# Colloquium

The colloquium of the Center for Mathematical Sciences, University of Lund, normally runs once a month. It is aimed at the entire Centre for Mathematical Sciences with overview talks by renowned experts about exciting mathematical topics.

Talks are usually scheduled for Wednesdays 15.15 - 16.15 in the Hörmander lecture hall placed on the ground floor of the Mathematics building. The room is equipped with 6 large blackboards and two projectors.

The purpose of our colloquium is twofold: firstly, it is to provide an inspiring overview of a specific field of mathematics, secondly, it is to bring together students and staff from the entire department and to serve as the proverbial waterhole where contacts are made and maintained.

The colloquium is organized by Yacin Ameur, Dragi Anevski, Magnus Goffeng, Tony Stillfjord and Karl Åström. Feel free to contact any one of us for questions or suggestions for colloquia speakers.

## Colloquia, Autumn 2023

### September 13

**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.

Reference:

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

### October 4

**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.

Reference:

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

### November 1

**Bernhard Keller** (Université Paris Cité)

Title: **From Coxeter-Conway friezes to cluster algebras**

Abstract:

TBA

### November 22

**Haluk Sengun **(University of Sheffield)

Title: **Some number theoretic aspects of the cohomology of arithmetic groups**

Abstract:

TBA

### December 13

TBA

Title: TBA

Abstract: TBA