Fourth Annual Meeting of Young Bulgarian Mathematicians

Event dates:

June 11-12, 2024

Event type:

In person

The International Center for Mathematical Sciences – Sofia is organizing for a fourth 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 11 – 12, 2024.

A special guest of the event is Prof. Mladen Savov. He is a Professor at Sofia University and the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences since 2020 and a corresponding member of the Bulgarian Academy of Sciences since 2021.

The conference’s main goal is to bring together young Bulgarian mathematicians working all over the world and present some of their most significant achievements in front of the mathematical community. The opportunities for their professional development in the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences and leading Bulgarian universities using the national and European scientific programs will be discussed at a virtual round table held during the first day. The event will enrich the relations between the young mathematicians in Bulgaria and the Bulgarian mathematical diaspora.

Invited speakers

Mladen Savov

Institute of Mathematics and Informatics – BAS and Faculty of

Mathematics and Informatics – Sofia University

Special guest

Marin Bukov

Max Plank Institute for the
Physics of Complex Systems
GERMANY

Evgeni Dimitrov

Department of Mathematics,
University of Southern California
USA

Dimitar Kodjabachev

Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences
BULGARIA

Martin Minchev

Faculty of Mathematics and Informatics,
Sofia University “St. Kliment Ohridski”
BULGARIA

Vladimir Mitankin

Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences
BULGARIA

Bogdan Stankov

Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences
BULGARIA

Dragomir Tsonev

Instituto de Ciências Exatas,
Universidade Federal do Amazonas
BRAZIL

Programme

Download Conference Poster

June 11, 2024

10:00 – 10:15 Opening session

Bernstein-gamma functions are a simple extension of the classical Gamma function and they have important role in modern probability theory especially in the context of study of some Markov semigroups.

In this talk, we are going to review their properties and reveal some of their applications. The talk is based on joint works with P. Patie (Cornell), M. Minchev (Sofia University) and others. 

11:15 – 11:45 Coffee Break

The celebrated Atiyah-Singer Index theorem, without a shadow of a doubt, stands as one of the deepest results of mathematics ever discovered. Its beauty and grace naturally permeate some of the fundamental areas of mathematics. It generalises a variety of deep theorems and has far reaching applications. And who knows whether or not it will not manifest itself again in some newer context. Both its statement and its different proofs necessitate an interdisciplinary mathematics, ergo, much time and effort are needed to penetrate into the essence and the details. Notwithstanding the latter difficulties, the goal of this short talk will be a humble attempt to bring forward, as much as possible, the geometry which lays behind this remarkable theorem. Hopefully this effort might instigate some interesting discussions among the participants.  

12:45 Lunch break

The ML revolution, fueled by unprecedented computational power and internet-driven data availability, intersects with arising quantum technologies, and may hold the key to resolving some of the outstanding challenges in developing modern quantum computers. Crucial for this is devising strategies to control quantum devices through feedback mechanisms, which fall within the framework of deep reinforcement learning (RL). RL’s adaptability to unexpected changes in the controlled system was key to its success in mastering complex games, and holds the promise to revolutionize different aspects in the way we interact with quantum systems. I will review recent advancements brought in by RL in quantum technologies, spanning a wide spectrum from state preparation, gate implementation, circuit synthesis, feedback control, and error correction, that have recently yielded notable successes like optimizing pulses for entangling gates and discovering new error-correcting codes.  

16:00 – 16:30 Coffee Break

In this talk, we will first present the exponential functionals of Lévy processes, which appear, for example, in the study of branching processes, financial mathematics, and self-similar processes. Following the work of Patie and Savov, we know that the complex moment generating function of these random variables (also known as the Mellin transform) can be represented in terms of a generalization of the gamma function. Using this result, and employing not only probabilistic arguments but also a variety of techniques such as Mellin inversion, Tauberian theorems, asymptotic analysis, and potential theory, we show that if the underlying Lévy process has a finite negative mean and a regularly varying tail, its exponential functional has a distributional limit. 

This is a joint work with Mladen Savov. 

18:30 Conference Dinner

June 12, 2024

–

I will discuss a two-parameter family of probability measures, that are called β-Krawtchouk corners processes. These measures are related to Jack symmetric functions, and can be thought of as integrable discretizations of β- corners processes from random matrix theory, or alternatively as non-determinantal measures on lozenge tilings of infinite trapezoidal domains. For such models we show that the height function asymptotically concentrates around an explicit limit shape and prove that its limiting fluctuations are described by a pull-back of the Gaussian free field, which agrees with the one for Wigner matrices. If time permits, I will discuss the main tools we use to establish our results, which are certain multi-level loop equations. The talk is based on joint work with Alisa Knizel. 

11:00 – 11:30 Coffee Break

А question of Mordell asks which integers are the sum of three cubes provided that there are no local obstructions to that. While Mordell’s question seems out of reach with the current machinery, it is widely believed that every integer not congruent to 4 or 5 mod 9 is a sum of three integral cubes. In this talk we will explain that if, instead of the sum of three cubes, a general ternary cubic form is considered local conditions no longer suffice for solubility, i.e. the integral Hasse principle may fail. We utilize the integral version of the Brauer-Manin obstruction to construct the first counter-examples to the integral Hasse principle in the diagonal setting. We will then explore in three different natural ways how such failures are distributed across the family of affine diagonal ternary cubics to emphasise why they are so hard to exhibit. This talk is based on a joint work with Harkaran Uppal and Julian Lyczak. 

12:30 Break

I will introduce the notion of a Galois extension, in the sense of Rognes, of commutative ring spectra and discuss its role in the computation of Picard groups via descent theory. I will illustrate the concepts with examples coming from complex K-theory and topological modular forms with level structure. 

 

15:30 – 16:00 Coffee Break

One tool used to encode the limit behavior of random walks is called the Poisson boundary. It is considered in particular in group theory due to a result that states that a group is amenable if and only if it admits a non-degenerate measure that induces a trivial Poisson boundary for the walk on its Cayley graph. For any action of that group one obtains a quotient of that boundary in the form of the Poisson boundary of the induced walk on the Schreier graph of that action. We present results relating to non-triviality of the Poisson boundary of the induced random walk for measures with finite first moment on a certain class of groups defined by actions on the real line, popularised by a paper of Monod, including in particular Thompson’s group $F$. 

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)”

and

Simons Foundation.