The aim of the seminar is to promote the exchange of knowledge, present new findings, encourage critical discussion, strengthen collaboration among researchers, and support the development of early-career researchers.
Organizers:
Alexander Meskhi (Chair)
Anzor Beridze
Tengiz Bokelavadze (Scientific Secretary)
Giorgi Chelidze
Besik Dundua
Giorgi Oniani
The seminar will be held in a hybrid form on Tuesdays at 17:00 by Georgian time. Other details will be announced in advance.
1. Title: On Growth and Complexity functions
March 31, 2026, Efim Zelmanov, SUSTech International Center for Mathematics, Southern University of Science and Technology, China
Abstract: History and recent results concerning: (i) growth functions of groups, algebras, monoids and languages, (ii) complexity functions of infinite sequences, will be discussed.
2. Title: Noncommutative Fujita exponent
April 28, 2026, Michael Ruzhansky, Ghent University (Belgium), Queen Mary University of London (UK), Imperial College London (UK).
Abstract: In this talk we will discuss the Fujita exponent for nonlinear heat equations in several noncommutative settings. In particular, we focus on unimodular Lie groups and Hörmander sums of squares.
3.Title: How close are we to solving one of the major problems in matrix factorisation?
May 6, 2026, Gennady Mishuris, Professor of Aberystwyth University (UK).
For more information please see the attached file.
Abstract. Factorising general nonsingular matrix functions on the unit circle is a longstanding open problem in mathematics. While solutions are known for certain special cases, no general method is currently available. Numerical approaches also face a major difficulty: small changes in a matrix function can lead to large and unpredictable changes in its factorisation. Stability can only be guaranteed under a restrictive condition—the Gohberg–Krein–Bojarski criterion—which requires the associated partial indices to differ by at most one. However, since there is no general way to compute these indices, this creates a significant obstacle in practice.
Recent progress has begun to address these challenges. In particular, exact factorisation methods have been developed for matrix polynomials, avoiding the need to determine partial indices explicitly [1–2]. In addition, a practical criterion has been proposed for identifying when stable factorisation is possible [3].
In this talk, we explain the main ideas behind these developments and apply them to a class of strictly nonsingular 2×2 matrix functions. We show how to determine when such matrices admit a canonical or stable factorisation.
Our approach provides a constructive condition ensuring that, beyond a certain stage in an approximation process, the matrix function enters a region where stable factorisation is guaranteed. This relies on an appropriate normalisation of the approximating matrices. We also present numerical examples illustrating the effectiveness of the method.
Interestingly, this approach suggests a possible pathway towards solving the general factorisation problem.
References
[1] L. Ephremidze, I. Spitkovsky (2020) On explicit Wiener–Hopf factorization of 2 × 2 matrices in a vicinity of the given matrix. Proc. R. Soc. A 476 (2238): 20200027.
[2] V. M. Adukov, N. V. Adukova, G. Mishuris (2022) An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realisations within ExactMPF package. Proc. R. Soc. A 478 (2263): 20210941.
[3] N. V. Adukova, V. M. Adukov, G. Mishuris (2024) An effective criterion for a stable factorisation of strictly nonsingular 2 × 2 matrix functions. Proc. R. Soc. A 480 (2299).
