**Upcoming Talk:**

Organizer: Yu-Wei Fan (YMSC)

Further information about the seminar can be found at: https://ywfan-math.github.io/GDS.html

Date and Time: Sep 25 (Wednesday) at 1:30-2:30pm

Zoom Meeting ID: 815 762 8413 (Passcode: BIMSA)

Speaker: Shizhuo Zhang (IBS-CGP)

Title: 1-nodal prime Fano threefolds parametrized by Bridgeland stable objects in Kuznetsov component of del Pezzo threefolds

Abstract:

Let $X$ be a smooth Fano threefold of index one and genus 8, a classical result tells us that $X$ is uniquely determined by a smooth cubic threefold $Y$ and a rank two instanton bundle on it. First, I will show that in the modern categorical language, $X$ is uniquely determined by its Kuznetsov component \Ku(X) and a distinguished object inside it. Then I will describe a conjectural picture for prime Fano threefolds of other genus. Second, I extend the conjectural picture from smooth cases to nodal prime Fano threefold cases and prove part of the conjecture. Namely, a 1-nodal maximally non-factorial prime Fano threefold of genus g=2d+2 coming from the so-called bridge construction is uniquely determined by a smooth del Pezzo threefold of degree d and an (acyclic extension) of a stable non-locally free instanton sheaf of rank two and charge d-1. Equivalently, each X is determined by \Ku(X) and a distinguished object inside the Kuznetsov component. All these facts support a conjecture that those Fano threefolds at most 1-nodal maximally non-factorial are parametrized by a Bridgeland moduli space of stable objects of character (d-1) multiple ideal sheaves of line in Kuznetsov component of degree d del Pezzo threefold. This talk is based on a joint work with Daniele Faenzi and Xun Lin.

**Past Talk:**

Date and Time: May 29 (Wednesday) at 1:30-2:30pm

Venue: Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Fei Yu (Zhejiang University)

Title: Hassett-Keel program and moduli spaces of Abelian differentials

Abstract:

The Hassett-Keel program seeks to offer a modular interpretation for the log minimal model program of the moduli space of stable curves. Alper, Fedorchuk, and Smyth have proposed a conjectural framework for this program by analyzing Gorenstein curve singularities under the action of G_m. Our work draws parallels between this approach and the study of the Kontsevich-Zorich conjectures pertaining to moduli spaces of Abelian differentials. We address pertinent questions in this area by employing test configurations and the Ree construction, methodologies that stem from the theory of K-stability. This research is a collaborative effort with Dawei Chen and is currently in progress.

Date and Time: May 22 (Wednesday) at 1:30-2:30pm

Venue: Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Xun Yu (Tianjin University)

Title: Automorphism groups of smooth hypersurfaces

Abstract:

In this talk, I will discuss some recent results about classifying automorphism groups of smooth hypersurfaces in the projective space. This talk is based on joint works with Keiji Oguiso, Li Wei, Song Yang, and Zigang Zhu.

Date and Time: May 15 (Wednesday) at 12:30-1:30pm (special time!)

Venue: Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Asilata Bapat (Australian National University)

Title: Categorical q-deformed rational numbers via Bridgeland stability conditions

Abstract:

We will discuss new categorical interpretations of two distinct q-deformations of the rational numbers. The first one, introduced by Morier-Genoud and Ovsienko in a different context, enjoys fascinating combinatorial, topological, and algebraic properties. The second one is a natural partner to the first, and is new. We obtain these deformations via boundary points of a compactification of the space of Bridgeland stability conditions on the 2-Calabi--Yau category of the A2 quiver. The talk is based on joint work with Louis Becker, Anand Deopurkar, and Anthony Licata.

Date and Time: May 8 (Wednesday) at 1:30-2:30pm

Venue: Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Hiroki Matsui (Tokushima University)

Title: The spectrum of a triangulated category and its applications to algebraic geometry

Abstract:

Tensor triangular geometry is an epoch-making theory initiated by Balmer. For a tensor triangulated category, Balmer defined the ringed space called the Balmer spectrum of T, whose underlying set consists of prime thick ideals of T. This theory has been successfully applied to the study of tensor triangulated categories appearing in various areas of mathematics. However, this theory cannot be applied to triangulated categories without tensor structures, and it is badly behaved concerning triangulated functors that do not preserve tensor structures. In this talk, we define the ringed space for a given triangulated category without using tensor structure and develop a tensor-free analog of Balmer’s tensor triangular geometry. Also, we apply the theory to study perfect derived categories of algebraic varieties, Fourier-Mukai partners, and reconstruction problems.

Date and Time: April 17 (Wednesday) at 1:30-2:30pm

Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Takumi Otani (Tsinghua University)

Title: The number of full exceptional collections for orbifold projective lines

Abstract:

The derived category of an orbifold projective line with positive Euler characteristic is equivalent to the one of an extended Dynkin quiver. For a Dynkin quiver, Obaid—Nauman—Shammakh—Fakieh—Ringel gave a counting formula for the number of full exceptional collections in the derived category. The number coincides with the degree of the Lyashko—Looijenga map for an ADE singularity. The equality of these numbers hints a consistency in some problems in Bridgeland stability conditions and mirror symmetry. In this talk, I will give a formula for the number of full exceptional collections for an orbifold projective line, which can be regarded as a generalization for Dynkin cases. Based on mirror symmetry, I will explain the relationship between the number and the degree of the Lyashko—Looijenga map for the orbifold projective line. This talk is based on a joint work with Yuuki Shiraishi and Atsushi Takahashi.

Date and Time: April 10 (Wednesday) at 1:30-2:30pm

Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Jun Zhang (University of Science and Technology of China)

Title: Metric geometry on Grothendieck groups in symplectic geometry

Abstract:

In this talk, we will introduce a new method to carry out quantitative studies on the Grothendieck group of a derived Fukaya category. This fits into a bigger algebraic framework called triangulated persistence category (TPC). This category unites the persistence module structure (from topological data analysis) and the classical triangulated structure so that a meaningful measurement, via cone decomposition, can be defined on the set of objects. In particular, a TPC structure allows us to define non-trivial pseudo-metrics on its Grothendieck group, which is the first time that people can study a Grothendieck group in terms of the metric geometry. Finally, we will illustrate how to use this method to distinguish classes from the Grothendieck group (of a derived Fukaya category) from a quantitative perspective. This is based on joint work with Paul Biran and Octav Cornea.

Date and Time: March 27 (Wednesday) at 1:30-2:30pm

Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Ziming Nikolas Ma (Southern University of Science and Technology)

Title: Deformation theory, Fukaya's conjecture and the Gross-Siebert program

Abstract:

In this talk, we review the story of constructing mirror Calabi-Yau manifold from a Lagrangrain torus fibration, beginning with Fukaya’s conjectural construction using gradient flow trees and relate it to the Gross-Siebert program via a construction of a Kodaira-Spencer type dgLa. If time allows, we will discuss the construction of B-model Frobenius manifold using this dgLa/dgBVa. This is base on a series of joint work with Kwokwai Chan and Naichung Conan Leung.

Date and Time: March 20 (Wednesday) at 1:30-2:30pm

Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Speaker: Yat-Hin Suen (Korea Institute for Advanced Study)

Title: Toric vector bundles, non-abelianisation, and spectral networks

Abstract:

Spectral networks and non-abelianization were introduced by Gaiotto-Moore-Neitzke and they have many applications in mathematics and physics. In a recent work by Nho, he proved that the non-abelianization of an almost flat local system over the spectral curve of a meromorphic quadratic differential is actually the same as the family Floer construction. Based on the mirror symmetry philosophy, it is then natural to ask how holomorphic vector bundles arise from spectral networks and non-abelianization. In this paper, we construct toric vector bundles on complete toric surfaces via spectral networks and non-abelianization arising from Lagrangian multi-sections.

Title: Isoresidual fibration and resonance arrangements

Speaker: Guillaume Tahar (BIMSA)

Date and Time: March 13 (Wednesday) at 1:30-2:30pm

Zoom Meeting ID: 928 682 9093 (Passcode: BIMSA)

Abstract:

Meromorphic 1-forms on the Riemann sphere with prescribed orders of singularities form strata endowed with period coordinates. Fixing residues at the poles defines a fibration of any stratum to the vector space of configurations of residues. In a joint work with Quentin Gendron, it has been proved that for strata of 1-forms with only one zero, the isoresidual fibration is a cover of the space of configurations of residues ramified over an arrangement of complex hyperplanes called the resonance arrangement. Using combinatorics of decorated tree and the dictionary between complex analysis and flat geometry, we give a formula to compute the degree of this cover and investigate its monodromy. In a more recent work with Dawei Chen, Quentin Gendron and Miguel Prado, we investigate the case of strata with two zeroes where isoresidual fibers are complex curves endowed with a canonical translation structure. Singularities of this structure provide topological invariants of the fibers that refine the Euler characteristic and still lack an interpretation in terms of enumerative geometry.