During the course of the last few years a number of startling connections between quantum invariants of knots and 3-manifolds and high-level number theory have emerged. Already the rigidity theorems of 3-dimensional hyperbolic topology, which have been known for many years, had a quite non-trivial arithmetic content, with the volume of every hyperbolic 3-manifold being linked via the dilogarithm to the so-called Bloch group and algebraic K-theory, and another connection comes from the Kashaev invariant, which is linked via his famous conjecture to the hyperbolic volume but also belongs to the so-called Habiro ring, which is a beautiful number-theoretical object that is not yet well known to number theorists.
From Knots to Number Theory, by Don Bernard Zagier
One of the key developments in combinatorics and algebra of recent years has been the discovery of Lefschetz principles beyond Hodge structures, resolving several long-standing conjectures. I will provide an overview of recent developments, and discuss joint work with Johanna Steinmeyer, Stavros Papadakis and Vasiliki Petrotou.
Lefschetz theorems beyond Hodge structures, by Karim Adiprasito
Let $C$ be a smooth projective curve of genus at least 2, and let $N$ be the moduli space of semistable rank-two vector bundles of odd degree on $C$. We construct a semi-orthogonal decomposition in the derived category of $N$ conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form $D(C_d)$ where $C_d$ are $d$-th symmetric powers of $C$, and the semi-orthogonal complement to these blocks is conjecturally trivial. In order to prove our result, we use the moduli spaces of stable pairs over $C$. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall. This is a joint work with J. Tevelev.
Windows and the BGMN conjecture

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 Acad. Julian Revalski, President of the Bulgarian Academy of Sciences.

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

# Consortium Distinguished Lecture Series

## Don Zagier

### Ramanujan International ChairInternational Centre for Theoretical Physics

Don Bernard Zagier is an American-German mathematician whose main area of work is number theory. He was a scientific member and one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the Collège de France in Paris, France. He has been associated with the ICTP in Trieste, Italy, since 2014 and now holds the Ramanujan International Chair there.

For his groundbreaking work Zagier was awarded the Cole Prize in Number Theory in 1987, the von Staudt Prize in 2001. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1997 and a member of the National Academy of Sciences (NAS) in 2017. He is also one of the two 2021 Fudan – Zhongzhi Science Award laureates.

# From Knots to Number Theory II

## Friday, October 29th, 2021, 4:30pm (Trieste) 10:30am (Miami)

Abstract: During the course of the last few years a number of startling connections between quantum invariants of knots and 3-manifolds and high-level number theory have emerged. Already the rigidity theorems of 3-dimensional hyperbolic topology, which have been known for many years, had a quite non-trivial arithmetic content, with the volume of every hyperbolic 3-manifold being linked via the dilogarithm to the so-called Bloch group and algebraic K-theory, and another connection comes from the Kashaev invariant, which is linked via his famous conjecture to the hyperbolic volume but also belongs to the so-called Habiro ring, which is a beautiful number-theoretical object that is not yet well known to number theorists. The more recent developments that I will describe concern even deeper links to algebraic number theory (construction of non-trivial units) and algebraic K-theory, to a generalization of the classical Habiro ring to Habiro rings associated to arbitrary algebraic number fields, and above all to surprising “quantum modularity” properties of the Kashaev invariant and its generalizations leading to a new type of object in the theory of modular forms. All of the work that will be reported on is joint with Stavros Garoufalidis, and some of it also with Rinat Kashaev and with Peter Scholze.

The lectures are intended for a general mathematical audience and do not require any prior knowledge of knot theory, K-theory, modular forms theory, or any other theory.

# Consortium Distinguished Lecture Series

## November 1-2, 2021, 5:00 pm (EEST, Sofia time)ICMS-Sofia, via Zoom

### Department of Mathematical Sciences, University of Copenhagen, Denmark

Prof. Adiprasito is known for his fundamental contributions to combinatorics and discrete geometry. Together with June Huh and Eric Katz he proved the Heron-Rota-Welsh conjecture regarding the log-concavity of coefficients of the characteristic polynomials of matroids. Prof. Adiprasito resolved the Hirsch conjecture for flag triangulations of manifolds, and McMullen’s g-conjecture for simplificial spheres.

Prof. Adiprasito was awarded the 2015 European Prize in Combinatorics, the 2019 New Horizon Prize for Early-Career Achievements in Mathematics associated with the Breakthrough Prize in Mathematics, and the 2020 Prize of the EMS.

# Women in Mathematics in South-Eastern Europe

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.

# The Annual Meeting of Young Bulgarian Mathematicians

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.

# Selected Lectures 2020-2021

## Sendov’s conjecture for sufficiently high degree polynomials

### by Terrence Tao, University of California, Los Angeles (UCLA) | January 26, 2021

Colloquium talk in memory of Acad. Blagovest Sendov by Terence Tao, University of California, Los Angeles (UCLA.) Winner of the Fields Medal 2006, the Breakthrough Prize in Mathematics 2014, the Crafoord Prize 2012.

## Hodge Theory and Local Systems

### by Carlos Simpson, CNRS, Université Côte d’Azur, Nice | March 3-5, 2020

Prof. Carlos Simpson, Alexander Efimov, Ludmil Katzarkov, Viсtor Przyjalkowski and Dmitry Kaledin gave series of lectures on Hodge theory and local systems in the period March 3-5, 2020

## ICMS-IMSA Working Seminar

### by Rene Mboro, Rodolfo Aguilar, ICMS-Sofia and IMSA | August 20, 2021

Prof Rene Mboro gave a talk On determinantal cubic hypersurfaces (after Iliev-Manivel, Beauville,…) Prof Rodolfo Aguilar gave a talk on Quantum representations of fundamental groups of curves with infinite image.

## Lectures of Steven Cantrel and Mina Teicher

### by Robert Steven Cantrel, IMSA and Mina Teicher, | June 30, 2021

Prof Robert Steven Cantrel gave a talk A Journey from Math to Biology and Back Again … and Again and prof. Mina Teicher gave a talk The Mathematics of Beauty

# News

#### Visiting fellowships for the WIMSA

June 14th, 2021|

The International Center for Mathematical Sciences – Sofia (ICMS-Sofia) in collaboration with the Institute of the Mathematical Sciences of the Americas at the University of Miami (IMSA) offers three fellowships for WIMSA 2021 - 2022 Programme.

#### Metric Geometry of Singularities

June 14th, 2021|

The goal of the meeting is to gather researchers interested in metric geometry of singularities and Lipschitz geometry. There will be two, three short courses and eight talks. The event will be held at the University of Chicago Center in Paris with some limited in person participation. All talks will be broadcasted through Zoom.

#### Recent Developments in Hodge Theory

March 24th, 2021|

The ICMS-Sofia invites you to attend the virtual conference Recent Developments in Hodge Theory which is going to be held on March 29 – April 02, 2021. Jointly organized with the Institute of the Mathematical Sciences of the Americas at the University of Miami (IMSA).

March 1st, 2021|