Humboldt University Berlin, Summer 2024
Time: Tuesdays 13:15 - 14:45
Venue: Room 3.006, Rudower Chaussee 25, 12489 Berlin
— See also the Algebraic Geometry Seminars at HU and FU —
| 22.10.2024 | Jean-Baptiste Teyssier (Paris) |
| Title: | Boundedness for Betti numbers of étale sheaves in positive characteristic |
| Abstract: | Let X be a smooth proper variety over an algebraically closed field of characteristic p>0 and let D be a divisor of X. In this talk, we will advertise the existence of bounds for the Betti numbers of a local system L on X-D depending only on local numerical data: the rank of L and the ramification conductors of L at the generic points of D. If time permits, we will explain a consequence of these bounds to a form of the wild Lefschetz theorem envisioned by Deligne. This is joint work with Haoyu Hu. |
| 05.11.2024 | Andrei Yafaev (UCL) |
| Title: | On the Andre-Pink-Zannier conjecture and its generalisations, part I |
| Abstract: | This is a joint work with Rodolphe Richard (Manchester). The Andre-Pink-Zannier conjecture is a case of Zilber-Pink conjecture on unlikely intersections in Shimura varieties. We will present this conjecture and a strategy for proving it as well as its proof for Shimura varieties of abelian type. In the second talk, which will be in the Algebraic Geometry Seminar on Wednesday, we present a `hybrid conjecture' combining the recently proved Andre-Oort conjecture and Andre-Pink-Zannier. It is motivated by its analogy with Mordell-Lang for abelian varieties. We will explain this analogy as well as the proof of the hybrid conjecture for Shimura varieties of abelian type. |
| 15.11.2024 | A day of Arithmetic Geometry on the occasion of the retirement of Jürg Kramer |
| Website: | A day of Arithmetic Geometry |
| Note: | This conference takes place on a FRIDAY (not the usual seminar slot). |
| 19.11.2024 | Ya Deng (Nancy) |
| Title: | Euler Characteristic of Algebraic Varieties |
| Abstract: | This talk is based on joint works with Botong Wang. A conjecture by Chern-Hopf-Thurston states that an aspherical closed real n-manifold \(X\) satisfies \( (-1)^n\chi(X) \geq 0 \), where \( \chi(X) \) denotes the Euler characteristic of \(X \). I will focus on the case where \(X\) has the structure of a complex algebraic variety, which implies that \(X\) has large fundamental group. Inspired by this, in 1995, Kollár proposed the following conjecture: a complex projective manifold \(X\) satisfies \(\chi(K_X) \geq 0\) if it has generically large fundamental group. In this talk, I will outline the proofs of both conjectures under the assumption that \(\pi_1(X)\) is linear. |
| 26.11.2024 | Pengfei Huang (Leipzig) |
| Title: | Parahoric reduction theory of formal connections |
| Abstract: | The celebrated reduction theory of formal connections is due to Hukuhara, Levelt, Turrittin, and Babbitt-Varadarajan, among others. In this talk, we will demonstrate the parahoric reduction theory of formal parahoric connections, which generalizes the aforementioned results and also extends Boalch’s result for the case of regular singularities. As applications, we will establish the equivalence between extrinsic and intrinsic definitions of regular singularities, as well as a parahoric version of Frenkel-Zhu’s Borel reduction theorem for formal connections. This is based on a recent joint work with Z. Hu, R. Sun, and R. Zong. |
| 14.01.2025 | Ning Guo (Harbin Institute of Technology) |
| Title: | Geometric presentations and applications |
| Abstract: | Various problems concerning cohomology sets and K-theory, including Gersten's injectivity and the Grothendieck-Serre conjecture, have affirmative answers in the equi-characteristic case. A helpful tool for dealing with mixed characteristic cases is geometric presentation theorems. Roughly speaking, these realize the local ring under consideration as a relatively smooth (or étale) object over a regular DVR or a field with a low relative dimension such that problems reduce to the DVR or field cases. In this talk, we will discuss several geometric presentation theorems (due to Gabber-Quillen, Colliot-Thélène, Ojanguren, etc) and their application to the Grothendieck-Serre conjecture in mixed characteristic. In particular, I will introduce the notion of weak elementary fibrations, which traces back to Artin's "bon voisinage" in SGA4. These are individual joint works with Ivan Panin, Fei Liu and Yisheng Tian. |
| 21.01.2025 | Michael Dettweiler (Bayreuth) |
| Title: | Regular Galois Realizations of Groups of Lie Type |
| Abstract: | We survey the inverse Galois problem and state some new and old result on Galois realizations of groups of Lie type over the rational function field. Most of these can be obtained by a suitable combination of tensor products, convolutions on the affine line, and l-adic Fourier transformations. |
| 28.01.2025 | Laurent Coté (Bonn) |
| Title: | Fukaya categories of conical symplectic resolutions |
| Abstract: | Conical symplectic resolutions are a rather loosely defined class of hyperkähler varieties arising from canonical constructions in representation theory. Important examples include hypertoric varieties, Nakajima quiver varieties and Hitchin spaces. Assuming as little background as possible in symplectic geometry, I will introduce a categorical invariant called the Fukaya category, which is defined using methods of global analysis. When specialized to conical symplectic resolutions, the Fukaya category turns out to be intimately related to a category of longstanding interest in geometric representation theory, called Category O. This talk will report on joint work with (subsets of) Benjamin Gammage, Justin Hilburn, Christopher Kuo, David Nadler and Vivek Shende. |
| 04.02.2025 | Mattia Cavicchi (Dijon) |
| Title: | tba |
| Abstract: | tba |