## Second International Conference

Women in Mathematics in

South-Eastern Europe

## December 9 – 10, 2021,

Sofia, Bulgaria

## organized by

**International Center for Mathematical Sciences – Sofia (ICMS-Sofia)** and the

**Institute of the Mathematical Sciences of the Americas at the University of Miami (IMSA)**

**T****he initiative Women in Mathematics of South-Eastern Europe**, aiming at promoting the role of female mathematicians, started in Dec 2020, when the inaugural conference took place. A special distinguished guest of the inaugural conference was prof. Mina Teicher from the Department of Mathematics and Gonda Brain Research Center, Bar-Ilan University, Israel, Director of Emmy Noether Institute for Mathematics. Information about the first conference is available at: https://icms.bg/women-in-math-2020/.

We intend to make these conferences annual. The main goal of this initiative is to celebrate women in Mathematics, to disseminate new results and create new long-term collaborations among scientists in South-Eastern Europe. We hope Women in Mathematics of South-Eastern Europe will attract the attention of young researchers and researchers from less-favoured countries.

The speakers are proposed by an international committee including Mina Teicher (Department of Mathematics and Gonda Brain Research Center, Director of Emmy Noether Institute for Mathematics), Phillip Griffiths (Professor Emeritus, Institute for Advanced Study, USA), Ludmil Katzarkov (Miami University, Co-director of the Institute of Mathematical Sciences of the Americas), Velichka Milousheva (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Deputy Director).

# Distinguished Invited Speakers

# Barbara Drinovec Drnovšek

Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

**Nadya Morozova**

### Institut des Hautes Études Scientifiques (IHES), France

# Invited Speakers

**Adela Mihai,**Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest and Transilvania University of Brasov, Interdisciplinary Doctoral School, Brasov, Romania**Azniv Kasparian**, Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, Bulgaria**Biljana Jolevska-Tuneska**, Ss. Cyril and Methodius University, Skopje, N. Macedonia**Danijela Rajter-Ćirić**, Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia

**Margarita Zachariou**, The Cyprus Institute of Neurology and Genetics, Cyprus**Mariya Soskova**, Department of Mathematics, University of Wisconsin–Madison, USA and Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”,

Bulgaria**Meral Tosun**, Department of Mathematics, Galatasaray University, Turkey**Mirjana Vukovic**, Academy of Science and Arts of Bosnia and Herzegovina, Bosnia and Herzegovina

# Programme

## December 9

**Mathematical modeling of cancer cells populations behavior**

*Abstract:* The talk will describe how mathematical modeling of biological processes can address the important questions which can not be solved by experimental approaches. Two examples from cancer biology will be discussed: (1) the phenomenon of Cancer Stem Cells population stabilization and (2) the phenomenon of post-irradiation induction of Cancer Stem Cells. Cancer Stem Cells (CSC) is a small population in heterogeneous cancer cells population which is resistant to conventional cancer therapies (chemo- or radio therapies), and thus responsible for tumor relapses. The model considers different possible modes of stem and non-stem cancer cells population behavior in different conditions with different sets of experimentally measured parameters. The results obtained by the model allow determination of time-varying corridors of probabilities for different cell fates of CSC in a given experimental system, and determination of cell-cell communication factors influencing these time-varying probabilities of cell behavior scenarios. The analysis of these results provides as theoretical insights into the phenomena of CSC behavior, so a set of biomedical suggestions, essential for cancer therapy.

**Simple Elliptic Singularities and Generalized Slodowy Slices**

*Abstract:* We will present a relation between the family of simple elliptic singularities which are complete intersections and finite dimensional Lie algebras. We will extend the concept of Slodowy slices to the case of these singularities.

**On proper holomorphic maps**

*Abstract:* The image of a proper holomorphic map from an open Riemann surface is an analytic subvariety of the ambient space. We will explain a method for constructing such maps, and we will present some recent related results.

**From Krasner’s corpoid and Bourbaki-Krasner’s graded rings to Krasner-Vuković’s paragraded rings**

*Abstract:*

Marc Krasner investigating valued fields and observing a connection between them and their valuation rings using equivalence of valuations came to the fundamental notion of a corpoid in a series C.R. of Academy of Sciences of Paris notes (1944-45), which represents the origin of the development forerunner of the general graded theory – a theory of homogroupoids, anneids and moduloids – more general than one given by *Bourbaki (Algèbre, Chap II Paris, 1962)*, since neither the associativity, nor the commutativity nor the existence of a neutral element is assumed in the set of grades, where corpoid, as a special case of an anneid, is viewed

as a homogeneous part of a graded field. So, the abstract notion of corpoid led Krasner to a development of general graded structures *(Anneaux gradués géneraux, 1980)*

Since the category of graded structures (groups, rings, modules) has no the property of closure with respect to the direct product and the direct sum, it was for M. Krasner and myself a motivation to go further with generalizations and to introduce the notion of paragraded structures: groups, rings, modules, which appeared for the first time, late in 1980s, in a series Krasner- Vuković’s Proceedings Japan Academy notes and in our monograph Structures Paragraduée (groupes, anneaux, modules) *(Queen’s Papers in pure and Applied Mathematics, Kingston, ONT. Canada)*. As we have already noted, these structures resolves the formentioned problem of closure. In this presentation we are particularly interested in the structure of a paragraded ring and their radicals, and we are going to present some results on the different types of paragraded radicals, introduced in my joint papers with my pupils, in the class of the paragraded rings. More precizely we will present prime and Jacobson radicals, discuss the general Kurosh-Amitsur theory of radicals of paragraded rings, characterise paragraded normal radicals and prove that all special paragraded radicals of paragraded rings can be described by the appropriate class of their paragraded modules and finally provide information about ongoing work regarding paragraded Brown-McCoy radicals, i.e. inspired by Halberstadt results about Jacobson’s radicals of graded rings, to introduce two versions of paragraded Brown-McCoy radicals: the Brown- McCoy radical and the large Brown-McCoy radical of a paragraded rings and to prove that the large Brown-McCoy radical of a paragraded ring coincides with the largest homogeneous ideal contained in the classical Brown-McCoy radical of that ring.

**Application of fractional calculus in solving some deterministic and stochastic PDEs**

*Abstract: *Fractional calculus has proven to be very useful in studying many real-life problems. It is especially very effective when problems of consideration include memory effect issues. Therefore, the application of fractional calculus is widespread in many different scientific areas. Here we present a specific approach in solving some PDEs by using fractional calculus. We consider reaction-advection-diffusion equations with space fractional derivatives, inhomogeneous fractional evolution equations, time and time-space fractional wave equations with variable coefficients, and finally, stochastic fractional heat equation with variable thermal conductivity

and multiplicative noise.

In all cases mentioned, we prove that there exists a unique solution to the problem within a certain Colombeau generalized function space. These results are obtained by using the theory of generalized uniformly continuous semigroups of operators.

## December 10

**Toroidal compactifications**

*Abstract:* We discuss some properties of the toroidal compactifications X of discrete quotients of the complex 2-ball and, more general, of the toroidal compactifications Z of the quotients of the bounded symmetric domains by lattices Γ of holomorphic automorphisms. The logarithmic canonical bundle of a specific X is shown to be very ample by an explicit construction of sections. For a finite Galois quotient X/H of X, which is a compactification of a ball quotient, the dimension and the codimension of the logarithmic canonical model of X/H are obtained by counting the dimension of the H-invariant sections of the logarithmic canonical bundle of X. The fundamental group of Z is described as a quotient group of Γ.

**Network models and Network-based Integration for the brain**

*Abstract:* The brain can be studied mathematically at different levels from the molecular and genetic level up to the neural activity and behaviour. Here, I will present two different approaches with which we study the brain using network models and network-based data integration. In the first part of the talk, I will present the development and validation of generative network models of gamma oscillations in the visual cortex utilising empirical constraints from measurements at multiple spatial scales and discuss how, by using the validated model, we identified a novel route to gamma oscillation instability that may underlie the gamma power decay at high inputs. In the second part of the talk, I will present the development of network-based integration approach for multi-scale and high-throughput biological data to detect disease-related clusters of molecular mechanisms and identify new or repurposed drugs for neurodegenerative diseases.

**New Characterizations of Rectifying Curves**

*Abstract: *An involute of a curve x is a curve y that lies on the tangent surface to x and intersects the tangent lines orthogonally. If the curve y is an involute of x, then x is said to be an evolute of y. We give new characterizations of rectifying curves by their involutes and evolutes.

Work supported by the internal project UTCB-CDI-2021-012 of the Technical University of Civil Engineering Bucharest, Romania.

*Abstract:* In this talk the connection between neutrix calculus and special functions will be given. Some results concerning neutrix calculus, incomplete gamma function, beta function and digamma function will be presented.

**Logic, degrees, and definability**

*Abstract:* In this talk I will present my take on one line of research in Computability Theory that studies logical properties of degree structures. There are different ways in which we can compare the algorithmic complexity between countable mathematical objects. A set A of natural numbers is Turing reducible to a set B if there is an algorithm to determine membership in A using B as data. A set A is enumeration reducible to B if there is an algorithm to list the members of A using any listing of B as data. Each reducibility gives rise to a degree structure: a partial order in which we identify sets that have equal algorithmic complexity. The Turing degrees and the enumeration degrees are closely related: the Turing degrees have an isomorphic copy inside the enumeration degrees, which we call the total degrees. We study these structures from three interrelated aspects: how complicated is the set of true algebraic statements in each partial order; what relations have structurally definable presentations; what does the automorphism group for each partial order look like.