# Conference on Geometry and Physics

The goal of this conference is to consolidate and disseminate new developments in Geometry and Physics.

## July 16, 2022

#### 10:00 - 10:50, Phillip Griffiths

Extended Period Mappings

This lecture will discuss completions of period mappings with applications to the boundary structure of moduli of general type algebraic surfaces.  It is based on joint work with Mark Green and Colleen Robles.

#### 11:00 - 11:50, Kenji Fukaya

Atiyah-Floer (2) functor

This talk is based on a joint work with A. Daemi and partially with M. Lipyanskiy. We discuss SO(3) version of Atiyah Floer conjecture (which is a relation between Instanton Floer homology of 3 manifold (based on gauge theory) and Lagrangian Floer theory).

#### 12:00 - 12:50, Jørgen Ellegaard Andersen

Generalized Hitchin Connections with applications to quantization of Moduli Space of parabolic vector bundles

We will review our approach to the generalization of Hitchin connection and demonstrate how it can be applied to construct a projectively flat connection in the bundle over moduli space of curves obtained by applying geometric quantization to the moduli spaces of parabolic vector bundles. For the case of genus zero and certain weights we recall our results that this projectively flat connection is projectively equivalent to the KZ-connection. The talk ends with a discussion of new results showing that our generalized Hitchin connection construction applies to the very general situation of arbitrary families of complex structures on symplectic manifolds.

Lunch

#### 15:00 - 15:50, Yong-Geun Oh

Contact instantons and Sandon-Shelukhin’s conjecture

In this talk,  we  will explain the proof of a conjecture by Sandon and Shelukhin which is a contact analog to the Arnold’s conjecture type: there exists a translated point for any contactomorphism isotopic to the identity whose oscillation norm is smaller than the period gap of compact contact manifold , and at least the Betti number of translated points if the  contactomorphism is nondegenerate. The main machinery we employ is combination of contact Hamiltonian geometry and analysis of Hamiltonian-perturbed contact instantons with the Legendrian boundary condition.

#### 16:00 - 16:50, Karim Adiprasito

Beyond positivity for lattice polytopes

The semigroup algebra of an IDP reflexive lattice polytope was shown to be Gorenstein by Hochster. We compute the fundamental class and conclude a Lefschetz property in char 0. In particular, the h* polynomial has unimodular coefficients.

#### 17:00 - 17:50, Tony Pantev

Generalized complex branes, doubling, and shifted symplectic geometry

I will describe a symplectic approach to generalized complex geometry which by design produces a natural category of gc branes (at the large volume limit) and allows us to define and work with gc branes with substrates which are not submanifolds. The approach utilizes shifted symplectic geometry and a new theory of homotopy complex structures on derived stacks. I will discuss briefly this theory and will explain its implications for gc geometry. This is a report of a joint work in progress with P. Safronov and B. Pym and on recent work of Y. Qin.

Nessebar Old Town Sightseeing and Dinner

## July 17, 2022

#### 09:00 - 09:50, Oscar García-Prada

Kähler–Yang–Mills equations and gravitating vortices

I will start introducing the Kähler-Yang-Mills equations on a holomorphic vector bundle over a compact complex manifold. These equations, inspired by the
Donaldson–Uhlenbeck–Yau correspondence for bundles and the Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metrics, intertwine the curvature of a Hermitian–Yang–Mills connection on the bundle  and the scalar curvature of a Kähler metric on the manifold. After this, I will consider special symmetric solutions on a compact Riemann surface known as gravitating vortices.

#### 10:00 - 10:50, Kyoung-Seog Lee

Vector bundles on elliptic surfaces and logarithmic transformations

Logarithmic transformation is an important operation introduced by Kodaira in the 1960s. One can obtain an elliptic surface with multiple fibers by performing logarithmic transformations on an elliptic surface without multiple fibers. On the other hand, vector bundles on elliptic surfaces are important objects in many branches of mathematics, e.g., algebraic geometry, gauge theory, mathematical physics, etc. In this talk, I will discuss how certain vector bundles on elliptic surfaces are changed via logarithmic transformations. This talk is based on a joint work in progress with Ludmil Katzarkov.

#### 11:00 - 11:50, Ernesto Lupercio

A Localization Principle for Orbifold Theories

In this talk we survey several manifestations of a general localization principle for orbifold theories such as K-theory, index theory, motivic integration, and elliptic genera. joint work with Tommaso de Fernex, Ernesto Lupercio, Thomas Nevins, and Bernardo Uribe

Lunch

#### 16:00 - 16:50, Dennis Borisov

Dg Quot-stacks equipped with -2-shifted symplectic forms

I will discuss construction of dg Quot-schemes and their relation to derived schemes. Dg Quot-stacks of coherent sheaves on Calabi-Yau 4-folds carry -2-shifted symplectic structures. Switching to smooth real valued functions one can construct Lagrangian distributions for these shifted symplectic structures, resulting in globally defined -1-shifted potentials. This should be related to the moduli space of Spin(7)-instantons.

Joint work with L.Katzarkov, A. Sheshmani, S-T Yau.

#### 17:00 - 17:50, Josef Svoboda

$$\Zhat_a-$$invariants of Seifert manifolds and splicing
Based on joint work with L. Katzarkov and S. Gukov, I will show how to simplify the formulas for Z-hat invariants of Gukov, Putrov, Pei, Vafa in the case of Seifert manifolds. Next I will explain why the sum of Z-hat series over spin-c structures only depends on less combinatorial data – the splicing diagram. This is a much more general statement. Finally I will comment on modularity properties of these functions.

## July 18, 2022

#### 09:00 - 09:50, Richard Paul Horja

Toric birational geometry and discriminants

Given a Gorenstein toric singularity, I will explain a proposal for the B-side categories appearing in toric homological mirror symmetry along the strata of the associated discriminant locus in the combinatorial language of Gelfand, Kapranov and Zelevinsky. A conjectural construction of the web of associated spherical functors put forward by Aspinwall, Plesser and Wang and some K-theoretic supporting evidence will be discussed. This is joint work with Ludmil Katzarkov.

#### 10:00 - 10:50, Meral Tosun

Singularities, a new root system and linear free divisors

Abstract: We present two relations between Lie algebras and singularities of surfaces. We construct a new root system and produce linear free divisors.

Lunch

## Participants

Alexander Petkov, Institute of Mathematics and Informatics, Sofia and Sofia University
Antoni Rangachev, Institute of Mathematics and Informatics, Sofia
Artan Sheshmani, Harvard University
Dennis Borisov, University of Windsor
Enrique Ruby Becerra, Institute of Mathematics and Informatics, Sofia and University of Miami
Erik Paemurru, Institute of Mathematics and Informatics, Sofia and University of Miami
Ernesto Lupercio, Cinvestav-IPN
Jørgen Ellegaard Andersen, Centre for Quantum Mathematics, University of Southern Denmark
Josef Svoboda, University of Miami
Julian Revalski, Bulgarian Academy of Sciences
Kenji Fukaya, Simons Center for Geometry and Physics at Stony Brook
Kyoung-Seog Lee, University of Miami
Ludmil Katzarkov, Institute of Mathematics and Informatics, Sofia and University of Miami
Marin Genov, Institute of Mathematics and Informatics, Sofia
Meral Tosun, Galatasaray University
Mina Teicher, University of Miami
Oscar García-Prada, Instituto de Ciencias Matemáticas (ICMAT)
Peter Boyvalenkov, Institute of Mathematics and Informatics, Sofia
Phillip Griffiths, Institute for Advanced Study
Rene Mboro, Institute of Mathematics and Informatics, Sofia and University of Miami
Richard Paul Horja, University of Miami
Robert Stephen Cantrell, University of Miami
Rodolfo Aguilar, Institute of Mathematics and Informatics, Sofia and University of Miami
Sebastian Torres, Institute of Mathematics and Informatics, Sofia and University of Miami
Stefan Ivanov, Institute of Mathematics and Informatics, Sofia and Sofia University
Tokio Sasaki, Institute of Mathematics and Informatics, Sofia and University of Miami
Tony Pantev, University of Pennsylvania
Velichka Milousheva, Institute of Mathematics and Informatics, Sofia
Yong-Geun Oh, IBS Center for Geometry and Physics

Supported by the Simons Foundation and the Ministry of Education and Science of the Republic of Bulgaria