Titles and abstracts
| Hanno Becker: Khovanov-Rozansky homology and Hochschild homology of Soergel bimodules
Khovanov-Rozansky homology is a categorification of the classical quantum 𝖘𝖑(k) link invariant that is defined via matrix factorizations, a concrete model for the singularity category of a hypersurface. In this talk, I will recall its construction and show how it can alternatively be described through Hochschild homology of Soergel bimodules known from representation theory. |
| Federico Binda: Algebraic cycles with modulus and applications
In this talk we will survey some recent results involving the so-called (higher) Chow groups with modulus for a pair (X,D), where X is a smooth variety over a field k and D is a (possibly non reduced) effective Cartier divisor on X, generalizing additive Chow groups studied by K. Rülling, J. Park and A. Krishna. These groups of algebraic cycles are a modified version of Bloch's higher Chow groups and are related to several non homotopy invariant objects. We will say how it is possible to construct, by modifying a method due to M. Levine, a cycle class map from the groups of higher 0-cycles with modulus to Quillen's relative K-theory groups for the pair (X,D). After that, we will sketch the construction over C of a regulator map to a relative version of Deligne cohomology, providing an Abel-Jacobi map to some intermediate Jacobians with additive part. Such Abel-Jacobi map satisfies a universal property that is the analogue of the one proved by Esnault-Srinivas-Viehweg for the Chow group of 0-cycles of a singular variety. This part is a joint work with S. Saito. |
| Xevi Guitart: Computing equations of elliptic curves over number fields via p-adic methods
It is conjectured that to every modular form over a number field which is a Hecke eigenform with rational coefficients is attached an elliptic curve with the same L-function. This is known for modular forms over the rationals; moreover, the Eichler-Shimura construction provides an explicit method for computing the curve from the modular form that has been used to produce extensive tables of curves over Q, such as Cremona's tables. The Eichler-Shimura construction does generalize (at least partially) to forms over totally real number fields, but over fields having complex places no explicit construction seems to be available. The aim of the talk is to describe a conjectural construction which applies under certain additional conditions and is a natural generalization of the p-adic uniformizations arising in the theory of Stark-Heegner points. I will also report on some numerical computations supporting the conjecture in the case when the number field has exactly one complex place. This is joint work with Marc Masdeu and Haluk Sengun. |
| Eugen Hellmann: Eigenvarieties and spaces of trianguline representations
I will explain the relation between eigenvarieties, i.e. certain p-adic families of automorphic representatiions, and p-adic families of certain Galois-representations, so called trianguline Galois-representations. In the case of GL2(Q) this relation is part of the p-adic Langlands correspondence. For unitary groups over CM fields one can establish this relation using the Taylor-Wiles patching method. This is joint work with Christophe Breuil and Benjamin Schraen. |
| Ariyan Javanpeykar: Arakelov invariants, Belyi degree, Shafarevich conjecture
Let X be a curve over a number field. We study invariants of X such as the Belyi degree d(X) and the stable Faltings height h(X). Our main result is an explicit inequality relating these invariants: h(X)<109d(X)6. As a first application we prove the 2011 conjecture of Edixhoven-de Jong-Schepers on the Faltings height of a cover of curves. Then we discuss applications to Szpiro's small points conjecture and the effective Shafarevich conjecture. In fact, in a joint work with Rafael von Kaenel, we prove the these conjectures for cyclic covers of the projective line. Moreover, we explain how von Kaenel uses the above theorem and certain modularity results to prove the effective Shafarevich conjecture for abelian varieties of product GL2-type. |
| Luigi Lombardi: Regularity of curves on abelian varieties
In this talk I will establish a bound on the Theta regularity of an arbitrary curve on an abelian variety in terms of its degree with respect to a polarization. This notion of regularity was introduced in a series of papers by Pareschi-Popa and measures the complexity of sheaves in terms of their cohomological properties on polarized abelian varieties (similarly to Castelnuovo-Mumford regularity for sheaves on projective spaces). As an application, we bound the degrees of defining equations of curves on abelian varieties and furthermore we identify which line bundles on the ambient space restrict to non-special line bundles on the curve. Moreover, I will deduce a Castelnuovo's type genus bound for curves on abelian varieties which improves, in some few cases, Debarre's genus bound. Time permitting, I will also discuss Theta regularity of Brill-Noether loci on Jacobians. This is a work in progress in collaboration with Wenbo Niu. |
| Helge Ruddat: Canonical Coordinates from Tropical Curves
Morrison defined canonical coordinates near a maximal degeneration point in the moduli of Calabi-Yau manifolds using Hodge theory. Gross and Siebert introduced a logarithmic-tropical algorithm to provide a canonically parametrized smoothing of a degenerate Calabi-Yau. We show that the Gross-Siebert coordinate is a canonical coordinate in the sense of Morrison. The coordinates are given by period integrals which we compute explicitly integrating over cycles constructed using tropical geometry. This is joint work with Siebert. |
| Axel Stäbler: Bernstein-Sato polynomials and test modules in positive characteristic
In complex geometry there is a relation between the birational notion of multiplier ideals and the D-module theoretic notion of Bernstein-Sato polynomials due to work of Budur and Saito. In positive characteristic Mustaţă has shown that the test ideal filtration relates to a notion of Bernstein-Sato polynomials defined using divided power Euler operators. I will report on joint work with M. Blickle where Mustaţă's correspondence has been extended from the structure sheaf to Cartier modules. |
| Ronan Terpereau: Moduli spaces of (G,h)-constellations
This talk focuses on a work initiated by Tanja Becker in her PhD thesis three years ago and completed recently by myself. Given a reductive group G acting on an affine scheme X, a Hilbert function h, and a stability condition θ, we explain how to construct the moduli space M of θ-stable (G,h)-constellations on X, which is a common generalization of the invariant Hilbert scheme after Alexeev and Brion and of the moduli space of θ-stable G-constellations for finite groups introduced by Craw and Ishii. The main tools for this construction are the geometric invariant theory and the invariant Quot schemes. Moreover, the moduli space M is naturally equipped with a morphism μ: M → X//G which turns to be a "nice" desingularization of the quotient X//G in many situations. |
| Zhiyu Tian: A geometric proof of the Hasse principle for quadrics over global function fields
The Hasse principle over a global function field states that one can solve a system of equations over the global function field if one can solve them locally over Laurent fields. This is true for quadrics and remains open even for cubic hypersurfaces. In this talk I will discuss a simple geometric criteria for the Hasse principle to hold for quadratic and cubic hypersurfaces and then show how to use this criteria to give a geometric proof of the classical result about quadrics. I will also discuss some cases of cubic hypersurfaces if time permits. |
| Rodolfo Venerucci: On the two-variable p-adic Birch and Swinnerton-Dyer conjecture
Let A/Q be an elliptic curve, corresponding to a weight-two newform f. Let p be a prime of ordinary reduction for A. The p-adic analogue of the Birch and Swinnerton-Dyer (BSD) conjecture, formulated by Mazur, Tate and Teitelbaum, relates the leading coefficient of the cyclotomic p-adic L-function of f to the determinant of the cyclotomic canonical height pairing, defined on the (extended) Mordell-Weil group of A. The exceptional zero situation, corresponding to the case where A has split multiplicative reduction at p, is of particular interest and leads to the celebrated exceptional zero conjecture. Hida theory allows us to view f as the weight-two term in a p-adic family of modular forms of varying weight. The cyclotomic p-adic L-functions of the members of the family can be packaged into a single two-variable (cyclotomic and weight) p-adic L-function, the so called Mazur-Kitagawa p-adic L-function. We discuss a two-variable p-adic BSD conjecture, relating the leading term of the Mazur-Kitagawa p-adic L-function to the determinant of a suitable two-variable cyclotomic height-weight pairing, whose definition relies on Nekovář's theory of Selmer complexes. We also outline a proof of the corresponding two-variable exceptional zero conjecture in the rank one case. As a crucial step in the proof, we compare p-adic Beilinson-Kato elements and Heegner points, thus establishing a conjecture of Perrin-Riou. |
| Maksim Zhykhovich: Integral Chow motives and their decompositions
In this talk I will discuss the properties of integral Chow motives of twisted flag varieties. I will explain a method which allows one to construct integral motivic decompositions out of those modulo primes. As an application, I will discuss the Krull-Schmidt principle for integral motives and provide a complete list of generalized Severi-Brauer varieties with indecomposable integral motive. This talk is based on a joint work with Nikita Semenov. |