**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: 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.

**Past Talk:**

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.