Location: Utrecht, Utrecht University, room 2.02 of the Minnaert building

13:00 - 14:00 Lisanne Taams (Radboud University Nijmegen):The Motive of the Moduli Stack of Vector bundles on a Stacky Curve.

Abstract: The language of stacky curves naturally  describes several different generalizations of vector bundles, such as G-equivariant bundles and Parabolic bundles. The advantage of working with stacky curves is that often we can directly generalize the methods used for classical curves. In this talk I will explain the basics of stacky curves and their vector bundles. Afterwards I will give some properties of the moduli stack, including a computation of the (virtual) motive.

14:15 - 15:15 Floris Vermeulen (KU Leuven): Scrollar invariants, syzygies and representations of the symmetric group

To a curve C equipped with a degree d morphism to P^1, one can associate d-1 integers which are called the scrollar invariants of C. Schreyer has shown how to construct a relative minimal resolution of C inside some natural ambient space. This resolution gives another list of invariants one can associate to C, called the splitting types. I will explain the construction of this resolution and use it to relate these splitting types to the scrollar invariants of various other curves arising from C. Along the way, we use Galois theory and representation theory of the symmetric group. This is joint work with Wouter Castryck and Yongqiang Zhao.

15:45 - 16:45 Aline Zanardini (Leiden University): Non-symplectic automorphisms of order multiple of seven on K3 surfaces

Abstract:  Automorphisms of K3 surfaces have been widely studied in recent years and a fundamental problem is to obtain a complete classification of non-symplectic automorphisms of finite order, for any possible order. In this talk I will report on some recent progress on the classification of non-symplectic automorphisms of K3 surfaces whose order is a multiple of seven. This is joint work in progress with R. Bell, P. Comparin, J. Li, A. Rincón-Hidalgo and A. Sarti.

Time: Friday December 10th, 2021, 14:00 - 17:00 (CET)   

Location (Zoom): https://uva-live.zoom.us/j/82881528544?pwd=Vk82N0o0SFM4R2ZaOWVEWlhVdzZBUT09

Meeting ID: 828 8152 8544

Passcode: 967634

14:00 - 14:05: Introduction

14:05 - 14:50: Przemek Grabowski (University of Amsterdam): Families of conics on P^2.

Abstract: This talk is about pencils of conics on P^2 and corresponding to them foliations, we do not assume the audience to know what that is. We will recall a discriminant of a family of quadrics, the characterization of singular fibres, and how to generate all such pencils easily and concretely. After that we will use those to discuss a problem of translating pencils into foliations. As such, the talk could be considered as a case-based introduction to foliations in algebraic geometry.

The main references are "Basic Algebraic Geometry 1" by Shafarevich and "Birational Geometry of Foliations" by Brunella."

15:00 - 15:45: Dhruva Kelkar (University of Amsterdam): The Langlands - Kottwitz method.

Abstract; The Langlands-Kottwitz method is an approach to describe the l-adic cohomology of Shimura varieties in terms of automorphic representations. Applications of this include constructing Galois representations associated to certain automorphic representations as predicted by conjectures in the Langlands program. In this talk, I will try to explain some aspects of this with the example of modular curves and give a general overview of the method

16:10 - 16: 55: Jasper van de Kreeke (University of Amsterdam): How to push a deformation through an unknown functor?

Abstract: A persistent pain in mirror symmetry is the lack of an explicit functor Db(Fuk X) → Db(Coh Y). Given a deformation of the left side, how would we even have a chance to transport it to the right? Similar questions appear all over algebraic geometry. In the case of mirror symmetry, a solution to this deformation problem is known in the commutative case, but a version for Bocklandt's noncommutative dimers is lacking. I will share 3 general remedies learnt at my summer camp 2020, and sketch how they apply in the case of dimers. You will have the opportunity to learn about Hochschild invariants and the DG Lie algebra approach to deformation theory.