# Young Researchers' Seminar

## Managers

• Casey Blacker

## Young Researchers' Seminar

 venue SPbU, Dept. of Mathematics & Computer Science Room 201 time biweekly, Friday 12:50 – 1:40 organizer Casey Blacker

The aim of this seminar is to provide a forum for young researchers to share their interests and showcase their results. Expository talks by senior faculty and academic visitors are also welcome.

Lunch will be provided. If you plan to attend, please indicate your lunch preferences here: https://public.eimi.ru/~cblacker/pizza.html

 Oct 1 Ivan Telpukhovskiy Geometry of Teichmüller space I will give a quick overview of the geometry of Teichmüller space with various metrics on it. In the end I will mention some open problems. 15 Olga Parshina A few words on combinatorics I'll cover some problems arising in combinatorial study of symbolic sequences. 29 Boris Zolotov Dynamic explicit Voronoi diagrams I'll extend on my struggles to create sublinear algorithms that maintain the explicitly stored graph of a Voronoi diagram under insertion of a new point. Nov 12 Konstantin Kalinin Viscous fingers and mixing zone growth rate estimates We will consider a system of PDEs that describes miscible displacement in porous media. If the initial and boundary conditions are such that a more viscous fluid displaces a less viscous fluid, their interface is unstable. Any small perturbation leads to viscous fingers. In practice, the following question plays an important role: is it possible to estimate the velocity of the fingers? Time permitting, we will talk about the applications of these estimates in the optimal injection profile problem. 26 Dmitrii Adler Jacobi forms and root systems In 1992 K. Wirthmuller proved an analogue of the Chevalley theorem for weak Jacobi forms associated with irreducible root systems, except $E_8$. However, generators in his proof are not constructed explicitly, but they are important in applications. In my talk, I will discuss the new simple approach to this problem in cases of root systems D_n and F_4. Following this method, it is possible to construct the generators of the corresponding algebras of weak Jacobi forms using only Jacobi theta functions and modular differential operators. Dec 17 Casey Blacker Reduction of multisymplectic manifolds A multisymplectic structure on a smooth manifold is a closed and nondegenerate differential form of arbitrary degree. In this brief presentation, we first review the Marsdeni–Weinstein–Meyer reduction theorem in the original symplectic setting, and then show how this result extends to multisymplectic manifolds. 24 Rahul Gangopadhyay Crossings in geometric hypergraphs We introduce the notion of a geometric hypergraph. We discuss the extremal result in the geometric hypergraph, i.e., how many hyperedges a uniform geometric hypergraph can have if it does not contain any crossing pair of hyperedges? We then discuss a lower bound on the number of crossing pairs of hyperedges in a complete geometric d-hypergraph. We also discuss the maximum crossing number of a complete geometric d-hypergraph for d<=4. We further discuss open questions in this regard and their connection to the polytope theory. Jan 14 Vladislav Makarov Cocke-Kasami-Younger-Schwartz-Zippel algorithm and its relatives I will talk about a fast algorithm for a "limited" version of equivalence problem for unambiguous grammars. Moreover, I will explain how the ideas behind the algorithm can be used in other contexts, for example, for proving lower bound results. 28 Dimitrios Chiotis Exterior products of operators and superoptimal analytic approximation For a given matrix-valued function G, which is expressible as the sum of a bounded analytic matrix-valued function on the unit disc D and a continuous matrix-valued function on the unit circle T, we present an algorithm that yields the superoptimal approximant AG of G. The algorithm employs operator theoretic results on exterior powers of Hilbert spaces along with the compactness of certain Hankel-type operators. The talk is based on joint work with Z.A. Lykova and N.J. Young. Feb 11 Vsevolod Evtushevskiy Ergodicity of the Martin boundary of the Young–Fibonacci graph Among central measures on the path space of the Young–Fibonacci lattice the so-called Plancherel measure has a special role. Its ergodicity was proved by Kerov and Gnedin. The goal of this paper is to prove that remaining measures from the Martin boundary of this graph (which were described by Kerov and Goodman) are also ergodic. The measures are parametrized with an infinite word of digits 1 and 2 and the parameter β∈(0,1] (the case β=0 corresponds to the Plancherel measure). 25 Boris Runov Quantum groups for classical PDEs I will review the algebraic structures behind integrable quantum field theories and demonstrate how the same structures emerge in theory of integrable PDEs. Mar 11 Ilnur Baybulov Spectrum of C* algebras of pseudodifferential operators We will look at a few examples of C*-algebras generated by pseudo-differential operators and see how local algebras can help investigate their properties. 25 Alexey Semchenkov On some problems in Combinatorial Number Theory Additive Combinatorics is an interdisciplinary subject that, roughly speaking, studies structural properties of sets in groups, related to operations + and \times. This area naturally involves tools of Number Theory, Combinatorics, Graph Theory, Ergodic Theory, Linear Algebra, Functional Analysis, and others. To give listeners a flavour of what such a mixture of methods could look like, without too much deepening into the subject, we will consider the two following problems of Combinatorial Number Theory: 1) Given natural c > 1, how many n with n - \varphi(n) = c does exist? 2) Given large prime p, how many residues sequence 1!, 2!, 3!, ... produces modulo p ? Their solutions involve Incidence Geometry, Graph Theory, Combinatorics, and Algebraic Geometry. Apr 8 Nikita Gaevoy CIRCUIT-SAT Algorithms I will talk about modern approaches to the CIRCUIT-SAT problem. I will describe how best modern solvers work and show promising directions of their improvement. 22 Sasha Kuznetsov May 6 Julius Julius 20 Ignacio Vergara Group actions on Banach spaces I will discuss different kinds of group actions on Banach spaces. Beginning with isometric actions on Hilbert spaces and the definition of the Haagerup property and Property (T), we will explore some of their generalisations. Jun 3 Dimitris Papathanasiou An introduction to linear dynamics We will discuss the notions of hypercyclicity and chaos for bounded linear operators and mention familiar operators that have those properties. 17 Andrey Melnikov Stability analysis of residually-stressed electroelastic tubes We will show and discuss conditions under which residually-stressed electroelastic tubes can bifurcate into axisymmetric and other modes.

#### May 2022

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