# Conference on Geometry and Physics

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

## Programme

## July 16, 2022

**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.

**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).

**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

**Contact instantons and Sandon-Shelukhin’s conjectur**e

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.

**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.

**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

**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.

**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.

**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

**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.

**$$\Zhat_a-$$invariants of Seifert manifolds and splicing**

## July 18, 2022

**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.

**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

**Karim Adiprasito,** University of Copenhagen

**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