Second Annual Meeting of Young Bulgarian Mathematicians

Event dates:

June 13-14, 2022

Event type:

In person

The International Center for Mathematical Sciences – Sofia is organizing for a second consecutive year a meeting of young Bulgarian mathematicians from around the world.

The conference will take place at the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (IMI-BAS) on June 13 – 14, 2022.

Special guest of the event is Prof. Yuri Tschinkel from Courant Institute, NYU and Simons Foundation Director, Mathematics and Physical Sciences. We envision two main outcomes of this meeting:

  • Enriching the relations between young mathematicians working in Bulgaria and the Bulgarian mathematical diaspora;
  • Enhancing the professional development of young Bulgarian mathematicians by presenting to them new career opportunities through national and European scientific programs.

The conclusion of the scientific part of the event on June 14 will be followed by a round table to discuss these opportunities. We are all elated by this exciting event and we are looking forward to seeing you at IMI-BAS.

Invited speakers

Iva Halacheva

Northeastern University,
Boston, USA

Kalina Mincheva

Tulane University,
Louisiana, USA

Konstantin Delchev

Institute of Mathematics and Informatics – BAS, Bulgaria

Marin Genov

Institute of Mathematics and Informatics – BAS, Bulgaria

Nikola Kamburov

Pontifica Universidad Catolica de Chile, Santiago, Chile

Yanitsa Pehova

London School of Economics (LSE)
London, UK

Yuri Tschinkel

Courant Institute, NYU and
Simons Foundation Director, Mathematics and Physical Sciences

Programme

June 13, 2022

09:00 – 10:00  Registration

Chair: Ludmil Katzarkov, IMI-BAS

Equivariant Birational Geometry

Abstract: In this talk, we will describe recent developments in birational geometry based on joint papers with Kontsevich and Pestun, and Hassett and Kresh.

11:15 – 11:30   Coffee break

Chair: Mladen Savov, Sofia University and IMI-BAS

One-phase free boundaries subject to topological constraints

Abstract: Free boundary problems (FBP) model various physical interfaces that arise from formally minimizing certain energy or cost functionals. The free boundaries that do minimize energy/cost have been fairly well studied and understood. Much less explored have been the ones that are simply critical points of the underlying functional. One natural approach to studying those is to impose a topological constraint on the interface and investigate what geometric rigidity is enforced on its shape. In this talk, I will discuss our joint project with David Jerison (MIT), in which we investigate the extent to which solutions to the One-Phase FBP in the disk can be characterized according to their topological complexity. Our results elaborate a fascinating connection that exists between one-phase free boundaries in the plane and minimal hypersurfaces in Euclidean 3-space. Additionally, I will present our rigidity results for globally defined solutions of a related semilinear variational problem in the plane.

Geometric and categorical actions on crystals

Abstract: One approach to studying the representation theory of Lie algebras and their associated quantum groups is through combinatorial shadows known as crystals. The braid group acts on the original representations, while a closely related group—the cactus group—acts on the corresponding crystals. I will describe how this combinatorial action can be obtained both geometrically, as a monodromy action coming from a family of shift of argument algebras, as well as categorically through the structure of certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.

13:30 – 15:00 Lunch

Prof. Tschinkel will share his experience of how private capital helps fund research in various fields.

The lecture will be held at the Intercontinental Hotel Sofia, Ballroom Hall, 4 Narodno Sabranie Square.

Yuri Tschinkel is an algebraic geometer who covers a broad spectra of subjects – from arithmetics to complex geometry. He was educated in Moscow State University and received his doctorate in mathematics from the Massachusetts Institute of Technology.

Tschinkel has obtained spectacular results in counting points over different fields and stable rationality. For his work he has been elected to Academia Europea and Leopoldina, the German National Academy of Sciences. He has been a speaker at the International Mathematical Congress 2006.

Yuri Tschinkel is a Director of Simons Foundation, Mathematics and Physical Sciences division. The Simons Foundation is a private foundation established in 1994 by Marilyn and Jim Simons. The foundation is one of the largest charitable organizations in the USA. It makes grants in four areas: Mathematics and Physical Sciences, Life Sciences, autism research (Simons Foundation Autism Research Initiative) and Outreach, Education and Engagement.

Programme:

17:00 – 17:30 – Welcoming
17:30 – 18:30 – Lecture of Prof. Yuri Tschinkel
18:30 – 20:00 – Cocktail

June 14, 2022

Chair: Аntoni Rangachev, IMI-BAS

Tropical Geometry and the Commutative Algebra of Semirings

Abstract:

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side – using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue.

In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences and what they remember about the geometry of a tropical variety. The talk will give some overview of recent results and work in progress.

Functions holomorphic over finite-dimensional commutative associative algebras

Abstract: We shall discuss the basic function theory arising from replacing the field of complex numbers by a more general finite-dimensional commutative associative algebra both in the domain and codomain in the definition of single-variable complex differentiability. What we get is a function theory that lives and thrives on the border between the kingdom of the classical single-variable complex analysis and the realm of Several Complex Variables.

Chair: Alexander Petkov, Sofia University

Substructures in graphs – an extremal perspective

Abstract: Extremal graph theory is an exciting and rapidly developing research area which can be roughly described as the study of relationships between local and global properties of graphs. Over the past 20 years we have seen proofs of many long-standing conjectures such as the existence of designs, Kelly’s conjecture on Hamilton decompositions of tournaments, Ringel’s conjecture, and more. In this talk I will outline some main ideas from extremal graph theory, present popular open problems and theorems, and give sketches of some central methods in the field. This talk requires only a basic understanding of graph theory.

Some recent results on two-distance codes

Abstract:

Two-distance q-ary codes are not a new entity in the field of coding theory. The linear case has been studied for more than fifty years, as we can see from the seminal paper of Calderbank and Kantor. Many infinite classes of linear two-weight codes have been constructed, but the complete classification is far from complete and might be difficult to obtain. We set our goals to understand the structures of a more general case – arbitrary, not necessarily linear two-distance codes, we consider their general properties and obtain classification results in special cases, such as when the two distances are consecutive or one of the distances being maximal. The talk is based on joint works with P. Boyvalenkov, D.V. Zinoviev and V.A. Zinoviev, which explore the topic both from the point of constructions and the point of bounds for existence.

Moderators: Peter Boyvalenkov, IMI-BAS and Parvan Parvanov, Sofia University

Enhancing Research Capacity in Mathematical Sciences

17:30 -19:00  Cocktail

Gallery

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