# Oberseminar Summer term 2023

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.

Termin | Vortragende*r | Titel |
---|---|---|

13.4.2023 | Amnon Neeman (ANU, Univ.Bielefeld) | Finite approximations as a tool for studying triangulated categories |

20.4.2023 | Brandon Levin (Rice University/Max-Planck Institute) | Moduli of Galois representations and Serre’s modularity conjecture |

27.4.2023 | Andreas Bode (Univ. Wuppertal) | Auslander regularity for completed rings of $p$-adic differential operators |

4.5.2023 | First Joint Symposium of GRKs 2240/2553 | |

11.5.2023 | Lukas Bröring | Question and Answer session for the research seminar |

25.5.2023 | Claudius Heyer (Münster) | The Geometrical Lemma for Smooth Representations in Natural Characteristic |

1.6.2023 | Rustam Steingart (Univ. Heidelberg) | Analytic cohomology of Lubin-Tate $(\varphi,\Gamma)$-modules |

15.6.2023 | Jean Fasel (Inst. Fourier Grenoble) | Vector bundles on smooth quasi-projective 3-folds |

22.6.2023 | Florent Schaffhauser (Univ. Heidelberg) | Hodge numbers of moduli spaces of principal bundles on curves |

29.6.2023 | Carolina Tamborini (Univ. Utrecht) | Symmetric spaces and geometry of the Torelli locus |

6.7.2023 | Yajnaseni Dutta (Univ. Bonn) | A Family of curves |

13.7.2023 | N.N. | tba |

## Abstracts

**Amnon Neeman, April 13: Finite approximations as a tool for studying triangulated categories**

A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We’ll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.

And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And some other ideas, mimicking constructions in real analysis, turn out to also be powerful.

And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.

And what makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a generalization of a theorem of Rouquier’s, a short, sweet proof of Serre’s GAGA theorem and a proof of a conjecture by Antieau, Gepner and Heller. And the most recent development, still work in progress, amounts to a major extension of an old result of Rickard’s.

**Brandon Levin, April, 20: Moduli of Galois representations and Serre’s modularity conjecture**

Serre’s modularity conjecture (1973) was proven by Khare-Wintenberger building on work of Kisin in the mid-2000s. The methods and problems surrounding the conjecture have played an important role in the field of modularity lifting and the $p$-adic Langlands program. One such development is the recent construction of moduli stacks of $p$-adic Galois representation by Emerton-Gee. I will explain the connection between Emerton-Gee’s space and Serre’s conjecture in the simplest case of two-dimensional representations of the Galois group of $\mathbb Q_p$. I will then go on to describe how this geometry helps us understand higher dimensional generalizations of Serre’s conjecture based on joint work with Daniel Le, Bao V. Le Hung, and Stefano Morra.

**Andreas Bode, April 27: Auslander regularity for completed rings of $p$-adic differential operators**

The theory of co-admissible $\overparen{\mathcal D}$-modules on a smooth rigid analytic variety $X$, introduced by Ardakov-Wadsley, offers a geometric way to study $p$-adic representations. In this talk, we show that locally, $\overparen{\mathcal D}$ can be built out of Auslander regular Banach rings of global dimension bounded by $2\dim(X)$, and equal to $\dim(X)$ ‘as $n$ goes to infinity’. We use this to derive adjunction and projection formulae for co-admissible $\overparen{\mathcal D}$-modules.

**Claudius Heyer, May 25: The Geometrical Lemma for Smooth Representations in Natural Characteristic**

In the theory of smooth representations of a $p$-adic reductive group, the Geometrical Lemma, due to Bernstein–Zelevinsky and Casselman, describes parabolic restrictions of parabolically induced representations by providing a filtration whose graded pieces can be described explicitly. Classically, the Geometrical Lemma works well whenever the coefficient field has characteristic different from $p$, but for mod $p$ representations the filtration degenerates to a single degree. As it turns out, this defect can be resolved by working on the level of derived categories: the missing graded pieces appear in various cohomological degrees, which are therefore invisible on the abelian level. In this talk I will present the derived version of the Geometrical Lemma and, if time permits, describe some applications.

**Rustam Steingart, June 1st: Analytic cohomology of Lubin-Tate $(\varphi,\Gamma)$-modules**

If $L$ is a non-trivial finite extension of $\mathbb{Q}_p$ there exist $L$-linear representations of the absolute Galois group $G_L$ which are not overconvergent. A sufficient condition to ensure overconvergence is $L$-analyticity. This makes it interesting to study analytic extensions of such modules or, more generally, analytic cohomology. Using $p$-adic Fourier theory we can, after passing to a large field extension of $L$, describe these cohomology groups in terms of an explicit “Herr-complex” which allows us to deduce finiteness and base change properties analogous to the results of Kedlaya-Pottharst-Xiao on continuous cohomology. We also obtain a variant of Shapiro’s Lemma for Iwasawa-cohomology in certain cases. The above results form the technical foundations for studying an “analytic” variant of the local epsilon-isomorphism conjecture.

**Jean Fasel, June 15: Vector bundles on smooth quasi-projective 3-folds**

In this talk, I will introduce the notion of motivic vector bundle on a smooth scheme, and explain how to classify them in the case of smooth quasi-projective 3-folds, using discrete invariants. I will then explain how to construct actual vector bundles with given invariants.

**Florent Schaffhauser, June 22, Hodge numbers of moduli spaces of principal bundles on curves**

The Poincaré series of moduli stacks of semistable $G$-bundles on curves has been computed by Laumon and Rapoport. In this joint work with Melissa Liu, we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way. As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé, on the maximality on moduli spaces of vector bundles over real algebraic curves.

**Carolina Tamborini, June 29, Symmetric spaces and geometry of the Torelli locus**

Riemannian symmetric spaces are Riemannian manifolds with special symmetry properties. They are important in various fields of geometry. In algebraic geometry, they appear naturally as spaces parametrizing certain Hodge structures. In particular, the moduli space $\mathscr A_g$ of principally polarized abelian varieties is a quotient of the Siegel space, which is, in fact, a (hermitian) symmetric space. In the talk, we consider the Torelli locus. This is the closure in $\mathscr A_g$ of the image of the moduli space $\mathscr M_g$ of smooth, complex algebraic curves of genus $g$ via the Torelli map $j\colon \mathscr M_g \to \mathscr A_g$. We describe the problem of studying the (local) geometry of the Torelli locus in $\mathscr A_g$ and its relation with symmetric subspaces of the Siegel space. Also, we explain how this is linked to a conjecture by Coleman and Oort.

**Yajnaseni Dutta, July 6, A Family of curves**

I will report on an on-going discussion with D. Huybrechts about a family of genus 4 curves over a smooth cubic 4-fold. The family is closely related to the intermediate Jacobian lagrangian fibration of the cubic 4-fold. I will present some geometric properties of this family. Finally, we will look at the 8-dimensional relative compactified Jacobian of this family of curves and a curious differential 2-form on it. I will discuss strategies of showing its non-degeneracy on certain loci and explain where it becomes degenerate.