Women in Mathematics in South-Eastern Europe

## December 10

**Mathematics for analyzing brain activity**

*Abstract:* Studying the brain and analyzing brain activity is on the forefront of science today. It is connected to Robotics, Artificial intelligence, brain disorders, artificial organs and more. For the last decades, brain scientists believed that the main model to which the brain is subject to is firing rate and did not believe in synchronization. In this talk we will describe a research project that sheds light on the theoretical question how does the brain work, and in particular proves synchronization in brain activity of behaving animals. Moreover, we shall mention briefly few projects in clinical medicine of the brain – epilepsy and sleep disorders.

**On Knotoids and applications**

*Abstract:* The theory of knotoids, introduced by Turaev in 2011, extends classical knot theory. In this talk we review some aspects of the theory of knotoids and present some recent developments with a focus on singular knotoids. We also discuss applications in the topological study of proteins.

**Inexact Restoration methods for Finite Sum Minimization**

*Abstract:* Convex and nonconvex finite-sum minimization arises in many scientific computing and machine learning applications. Recently, first-order and second-order methods where objective functions, gradients and Hessians are approximated by randomly sampling components of the sum have received great attention. We discuss a class of methods which employs suitable approximations of the objective function, gradient and Hessian built via random subsampling techniques. The choice of the sample size is deterministic and ruled by the Inexact Restoration approach. Local and global properties for finding approximate first- and second-order optimal points and function evaluation complexity results are discussed in the framework of line search and trust region methods.

**On a non-smooth problem of the calculus of variations**

*Abstract:* The specificity of the basic problem of the calculus of variations considered as a constraint optimization problem on an infinite-dimensional space is discussed. A sufficient condition for tangential transversality involving measures of non-compactness as well as a Lagrange multiplier theorem for the infinite-dimensional optimization problem are obtained. The relation of the obtained results to the basic problem of calculus of variations is discussed.

The talk is based on a joint work with Mikhail Krastanov.

## December 11

**Magnetic trajectories on almost contact metric manifolds **

*Abstract:* This presentation consists in a collection of results obtained so far in the study of magnetic trajectories as well as some future work. As the study of magnetic trajectories was intensively developed in Kaehler manifolds, where the Kaehler 2-form is closed and hence defines a magnetic field, we investigate the magnetic curves in quasi-Sasakian manifolds. In

particular, the magnetic curves in Sasakian and cosymplectic manifolds of arbitrary dimension are classified. The 3-dimensional case is quite important, as it is well known that the geometry of quasi-Sasakian 3-manifolds is rather special and we will present the results obtained in the study of magnetic trajectories.

**The theory and the geometric modelling of curves and surfaces in Euclidean spaces**

*Abstract:* In this talk we give the theory of the curves and surfaces in Euclidean spaces. We give some basic concepts of the surfaces especially in Euclidean four space. In this study, the well- known geometric modeling and interpolation methods for curves and surfaces will be emphasized and also the studies in this area will be mentioned. Some well-known surfaces (such as rotational surface family) will be discussed and examples of these surfaces will be given. Also we present our recent works on the geometric modelling of the biological plants. Further, we discuss the Bezier interpolation methods with curves in Euclidean 4-space.

**Margarete Wolf, Symmetric Polynomials in Noncommuting Variables and Noncommutative Invariant Theory**

Satellite talk to the webinar

*Abstract:* In 1936 Margarete Caroline Wolf published a paper where she proved that the symmetric polynomials in the free associative algebra form a free subalgebra and described the system of free generators. The purpose of the talk is to present these results from the modern point of view and their relations with other results in the frames of commutative and noncommutative invariant theory.

*Joint project with Vesselin Drensky.*

**Marginally trapped (quasi-minimal) surfaces in pseudo-Euclidean 4-spaces**

*Abstract:* A surface in a pseudo-Riemannian manifold is called quasi-minimal if its mean curvature vector is lightlike at each point of the surface. When the ambient space is the Lorentz-Minkowski space, the quasi-minimal submanifolds are also called marginally trapped – a notion borrowed from General Relativity. The concept of trapped surfaces was first introduced by Sir Roger Penrose in 1965 in connection with the theory of cosmic black holes.

Marginally trapped surfaces in spacetimes satisfying some extra conditions have recently been

intensively studied in connection with the rapid development of the theory of black holes in Physics. Most of the results give a complete classification of marginally trapped surfaces under some additional geometric conditions, such as having positive relative nullity, having parallel mean curvature vector field, having pointwise 1-type Gauss map, being invariant under spacelike rotations, under boost transformations, or under the group of screw rotations.

Quasi-minimal surfaces in the pseudo-Euclidean 4-space with neutral metric satisfying some additional conditions have also been studied actively in the last few years. Most of the results are due to Bang-Yen Chen and his collaborators.

In this talk we will give an overview of these classification results and present the Fundamental existence and uniqueness theorem for the general class of quasi-minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric.

**Integral transforms through the prism of distribution theory**

*Abstract:* The Fourier transform is probably the most widely applied signal processing tool in science and engineering. It reveals the frequency composition of a time series by transforming it from the time domain into the frequency domain. However, it does not reveal how the signals frequency contents vary with time. A straightforward solution to overcoming the limitations of the Fourier transform the concept of the short-time Fourier transform (STFT). The short-time Fourier transform is a very effective device in the study of function spaces. However, significant barrier in application of the STFT is the fact that the fixed window function has to be predefined, which leads to a poor time-frequency resolution and, in general, the absence of a sufficiently good reconstruction algorithm. The Wavelet transform (WT) is used to overcome some of the shortcomings of the STFT. With the dilatation and translation of the window function, the WT has better phase modulation in the spectral domain. However, the self-similarity caused by the translation and the overlap in the frequency domain becomes non-avoidable since they do not permit straightforwardly the transfer of scale information into proper frequency information. The Stockwell transform (ST) also decomposes a signal into temporal and frequency components. In contrast to the WT, the ST exhibits a frequency-invariant amplitude response and covers the whole temporal axis creating full resolutions for each designated frequency. It is invertible, and recovers the exact phase and the frequency information without reconstructing the signal. The problem with the ST is its redundancy. But, there have been different strategies in order to improve the performance and the application of the ST.

On the other hand, the STFT, as a tool of the time-frequency analysis, contains localized time and frequency information of a function. Another idea is to localize information in time, frequency, and direction, which leads to directionally sensitive variant of STFT, which gives the Directional short time Fourier transform (DSTFT).

In mathematics, distributions extend the notion of functions. Distribution theory is a power tool in applied mathematics and the extension of integral transforms to generalized function spaces is an important subject with a long tradition. The theory is developed by proving that these transforms are well defined on the appropriate spaces of distribution. These is done by proving continuity results for these transforms on so called test function spaces, and then extending the definitions on distributions. In this talk, i consider several integrals transforms (STFT, WT, ST, DSTFT) and try to make short survey on their behaviour on distributions.

There are several approaches to the theory of distributions, but in all of them one quickly learn that distributions do not have point values, as functions do, despite the fact that they are called generalized functions. Natural generalization of this notion is the quasiasymptotic behavior of distributions. It is an old subject that has found applications in various fields of pure and applied mathematics, physics, and engineering. In the second part of my talk, I use Abelian and Tauberian ideas for asymptotic analysis of the mentioned integral transforms to characterize the asymptotic properties of a distribution.

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

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