## Monthly Online Colloquium

## Mathematics in Lebanon and beyond

### Institute of Mathematics Potsdam

This colloquium will serve as a meeting place for scientific exchange, bringing together mathematicians from the Lebanese diaspora, friends of Lebanon from around the world, and researchers based in Lebanon. By nature, the colloquium will be interdisciplinary and speakers will be asked to prepare pedagogical talks addressed to a wide audience including non experts in the subject. Presentations can be given in French or English.

Farewell on February 20th 2024:

A year after the opening session (on February 7th 2023) of the monthly online colloquium

**Mathematics in Lebanon and beyond**

which was supposed to serve as a meeting place for scientific exchange, bringing together mathematicians from the Lebanese diaspora, friends of Lebanon from around the world, and researchers based in Lebanon, we unfortunately have to come to the sad conclusion that it has not served its purpose.

Indeed, in spite of the great efforts made by the speakers to deliver talks accessible to the lay person, the attendance has been very low and decreasing with time. The speakers made a special effort which resulted in many beautiful talks!

We have therefore decided to close the colloquium since we do not think it is fair to ask speakers to put so much effort into preparing a presentation of their research in a way that is accessible to a layman, knowing that the audience is so small and may even be reduced to the two organizers.

Lebanon is going through very tough times, which on the one hand motivated the idea of having a colloquium, and on the other hand, also partly explains why mathematicians in Lebanon do not have the time and energy to participate such a colloquium on top of their many teaching duties and own research activities.

Let us say our colloquium has gone into hibernation as spring approaches, and could be reactivated later at a more opportune time!

We thank those among you who contributed to the colloquium with a talk and those who supported the colloquium by a faithful attendance.

Thank you to all,

Take care,

Sylvie and Georges

**Past talks:**

Monday February 19th 2024

- 3pm (CET)
- Barbara Nelli (University of l'Aquila, Italy): "The Jenkins-Serrin theorem reloaded"

Abstract:

In the 1960s, H. Jenkins and J. Serrin established a well-known theorem concerning minimal graphs in Euclidean 3-space with infinite boundary values. Over the past two decades, the Jenkins-Serrin problem has experienced a renewed interest. In this talk I will revisit the classical results and extend the discussion to include a very recent outcome: the solution to the Jenkins-Serrin problem in a 3-manifold with a Killing vector field.

- 3:30pm (CET)
- Kamal Khuri Makdisi (AUB, Beirut): "Meromorphic functions and projective embeddings of abelian varieties"

Abstract:

A g-dimensional abelian variety over the complex numbers is a special kind of complex torus C^g/L, where L is a lattice of rank 2g. The case g=1 is called an elliptic curve. This talk will mostly be expository, about using theta functions to construct meromorphic functions on abelian varieties, and the corresponding embeddings of abelian varieties into projective space. If time permits, I will conclude with a description of the Edwards model of elliptic curves, and its generalization to abelian surfaces in a recent collaboration between E. V. Flynn and myself.

Slides

- 3:30pm
- Marwa Banna (Saarland University and New York University of Abu Dhabi): "Deriving quantitative estimates on random matrices using free probability tools"

Abstract:

In this talk, I start with a brief introduction to random matrices and free probability theory and highlight the connection between the two domains. I will then show how free probability tools can be used to obtain regularity properties of limiting spectral distributions and how to quantify such convergences for some random matrix models. In particular, we consider a class of random block matrices and derive quantitative estimates on the level of Cauchy transforms that can be passed to the Kolmogorov distance in some cases. The content of the talk is based on results from joint works with G. Cébron and T. Mai.

as part of the conference

"At the confluence of Geometry, Analysis and Mathematical Physics"

October 10-12, 2023, at the American University of Beirut

https://sites.aub.edu.lb/humboldt/workshops/meeting-2023/"

- 2pm CET (3pm Beirut)
- Isabelle Chalendar (Université Gustave Eiffel, Marne-La-Vallée): "Discrete vs Continuous for linear and continuous operators on Banach spaces"

Abstract:

Given a linear and continuous operator T on a Banach space X, one can consider the discrete semigroup from its (positive) integer powers and a delicate problem consists in finding whether T can be embedded in a continuous semigroup T_t parametrised by a non-negative parameter t. There are obvious sufficient conditions involving spectral properties of T but in general we only have a few examples such as the famous Volterra operator V on various Banach spaces. This operator will be at the center of the talk and compactness properties will be analyzed.

This is joint work with Ihab Al Alam, Fida El Chami, Emmanuel Fricain, Georges Habib and Pascal Lefèvre.

Our collaboration was funded by "Théorale", Cèdre, Project Hubert Curien.

- 3pm CET:
- Wael Bahsoun (Loughborough University): "Statistical properties of chaotic dynamical systems via spectral methods" Abstract:

Smooth ergodic theory aims to analyse the long-term statistics of chaotic dynamical systems. There are several analytic and probabilistic tools that are used to answer such questions. Each of these approaches has its advantages and its shortcomings, depending on the system under consideration. In this talk, I will focus on transfer operator techniques and spectral methods. In the first half of this talk, I will explain ideas behind this approach through simple, yet important examples. In the second half of the talk, I will discuss a recent joint work with C. Liverani, whose long-term goal is to provide a good spectral picture for pricewise hyperbolic systems with singularities (e.g. billiard maps) in any dimension.

- 3pm CET:
- Pierre Youssef (New York University, Abu Dhabi): "Phenomena in High Dimensions" Abstract:

What do high dimensional convex bodies look like? How does the spectrum of a random matrix behave as its dimension grows? How does the connectivity of a graph evolve as its size grows? In this talk, I will try to discuss some phenomena that arise as the "dimension" grows in several related contexts.

- 3pm CET:
- Joseph AYOUB (University of Zurich): "Symmetries in algebraic geometry and motives" Abstract:

Starting from the general notion of `symmetry' in mathematics, we give a gentle introduction to the theory of motives, which is the universal cohomology theory for algebraic varieties.

Slides

- 3pm CET:
- Abdallah ASSI (University of Angers): "Numerical semi-groups and applications" Abstract:

A numerical semigroup (NSG for short) is a monoid S of the set N of non negative integers such that the group generated by S in Z is Z (that is, S is stable under addition, 0 is an element of S, and the GCD of the elements of S is 1). Given a NSG S, there exists a finite number of elements, a_1,...,a_n, of S such that S=a_1N+....+a_nN. Moreover, there exists an integer in N\S, denoted F(S), such that every integer > F(S) is in S. We call F(S) the Frobenius number of S. Then we can associate with S many numerical invariants, and this gives rise to many combinatorial problems and conjectures. In addition, NSGs appear in many areas in mathematics, in particular they give us an arithmetic tool in the problem of classification of singularities of plane algebraic curves. The purpose of this talk is to give a review of the results, problems, conjectures, and applications related to this field.

- 4pm-4:50pm CET:
- Karma Dajani (Utrecht University): "Interactions between ergodic theory and number theory: from beta-expansions to the Sierpinski gasket" Abstract

Opening session: Tuesday February 7th 2023

- 3pm-3:50pm CET:
- George Saliba (American University of Beirut): "Revisiting Islamic Science and Renaissance Europe"

- 4pm:-4:50pm CET:
- Hussein Mourtada (IMJ, Université Paris-Cité): "Singularities and integer partitions" Abstract:

An integer partition of an integer number is simply a decreasing sequence of integers whose sum is equal to n. Naturally, integer partitions and their theory are ubiquitous in mathematics. I will report on a link between an algebro-geometric invariant and the theory of integer partitions in the spirit of Ramanujan. The talk is aimed at a wide audience of mathematicians.

**Contact:**

- Lisa Franz (Potsdam) lfranz@uni-potsdam.de

Zoom link:

Join Zoom Meeting

https://uni-potsdam.zoom.us/j/8733076196

Meeting ID: 873 307 6196

Passcode: 29634559

**Organising committee:**

- Georges Habib (Beirut)
- Sylvie Paycha (Potsdam)

Impressum