# Colloquia, Autumn 2023

## Colloquia, Autum 2023

### September 13

#### Speaker

Per Enflo (Kent State University, emeritus)

#### Title

On the Invariant Subspace Problem in Hilbert Space

#### Abstract

A method to construct invariant subspaces for a general operator T on Hilbert space is presented. It represents a new direction of my "method of extremal vectors", first presented in 1998 in Ansari-Enflo [1].

The Main Construction of the method gives a non-cyclic vector of T by gradual approximation by "almost non-cyclic" vectors. There are reasons, why the Main Construction cannot work for some weighted shifts. But when the Main Construction fails, one gets, by using the information obtained, invariant subspaces of T, similar to those of weighted shifts.

The sequence y(n) of "almost non-cyclic" vectors follows the formula y(n+1) = y(n) + r(T) y(n), which will allow for efficient use of elementary Fourier Analysis. The more general formula y(n+1) = y(n) + z would lead to difficult problems concerning the relation between T and T*, problems which may be of interest in themselves.

#### References

- S. Ansari, P. Enflo, "Extremal vectors and invariant subspaces", Transactions of Am. Math. Soc. Vol. 350 no.2, 1998, pp.539-558.

### October 4

#### Speaker

Martin Gander (University of Geneva)

#### Title

The history of iterative methods for linear systems

#### Abstract

Iterative methods for linear systems were invented for the same reasons as they are used today, namely to reduce computational cost. Gauss states in a letter to his friend Gerling in 1823: "you will in the future hardly eliminate directly, at least not when you have more than two unknowns".

After a historical introduction to such classical stationary iterative methods, I will explain how the idea of extrapolation leads to Krylov methods, which are in fact not solvers but convergence accelerators.

I will then introduce modern iterative methods for solving partial differential equations, which come in two main classes: domain decomposition methods and multigrid methods. These methods develop their full potential when used together with Krylov methods, namely as preconditioners.

#### References

- A History of Iterative Methods, Martin J. Gander, Philippe Henry and Gerhard Wanner, in preparation, 2023

### November 1

#### Speaker

Bernhard Keller (Université Paris Cité)

#### Title

From Coxeter-Conway friezes to cluster algebras

#### Abstract

Since their invention by Fomin-Zelevinsky in 2002, cluster algebras have shown up in an ever growing array of subjects in mathematics (and in physics). In this talk, we will approach their theory starting from elementary examples. More precisely, we will see how the remarkable integrality properties of the Coxeter-Conway friezes and the Somos sequence find a beautiful unification and generalization in Fomin-Zelevinsky's definition of cluster variables and their Laurent phenomenon theorem. Motivated by the periodicity of Coxeter-Conway friezes, we will conclude with a general periodicity theorem, whose proof is based on the interaction between discrete dynamical systems and quiver representations through the combinatorial framework of cluster algebras.

### November 22 (CANCELLED)

Haluk Sengun (University of Sheffield)

#### Title

Some number theoretic aspects of the cohomology of arithmetic groups

#### Abstract

Put roughly, an arithmetic group is a group of matrices with integral entries defined by polynomial equations. The modular group SL(2,Z) is a prime example. The complex cohomology of arithmetic groups have fascinating connections with number theory and in this expository talk, I will try to illustrate a couple of these connections. I will start with some well-known connections and will make my way slowly towards more recent works that aim to bring in torsion classes in the integral cohomology into the picture.

### December 13

#### Speaker

Andrey Ghulchak (Lund University)

#### Title

The double bubble problem and beyond

#### Abstract

A single soap bubble quickly finds the least-surface-area way to enclose a fixed volume of air — the round sphere. This fact has been around for thousands of years since Zenodorous tried to prove it. Despite the fact that the problem seems so apparent, a complete proof was not found until the 1890s when H.Schwarz proved existence of the solution.

In case of two fixed volumes, Plateau's observations of soap bubbles in 1873 suggested that the surface had to be the standard double bubble — three spherical parts meeting along a circle. To prove it turned out to be a challenge and required new deep results now known as Geometrical Measure Theory. It was not resolved until around year 2000.

Furthermore, the case with three given volumes is so much harder that one of the authors of the double bubble proof called it "inaccessible", and a great surprise arose when the triple bubble conjecture was claimed proven in 2022 by two young mathematicians.

In this talk, we will follow the ideas behind the problem development since the very origin until the last year.