# Edging Higher

# ICMS-Sofia Inaugural Conference

July 11 – 14, 2022

Grand Hotel Sofia

Sofia, Bulgaria

ICMS – Sofia Inaugural Conference

# ICMS – Sofia Inaugural Conference

The International Center for Mathematical Sciences (ICMS-Sofia) is a dynamic research unit for developing and dissemination of cutting edge new directions in Mathematics.

The Center was created following inspirational discussions with many members of the mathematical community in Bulgaria and the Bulgarian mathematical diaspora. In July 2019, ICMS-Sofia initiated its full-scale presence on the European mathematical scene with the full support of Acad. Julian Revalski, President of the Bulgarian Academy of Sciences. We have secured funds from the Ministry of Education and Science of the Republic of Bulgaria and Simons Foundation. We are very grateful for their generous help.

This conference will serve as a unique opportunity for consolidation of our efforts and lifting our activities to the next level.

We are looking forward to meeting the challenges of our time and transforming ICMS into a major regional center of cutting edge collaborations, transfer of knowledge and reintegration of Bulgarian scientists.

Supported by the Ministry of Education and Science of the Republic of Bulgaria through the Scientific Programme “Enhancing the Research Capacity in Mathematical Sciences (PIKOM)”, Agreement № DO1-67/05.05.2022

We can offer financial support (up to 600 EUR) to PhD students and young researchers wishing to participate in the conference. The applications should be sent to Lilia Simeonova, e-mail: liliasim@math.bas.bg by **June 27, 2022**. Preferences will be given to displaced scientists.

We will be happy to accept participants supported by their own funds.

Invited Speakers

# Invited Speakers

**Andras Nemethi**, Alfréd Rényi Institute of Mathematics, Hungary**Dennis Gaitsgory**, Harvard University, USA**Ernesto Lupercio**, Cinvestav-IPN, México**Jørgen Ellegaard Andersen**, Centre for Quantum Mathematics, University of Southern Denmark**Jean-Pierre Bourguignon**, Honorary Professor, Institut des Hautes Études Scientifiques, Bures-sur-Yvette (France)**John Pardon**, Princeton University, USA**Karim Adiprasito**, University of Copenhagen, Denmark**Kenji Fukaya**, Simons Center for Geometry and Physics at Stony Brook, USA**Liana David,**Institute of Mathematics “Simion Stoilow”, Romanian Academy, Romania**Maryna Viazovska**, École Polytechnique Fédérale de Lausanne, Switzerland**Maxim Kontsevich**, Institut des Hautes Études Scientifiques, France**Mina Teicher**, University of Miami, USA**Oscar García-Prada,**Instituto de Ciencias Matemáticas (ICMAT), Spain**Phillip Griffiths**, Institute for Advanced Study**Robert Stephen Cantrell,**University of Miami, USA**Tony Pantev,**University of Pennsylvania, USA**Vestislav Apostolov**, Université du Québec à Montréal, Canada**Vivek Shende**, Centre for Quantum Mathematics, Syddansk Universitet, Denmark**Yong-Geun Oh**, IBS Center for Geometry and Physics, Korea

# Programme

## July 11, 2022 (Monday)

09:00 – Registration

10:30 – Opening and Award Ceremony for Maxim Kontsevich, Foreign Member of the Bulgarian Academy of Sciences

**Matrix models in the planar limit**

This talk is inspired by recent results by V.Kazakov and Z.Zheng on the analytic bootstrap for matrix models.

Observables in the planar limit satisfy an infinite system of quadratic equations, called loop equations (analogs of Schwinger-Dyson equations/Ward identities, meaning that the integral of a total derivative is zero.)

Naive counting gives more equations than variables, i.e. the system of loop equations is over-determined. Nevertheless, it has non-trivial solutions, depending on a finite or infinite number of parameters (for different models). I’ll talk about the cohomological mechanism behind this phenomenon.

12:00 – Wellcome Drink

**Hodge theory: What is it? How can it be used in algebraic geometry?**

This will be an informal, expository talk on the subjects in the title. Emphasis will be on how Hodge theory may be used to construct something geometric.

15:00 – Coffee Break

**Revisiting the Scalar Curvature**

16:30 – Break

**Moduli of flat connections on smooth varieties**

I will discuss the derived geometry of the moduli of local systems and flat bundles on a smooth but not necessarily proper complex algebraic variety.

In the Betti case I will show that these moduli carry natural Poisson structures, generalizing the well known case of curves. I will construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity, and show that a new feature, called strictness, appears as soon as the divisor at infinity has non-trivial crossings.

In the de Rham case I will introduce the notion of a formal boundary of the variety

and will explain how to define a restriction to the boundary map between derived moduli of flat bundles. I will discuss representability results for the geometric fibers of the restriction map and will explain why the restriction map is equipped with a canonical shifted Lagrangian structure.

This is a joint work with Bertrand Toen.

## July 12, 2022 (Tuesday)

**The stack of local systems with restricted variation and geometric Langlands with nilpotent singular support**

We will introduce a new geometric object: the stack of local systems with restricted variation. Using it, we will be able to formulate a version of the geometric Langlands conjecture that makes sense for etale sheaves over an arbitrary ground field; the geometric of the conjecture is the category of automorphic sheaves with nilpotent singular support. We will combine it with a Trace Isomorphism Theorem, to give a description of the space of unramified automorphic functions in terms of Langlands parameters.

This is a joint work with Arinkin, Kazhdan, Raskin, Rozenblyum and Varshavsky.

11:00 – Coffee Break

**Higgs bundles and higher Teichmüller spaces**

It is well-known that the Teichmüller space of a compact real surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R).

Higher Teichmüller spaces are generalizations of this, where PSL(2,R) is replaced by certain simple non-compact real Lie groups of higher rank. As for the usual Teichmüller space, these spaces consist entirely of discrete and faithful representations. Several cases have been identified over the years. First, the Hitchin components for split groups, then the maximal Toledo invariant components for Hermitian groups, and more recently certain components for SO(p,q). In this talk, I will describe a general construction of all possible higher Teichmüller spaces, and a parametrization of them using the theory of Higgs bundles, given in joint work with Bradlow, Collier, Gothen and Oliveira.

12:30 – Break

**Nonequilibrium thermodynamics as a contact reduction**

Starting from Hermann, contact geometry is utilized in the geometric formulation of the thermodynamics, especially of equilibrium thermodynamical process in mathematical physics. It has been observed in this formulation that the state of thermodynamic equilibrium can be interpreted as a Legendrian submanifold in the thermodynamic phase space (TPS) equipped with a contact form.

In this lecture, we will explain the origin of aforementioned contact structure in TPS as a contact reduction of the canonical contact structure associated to the classical phase space of probability distributions on the statistical phase space of many-body systems (SPS), which we call the kinetic theory phase space (KTPS). A particular role of Gibbs’ entropy (or Shanon’s entropy in Information Theory) will be emphasized in this derivation. This is based on a joint work with my student Jinwook Lim.

(If time permits, I will also relate how this work is motivated by and related to the study of Hamiltonian-perturbed contact instantons and its applications to the study of contact topology and contact Hamiltonian dynamics.)

15:00 – Coffee Break

**Generating function and Immersed Lagrangian Floer theory**

Generating function is a way to obtain Lagrangian submanifold of the cotangent bundle.

Various people studies such a Lagrangian submanifold by using Morse theory of the generating function. In the case Lagrangian submanifold is embedded it is an exact Lagrangian submanifold and one can use Floer theory also. In this talk I will explain how we can extend this story to the immersed cases.

16:30 – Break

**Cosheaves of wrapped Fukaya categories**

Joint work with Sheel Ganatra and Vivek Shende established a “descent formula” for wrapped Fukaya categories. This states that the wrapped Fukaya category of a Liouville manifold can be recovered from the wrapped Fukaya categories of the elements of an open cover and their multiple intersections, using a certain homotopy colimit construction. I will discuss our more recent work on how to reformulate this result as the assertion that the wrapped Fukaya category is a cosheaf with respect to a certain “topology” on Weinstein manifolds/sectors.

## July 13, 2022 (Wednesday)

**From neurotech to fintech **

In the talk I’ll explain the common features between brain activity and financial markets and how to bridge these two complex systems.

In particular, I’ll explains *black swan phenomena*, and present an algorithm for detection.

11:00 – Coffee Break

**Criticality in Artificial Models of Life**

In this talk, aimed at a general audience, we will do a very brief review of the fields of artificial life and self-organized criticality and finally see how the second informs the first, enhancing our understanding of biological systems. This talk is based on joint work with Kalinin and Shkolnikov, and with Cruz, Muñoz, Tabares, and Viafara.

12:30 – Break

**Resource matching in spatial ecology and evolutionary advantage**

A convergence of concepts from game theory (evolutionary stable strategy), ecological theory (the ideal free distribution), and mathematics (line sum-symmetry and its functional analytic generalizations) combine to explain how resource matching in spatially heterogeneous but temporally constant habitats can convey evolutionary advantage robustly across a range of modeling formats. The ideal free distribution (IFD) is an ecological construct that posits that when organisms have full knowledge of the landscape they inhabit (ideal) and are able to locate themselves as they wish (free), they will locate themselves to maximize reproductive fitness. The IFD can readily be translated into mathematical terms for models wherein the environment is spatially varying but temporally constant. In this talk we will discuss how this is done across a range of modeling formats and how it consequently leads to predictions of evolutionary advantage in such modeling formats. We then report on ongoing efforts to define the ideal free distribution mathematically in cases when the environment varies in both space and time, focusing on the case wherein habitats vary periodically in time.

15:00 – Coffee Break

**Perverse microsheaves**

I will review the construction of the category of microsheaves, which is an invariant associated to contact (or exact symplectic) manifolds, and its dependence on Maslov data. Then I will explain how for complex contact (or exact symplectic) manifolds, there is a canonical Maslov datum, and, for similar reasons, a t-structure. This t-structure agrees locally with the middle perverse t-structure on constructible sheaves.

16:30 – Break

**A survey of combinatorial algebraic geometry**

19:00 – Gala Dinner

## July 14, 2022 (Thursday)

**Fourier interpolation pairs and their applications**

Abstract: This lecture is about Fourier uniqueness and Fourier interpolation pairs, some explicit constructions of such pairs and their applications. Suppose that we have two subsets X and Y of the Euclidean space. Can we reconstruct a function f from its restriction to the set X and the restriction of its Fourier transform to the set Y? We are interested in the pairs (X,Y) such that the answer to the question above is affirmative. In this talk I will give an overview of recent progress on explicit constructions and existence results for Fourier interpolation pairs and corresponding interpolation formulas. Also I will try to convince you that explicit Fourier interpolation is a useful gadget in solving optimization problem and analysing differential equations.

11:00 – Coffee Break

**Bn-generalized pseudo-Kahler structures**

I will start with a short review on generalized Kähler geometry on exact Courant algebroids and its relation to bi-Hermitian geometry. Then I will define the notion of Bn – generalized (pseudo)-Kähler structure on odd exact Courant algebroids. I will express Bn generalized (pseudo)-Kähler structures in terms of classical tensor fields on the base of the Courant algebroid. I will end my talk with left invariant examples over Lie groups of dimension two, three or four. This is joint work with Vicente Cortes.

12:30 – Break

**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.

15:00 – Coffee Break

**Weighted k-stability of Kähler manifolds with application to scalar-flat Kähler cones**

In this talk, I will review some recent developments in the theory of constant scalar curvature Kähler metrics, and discuss an equivalence between scalar-flat affine Kähler cones and weighted extremal Kähler metrics in the sense of A. Lahdili. In the case of Calabi-Yau cones, this correspondence yields a special case of a v-soliton on a Fano variety, studied by Berman-Witt Nystrom and Han-Li. This provides an alternative approach — entirely within the framework of Kähler geometry — to the K-stability of affine complex cones associated to Sasaki polarizations, a notion originally defined by Collins and Székelyhidi. We will use this and a recent work by Han-Li to show that the variational approach to special Kähler metrics can be applied to prove that scalar-flat cones are uniformly K-stable in a weighted sense, thus improving upon the previously known K-semistability. Based on joint works with D. Calderbank and E. Legendre, and with A. Lahdili and S. Jubert.

16:30 – Break

**Lattice cohomology of curve singularities**

We introduce a graded Z[U]-module, the “lattice cohomology of a reduced curve singularity”, associated with the analytic type of the singularity. It is the categorification of the delta invariant. In the case of plane curve singularities it can be recovered from the embedded topological type too. Analytic flat deformations (of any embedded dimension) induce graded Z[U]-module morphisms between the corresponding cohomology modules.

The theory is part of a series of constructions, where we define the analytic lattice cohomology of any isolated singularity of any dimension. In the surface case it is the analytic pair of the topological lattice cohomology, which is the categorification of the Seiberg-Witten invariant (and it coincides with the Heegaard Floer theory).

It is a joint work with T. Agoston.

Registered Participants

# Registered Participants

**Alexander Petkov,** Sofia University, Bulgaria

**Andras Nemethi**, Alfréd Rényi Institute of Mathematics, Hungary

**Angela Slavova**, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

**Anna Branzova,** Institute of Mathematics and Informatics, Bulgaria

**
Antoni Rangachev,** ICMS-Sofia, Bulgaria

**Harvard University, USA**

Artan Sheshmani,

Artan Sheshmani,

**Cornelia-Livia Bejan,** Gh. Asachi Technical University Iasi, Romania

**
Daniela Nikolova,** Florida Atlantic University, USA

**University of Windsor, Canada**

Dennis Borisov,

Dennis Borisov,

**Harvard University, USA**

Dennis Gaitsgory,

Dennis Gaitsgory,

**Faculty of Mathematics, University of Belgrade, Serbia**

Dimitrije Špadijer,

Dimitrije Špadijer,

**Dimo Rezashki,** American University in Bulgaria

**Enrique Ruby Becerra, **University of Miami, USA

**
Erik Paemurru,** Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and University of Miami

**Cinvestav-IPN, México**

Ernesto Lupercio,

Ernesto Lupercio,

**University of Southern California, USA**

Greta Panova,

Greta Panova,

**Hristo Sariev,** Institute of Mathematics and Informatics, BAS, Bulgaria

**
Ilia Gaiur,** School of Mathematics, University of Birmingham, UK

**Bulgarian Academy of Sciences (INRNE)**

Ivan Todorov,

Ivan Todorov,

**Institut des Hautes Études Scientifiques (IHÉS)**

Jean-Pierre Bourguignon,

Jean-Pierre Bourguignon,

**John Pardon,** Princeton University, USA

**Jørgen Ellegaard Andersen,** Centre for Quantum Mathematics, University of Southern Denmark

**
Josef Svoboda,** University of Miami, USA

**Institute of Mathematics and Informatics, BAS, Bulgaria**

Julian Revalski,

Julian Revalski,

**University of Copenhagen**

Karim Adiprasito,

Karim Adiprasito,

**Faculty of Mathematics, University of Belgrade, Serbia**

Katarina Lukić,

Katarina Lukić,

**Simons Center for Geometry and Physics at Stony Brook, USA**

Kenji Fukaya,

Kenji Fukaya,

**Konstantin Delchev,** Institute of Mathematics and Informatics, BAS, Bulgaria

**Krasimira Ivanova,** Institute of Mathematics and Informatics, BAS, Bulgaria

**
Kyoung-Seog Lee,** University of Miami, USA

**Liana David,** Institute of Mathematics “Simion Stoilow”, Romanian Academy

**
Ludmil Katzarkov, **Institute of Mathematics and Informatics, BAS, Bulgaria

**Marin Genov,** Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

**
Maryna Viazovska,** École Polytechnique Fédérale de Lausanne, Switzerland

**Institut des Hautes Études Scientifiques, France**

Maxim Kontsevich,

Maxim Kontsevich,

**Maya Stoyanova,** Sofia University, Bulgaria

**
Meral Tosun,** Galatasaray University

**Bulgarian Academy of Sciences**

Milen Ivanov,

Milen Ivanov,

**University of Miami, USA**

Mina Teicher,

Mina Teicher,

**Mirjana Djorić,** University of Belgrade, Serbia

**Mladen Savov,** Institute of Mathematics and Informatics, BAS, Bulgaria

**
Nikolai Nikolov,** Institute of Mathematics and Informatics, BAS, Bulgaria

**Nikolay Nikolov,** INRNE-BAS, Bulgaria

**Ognyan Kounchev,** Institute of Mathematics and Informatics, BAS, Bulgaria

**
Oleg Mushkarov,** Institute of Mathematics and Informatics, BAS, Bulgaria

**Instituto de Ciencias Matemáticas (ICMAT), Spain**

Oscar García-Prada,

Oscar García-Prada,

**Galatasaray University**

Öznur Turhan,

Öznur Turhan,

**Institute of Mathematics and Informatics, BAS, Bulgaria**

Peter Boyvalenkov,

Peter Boyvalenkov,

**American University in Bulgaria (AUBG) and IMI-BAS**

Peter Dalakov,

Peter Dalakov,

**Peter Petrov,** Institute of Mathematics and Informatics, BAS, Bulgaria

**Peter Rashkov,** Institute of Mathematics and Informatics, BAS, Bulgaria

**Petko Nikolov**, Sofia University, Bulgaria

**
Phillip Griffiths, **Institute for Advanced Study

**Rene Mboro,** University of Miami and IMI-BAS

**Richard Paul Horja,** University of Miami, USA

**
Robert Stephen Cantrell,** University of Miami, USA

**ICMS and IMSA**

Rodolfo Aguilar Aguilar,

Rodolfo Aguilar Aguilar,

**ICMS-Sofia, IMSA-Miami**

Sebastian Torres,

Sebastian Torres,

**California Institute of Technology, USA**

Sergei Gukov,

Sergei Gukov,

**Svetoslav Zahariev,** City University of New York, U.S.A.

**Stefan Ivanov, **Sofia University, Bulgaria

**Teodor Boyadzhiev,** Institute of Mathematics and Informatics, Bulgaria

**Todor Branzov,** Institute of Mathematics and Informatics, Bulgaria

**Tokio Sasaki**, University of Miami & IMI-BAS

**
Tony Pantev,** University of Pennsylvania, USA

**Valdemar Tsanov**, Jacobs University Bremen

**
Velichka Milousheva,** Institute of Mathematics and Informatics, BAS, Bulgaria

**Université du Québec à Montréal, Canada**

Vestislav Apostolov,

Vestislav Apostolov,

**Centre for Quantum Mathematics, Syddansk Universitet, Denmark**

Vivek Shende,

Vivek Shende,

**Vladimir Dobrev,** INRNE, BAS

**Yacine Barhoumi,** University of Bochum, Germany

**Yong-Geun Oh,** IBS Center for Geometry and Physics, Korea

**
Zoran Rakic,** Faculty of Mathematics, University of Belgrade

## Programme Committee

## Programme Committee

- Ludmil Katzarkov (Chair)
- Phillip Griffiths
- Robert Stephen Cantrell
- Mina Teicher
- Julian Revalski
- Oleg Mushkarov

## Organizing Committee

## Organizing Committee

- Peter Boyvalenkov (Chair)
- Nikolai Nikolov
- Velichka Milousheva
- Krasimira Ivanova
- Alexander Petkov
- Anna Branzova

# Venue

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