Categorical and analytic invariants in algebraic, symplectic and complex geometry

3 Feb 2025, 10:00 → 7 Feb 2025, 18:00 Asia/Tokyo

Lecture Hall (Kavli IPMU)

Dates: February 3 - 7, 2025

Venue: Kavli IPMU, Lecture Hall

Overview:

The goal of the conference is to gather the world leading experts and young scientists working in various areas of geometry and related algebra. The aim is to boost the interaction of mathematicians working in algebraic, symplectic and complex geometry by categorical and analytic methods in order to transfer ideas between these areas of geometry, using category theory as the unifying tool. Mirror Symmetry, originated in QFT and developed by mathematicians as a deep relation between complex and symplectic geometries, is clearly the guiding thread in this activity. The extension of the categorical description to non-commutative algebraic geometry will also be one of the focuses of the conference. 

Time schedule:

Click here

Invited speakers: 

• Rina Anno (Kansas State University)
• Huijun Fan (Peking University)
• Wahei Hara (Kavli IPMU)
• Lutz Hille (University of Münster)
• Osamu Iyama (University of Tokyo)
• Mikhail Kapranov (Kavli IPMU)
• Dogancan Karabas (Kavli IPMU)
• Tatsuki Kuwagaki (Kyoto University)
• Timothy Logvinenko (Cardiff University)
• Alexei Lvov (St. Petersburg University)
• Katherine Maxwell (Kavli IPMU)
• Todor Milanov (Kavli IPMU)
• Shinnosuke Okawa (Osaka University)
• Kyoji Saito (RIMS)
• Atsushi Takahashi (Osaka Univeristy)
• Ryo Takahashi (Nagoya University)
• Sofia Tirabassi (Stockholm University)
• Yukinobu Toda (Kavli IPMU)

 

Organizing Committee Members:

• Agnieszka Bodzenta (University of Warsaw)
• Alexey Bondal (Kavli IPMU, Steklov Math. Institute)
• Shinnosuke Okawa (Osaka University)

---

Address:

Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),

the University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa City, Chiba 277-8583, Japan

---

 

program-conf-feb2025.pdf

Monday 3 February

10:00 → 11:00 TBA (Osamu Iyama)

Speaker: Osamu Iyama

11:00 → 11:30 Tea break

11:30 → 12:30 TBA (Shinnosuke Okawa)

Speaker: Shinnosuke Okawa

12:30 → 14:00 Lunch break

14:00 → 15:00 TBA (Lutz Hille)

Speaker: Lutz Hille

15:00 → 16:00 Tea break

16:00 → 17:00 Finiteness of Orlov spectra of singularity categories

The Orlov spectrum of a triangulated category is the set of generation times of strong generators. Ballard, Favero and Katzarkov proved that the singularity category of a hypersurface isolated singularity has finite Orlov spectrum. In this talk, we will introduce the new notion of uniformly dominant local rings. We will show that the singularity category of a uniformly dominant isolated singularity has finite Orlov spectrum, and consider when a given local ring is uniformly dominant.

Speaker: Ryo Takahashi

Tuesday 4 February

10:00 → 11:00 Semi-infinite Hodge structure associated with hyperbolic root systems

It is well-known that there exist semi-infinite Hodge structure associated to finite or
elliptic root systems (which describes the lattice of vanishing cycles for either simple
or elliptic root systems). Recently, we found that the semi-infinite Hodge structure
exist for hyperbolic root systems of rank 2. This is a surprise, since the hyperbolic
root systems do not have geometric origin so the they behaves quite differently than
the above classical cases (e.g. some eigenvalues of monodromy are not root of unity
but real). In the present talk, we will describe the construction down to the earth.

Speaker: Kyoji Saito

11:00 → 11:30 Tea break

11:30 → 12:30 TBA (Atsushi Takahashi)

Speaker: Atsushi Takahashi

12:30 → 14:00 Lunch break

14:00 → 15:00 Fukaya category of Landau-Ginzburg model via Witten equation

Landau-Ginzburg model has become a cornerstone theory of global mirror symmetry.
The closed string A-theory of a LG model has already been built, and is well-known
as the quantum singularity theory (or FJRW theory). An open string theory of a LG
model has also been treated in the paper “Fukaya Category of Landau-Ginzburg
model, arXiv:18012.11748v1”, but with not much attention. In this talk, I will recall
the construction in this paper, which is related to the boundary value problem of the
Witten equations arising from Landau-Ginzburg model, and mention the Maurer-
Cartan element conjecture proposed by Gaiotto-Moore-Witten (or Kapranov-
Kontsevich-Soibelman).

Speaker: Huijun Fan

15:00 → 16:00 Tea break

16:00 → 17:00 Genus-0 permutation-equivariant KGW invariants of the point

K-theoretic Gromov--Witteh (KGW) theory was introduced by Givental and Y.P. Lee as a generalization of Gromov--Witten theory. Recently, Givental realised that if we want to compute KGW invariants via fixed-point localization methods, we have to consider a more general theory, i.e., the permutation equivariant version of KGW theory. I would like to give an introduction to this topic and to explain how to compute the invariants in genus-0 for the simplest possible target -- the point.

Speaker: Todor Milanov

Wednesday 5 February

10:00 → 11:00 Effective characterizations of semi-abelian varieties

I will show how three logaritmic plurigenera and the logarithic irregularity are
enough to characterize semi-abelian surfaces among the quasi-projective
surfaces. I will also present some results for higher dimensional varieties in a
very special case. This is joint work with Mendes Lopes and Pardini and a work in progress with J. Baudin.

Speaker: Sofia Tirabassi

11:00 → 11:30 Tea break

11:30 → 12:30 Derived equivalence for the simple flop of type $G_2^{\dagger}$

In this talk we discuss an example of a simple flop that was found by Kanemitsu,
from the point of view of derived categories. A simple flop is a flop between two
smooth varieties that is connected by one smooth blow-up and one smooth blow-
down, and those flops were partially classified by Kanemitsu, using Dynkin data.
The exceptional divisor of the blow-ups has two projective bundle structures of the
same rank, and is called a roof. The simple flop of type $G_2^{\dagger}$, which we
discuss in this talk, is the only known example of a simple flop that has the non-
homogeneous roof. The main theorem of the talk is that the simple flop of type
$G_2^{\dagger}$ gives a derived equivalence. The proof is done by using tilting
bundles, and hence it also produces a noncommutative crepant resolution that is
derived equivalent to both sides of the flop. Despite its Dynkin label, the construction
of the tilting bundles is related to rational homogeneous manifolds of Dynkin type
$B_3$ and $D_4$.

Speaker: Wahei Hara

12:30 → 17:00 Free afternoon

Thursday 6 February

10:00 → 11:00 TBA (Mikhail Kapranov)

Speaker: Mikhail Kapranov

11:00 → 11:30 Tea break

11:30 → 12:30 TBA (Rina Anno)

Speaker: Rina Anno

12:30 → 14:00 Lunch break

14:00 → 15:00 Dolbeault Geometric Langlands conjecture via quasi-BPS categories

In this talk, I will introduce the notion of `limit category' for cotangents
of smooth stacks, which is expected to give a categorical degeneration of the category
of D-modules on them. I show that the limit category for the moduli stack of Higgs
bundles admits a semiorthogonal decomposition into products of quasi-BPS categories,
which are categorifications of BPS invariants of some non-compact Calabi-Yau 3-folds.
I propose the formulation of Dolbeault Geometric Langlands conjecture using the
limit category, which is regarded as a classical limit of Geometric Langlands correspondence.
I also show that the limit category admits Hecke operators. This is a joint work in
progress with Tudor Padurariu.

Speaker: Yukinobu Toda

15:00 → 16:00 Tea break

16:00 → 17:00 TBA (Timothy Logvinenko)

Speaker: Timothy Logvinenko

Friday 7 February

10:00 → 11:00 TBA (Katherine Maxwell)

Speaker: Katherine Maxwell

11:00 → 11:30 Tea break

11:30 → 12:30 TBA (Alexei Lvov)

Speaker: Alexei Lvov

12:30 → 14:00 Lunch break

14:00 → 15:00 Wrapped and compact Fukaya categories of plumbings

Given any finite quiver Q, where each vertex corresponds to a fixed Lagrangian $L_v$, I will describe an associated symplectic manifold known as the plumbing of $T^*L_v$'s along Q. Using a local-to-global approach, I will explain how their wrapped Fukaya category can be expressed as a Ginzburg dg algebra with based loop space coefficients or a derived multiplicative preprojective algebra. In the second part of my talk, I will demonstrate that microlocal sheaves on the union of $L_v$'s recover the compact Fukaya category of the plumbing, generalising the Nadler-Zaslow correspondence for cotangent bundles. The first part is joint work with Sangjin Lee (arXiv:2405.10783), and the second part is ongoing work with Sangjin Lee and Wonbo Jeong.

Speaker: Dogancan Karabas

15:00 → 16:00 Tea break

16:00 → 17:00 Hodge microsheaves on cotangent bundles and plumbings

The theory of Hodge microsheaves aims at generalizing the theory of mixed Hodge modules in twofold: (1) "infinite-dimensional" like wrapped sheaves of Nadler, (2) "microlocal" in the style of Bezrukavnikov-Kapranov. In this talk, I'll explain some background philosophy and some nontrivial computational results in the theory, based on joint work with Takahiro Saito.

Speaker: Tatsuki Kuwagaki