# Second Annual Meeting of Young Bulgarian Mathematicians

This meeting is the inaugural event of a series of annual meetings ICMS initiates. The series, which commemorates the brightest Bulgarian holiday *the Day of Bulgarian Enlightenment and Culture and the Slavоnic Alphabet* May 24, has as its main objective bringing together young Bulgarian mathematicians working all over the world.

We envision two main outcomes:

- enriching relations between young mathematicians working in Bulgaria and the Bulgarian mathematical diaspora;
- enhancing professional development of young Bulgarian mathematicians by presenting new opportunities using the national and European scientific programmes.

Following the conclusion of the scientific part of the event on May 19, we plan to organise a special round table to discuss these opportunities.

We are all elated by this exciting event and we are looking forward to seeing you there. This year the talks will happen via Zoom. We all hope to have an actual onsite live event starting next year.

## Invited speakers

## Programme

## May 19, 2021

## Afternoon session

Chair: **Martin Kassabov**, Cornell University, Ithaca, USA

**Counting Irreducible Sparse Polynomials of a Given Degree over a Finite Field**

*Abstract: *A classical result of Gauss states that among all monic polynomials of degree d over a finite field, approximately 1/d are irreducible. Extending previous results in the literature, we prove that under a mild assumption, the proportion of irreducible polynomials does not change even if only the last two coefficients are allowed to vary. Our approach is geometric. The talk will be nontechnical and accessible to a broad audience.

**Optimal Control for the Evolution of Deterministic Multi-agent Systems**

*Abstract: *We investigate an optimal control problem with a large number of agents (possibly infinitely many). Although the dynamical system (a controlled ordinary differential equation) is of the same type for every agent, each agent may have a different control. So, the multi-agent dynamical system has two levels: a microscopic one, which concerns the control system of each agent, and a macroscopic level, which describes the evolution of the crowd of all agents. The

state variable of the macroscopic system is the set of positions of the agents. We define and study the evolution of such a global dynamical system whose solutions are called solution tubes. We also consider a minimization problem associated with the multi-agent system and we give a new characterization of the corresponding value function as the unique solution of a Hamilton- Jacobi-Bellman equation stated on the space of compact subsets of $R^d$.

The talk is based on joint work with Marc Quincampoix.

**What is the smallest algebraic integer?**

*Abstract: * We survey Lehmer’s problem on the smallest Mahler measure of an integer non-cyclotomic polynomial, synthesizing an introduction to this subject along with the state-of-art results known today. Then we will present the solution of a closely related conjecture of Schinzel and Zassenhaus, and propose some answers towards the question of our title. Time permitting, we discuss also some related recent results on the geometry of integer polynomials.

## Round table

Moderators: **Oleg Mushkarov** and **Antoni Rangachev**

## May 20, 2021

## Afternoon session

Chair: **Kiril Datchev**, Purdue University, USA

**Rigidity in Dimension 2**

*Abstract:* I will discuss an approach to proving the conjecture that a normal rigid surface is smooth. The approach is based on the notion of deficient conormal singularities introduced by the speaker.

**Coxeter quotients of knot groups**

*Abstract:* A knot is a smooth embedding of a circle into the three-sphere. Knots have seduced the imagination since at least ancient Rome, and the classification of knots up to continuous deformations preoccupied even Gauss. Knots can be distinguished, and studied, using the fundamental groups of their complements in $S^3$. I will describe a method for computing the bridge number of a knot $K$ — a geometric invariant, which I will define — by using homomorphisms of $\pi_1(S^3\backslash K)$ onto Coxeter groups. This settles a conjecture by Cappell and Shaneson from the 70s for some infinite families.

Joint work with Sebastian Baader and Ryan Blair.

**The Almost Schur Lemma in Quaternionic Contact Geometry**

*Abstract:* In this talk we consider the quaternionic contact (qc) version of the almost Schur Lemma. Namely, we derive an integral inequality that informally states that on a compact qc manifold of dimension bigger than seven, satisfying some positivity condition, if the traceless qc Ricci tensor and some traces of the qc torsion tensor are close to zero, then the qc scalar curvature is close to a constant.

## Evening session

Chair: **Greta Panova**, University of Southern California, USA

**Orthogonal Tensor Deco mposition**

*Abstract: *Tensor decompositio

**Pólya Sequences with Dominant Colors**

*Abstract:*

We introduce a class of *“Pólya Sequences with Dominant Colors”*, which can be described as randomly reinforced urn processes with color-specific random weights and unbounded number of possible colors. Let ${D}$ be the set of colors for which the expected random reinforcement attains its maximum value; in this sense, we say that the colors in ${D}$ are *dominant*. Under fairly mild conditions, we show that the predictive probability of observing a dominant color, $P_n({D})$, converges a.s. to one. Moreover, there exists a random probability measure $\tilde{P}$ with $\tilde{P}({D})=1$ such that the predictive and the empirical distributions converge weakly a.s. to $\tilde{P}$. The latter implies, in particular, that the urn process is asymptotically exchangeable with limit directing random measure $\tilde{P}$. In the general case, for any $\delta$-neighborhood ${D}_\delta$ of ${D}$, the predictive probabilities $P_n({D}_\delta)$ and the empirical frequencies $\hat{P}_n({D}_\delta)$ converge a.s. to one. As a result, the distance between the observed color and ${D}$ converges in probability to zero. We refine the above results with rates of convergence and central limit theorems. We further hint examples of the potential use of our model in randomized clinical trials and species sampling.

The conference will be held via Zoom. A direct link to the virtual hall, the conference protocol and some technical tips on Zoom are available below.

All times are EET (UTC+2) – Sofia local time.