# About us

The newly established **International Center for Mathematical Sciences – Sofia** is a dynamic research unit for developing and dissemination of cutting edge new directions in Mathematics. It is affiliated with the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences which is providing the infrastructure for the activities of the Center. The Center is supported by the Ministry of Education and Science of the Republic of Bulgaria. The Activities of the ICMS – Sofia are carried out in collaboration with the Institute of the Mathematical Sciences of the Americas at the University of Miami (IMSA) and Higher School of Economics, National Research University, Moscow (HSE University). The ICMS-Sofia is also working in collaboration with Bulgarian universities and institutes of the Bulgarian Academy of Sciences.

The Center was created following inspirational discussions with many members of the mathematical community in Bulgaria and the Bulgarian mathematical diaspora. In July 2019, ICMS-Sofia initiated its full-scale presence on the European mathematical scene with the full support of the Presidency of the Bulgarian Academy of Sciences.

The first director of ICMS-Sofia was Acad. Blagovest Sendov.

# Simons Foundation

Grant 2022-2025

The Simons Foundation International, Ltd., (SFI) is funding a project of the International Center for Mathematical Sciences for the period 2022 – 2025.

The grant is aimed at supporting displaced scientists from Ukraine and other countries, organizing international scientific events with the participation of world-renowned and established mathematicians, and opening new research positions for young scientists.

# On focus:

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

**presents**

## ICMS Seminar

9.04.2024, 14:00 Sofia time,

ICMS-Sofia, Room 403

# The Specht property for varieties of $Z_n$-graded Lie algebras

##
Daniela Martinez Correa

Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”

Daniela Martinez Correa

Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”

**Abstract:**

Let $UT_n(K)$ be the algebra of the $ntimes n$ upper triangular matrices and denote $UT_n(K)^{(-)}$ the Lie algebra on the vector space of $UT_n(K)$ with respect to the usual bracket (commutator), over an infinite field $K$. In this talk, we give a positive answer to the Specht property for the ideal of the $\mathbb{Z}_n$-graded identities of $UT_n(K)^{(-)}$ with the canonical grading when the characteristic $p$ of $K$ is 0 or is larger than $n-1$.

Moreover, if $K$ is a field of positive characteristic $p$, we construct three varieties of $Z_{p+1}$-graded Lie algebras which do not have a finite basis of their graded identities and satisfy the graded identities which in the case of infinite field define the variety generated by $UT_{p+1}^{(-)}(K)$. The first variety contains the other two. The second one is locally finite. The third variety is generated by a finite dimensional algebra over an infinite field.

This is a joint work with Vesselin Drensky (IMI-BAS) and Plamen Koshlukov (Unicamp, Brazil).

# Events and activities

The mission of the initiative * Women in Mathematics in South-Eastern Europe *is to promote the role of female mathematicians.

The mission will be achieved through series of annual conferences with the main goal 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 inaugural conference of the initiative Women in Mathematics in South-Eastern Europe took place in Dec 2020. 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. Prof Teicher is also Director of Emmy Noether Institute for Mathematics.

This meeting 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 the professional development of young Bulgarian mathematicians by presenting new opportunities using the national and European scientific programmes.

The inaugural event of the series was held May 19-20 2021 and commemorated the brightest Bulgarian holiday the Day of Bulgarian Enlightenment and Culture and the Slavоnic Alphabet May 24.

# News and Announcements

#### The Milnor number of a smoothable curve, talk by Antony Rangachev

In this talk I will derive an algebraic formula for the Milnor number of a smoothable complex analytic curve singularity X by relating it to the Euler characteristic of its smoothing, which in turn I will relate to the multiplicity of the Jacobian ideal of X and and the multiplicity of X at its singular point. If time permits I will discuss generalizations to higher dimensions. This is a report on a joint work with Gaffney and Bengus-Lasnier.

#### Stratified Morse theory and the critical locus of a linear functional, talk by Antony Rangachev

In this talk I will use stratified Morse theory to relate the number of critical points of a generic linear functional on a complex analytic manifold M to the Euler characteristics of M and a generic hyperplane slice of M.

#### Recent developments in the study of Følner functions, talk by Bogdan Stankov

The Følner function of a group is defined on positive integers n as the smallest size of a Følner set, the boundary of which is at most 1/n times the size of the set. Its values are then finite if and only if the group is amenable. It can be thought of as encoding "how amenable a group is". We will give an overview of how our understanding of Følner functions has progressed. We will mostly talk about two major types of development. The first one concerns proving, for a given type of function, the existence of a group that has a Følner function of that type. The other one is connections between the asymptotics of Følner functions and those of the growth function.

#### de Sitter (dS) Relativity versus Poincaré Relativity, talk by Hamed Pejhan

This presentation introduces a novel holographic correspondence in d-dimensional de Sitter (dS_d) spacetime, connecting bulk dS_d scalar unitary irreducible representations (UIRs) with their counterparts at the dS_d boundary, all while preserving reflection positivity. The proposed approach, with potential applicability to diverse dS_d UIRs, is rooted in the geometry of the complex dS_d spacetime and leverages the inherent properties of the (global) dS_d plane waves, as defined within their designated tube domains.