# Colloquia, Autumn 2023

## Colloquia, Autum 2023

### February 1

#### Speaker

Klas Modin (Chalmers and University of Gothenburg)

#### Title

A Brief History of Geometric Mechanics

#### Abstract

In this talk I wish to give an overview of the tight connection between differential geometry and classical mechanics. I shall take a historical route, going from Newton and Euler, via Lagrange and Hamilton, Poincaré and the Prize of King Oscar II, to Arnold and the Riemannian description of hydrodynamics.

### March 1

#### Speaker

Alain Valette (University of Neuchâtel)

#### Title

The Wasserstein distance on metric trees

#### Abstract

Let (X,d) be a metric space. The Wasserstein distance, or earthmover distance, is a metric on the space of probability measures on X, that originates from optimal transportation: if \mu,\nu are probability measures on X, the Wasserstein distance between \mu and \nu intuitively represents the minimal amount of work necessary to transform \mu into \nu. Since the definition of the Wasserstein distance involves an infimum, it is not expected in general that there is a closed formula for this distance. Such a closed formula however exists for metric trees (i.e. combinatorial trees where the length of an edge can be any positive real number), and this closed formula has an interesting history that makes it suitable for a colloquium talk: it appeared first in computer science papers (Charikar 2002), then surfaced again in bio-mathematics (Evans-Matsen 2012), before catching the interest of pure mathematicians. In joint work with M. Mathey-Prévôt, we advocate that the right framework for this closed formula is real trees (i.e. geodesic metric spaces with the property that any two points are connected by a unique arc); we give two proofs of the closed formula, one algorithmic, the other one connecting with Lipschitz-free spaces from Banach space theory.

### March 29

#### Speaker

Jeff Steif (Chalmers and University of Gothenburg)

#### Title

Boolean Functions, Critical Percolation and noise sensitivity

#### Abstract

I will introduce and discuss the notion of noise sensitivity for Boolean functions, which captures the idea that certain events are very sensitive to small perturbations. While a few examples will be given, the main example which we will examine from this perspective is so-called 2-dimensional critical percolation from statistical mechanics. There will also be some connections to combinatorics and theoretical computer science. The mathematics behind the story includes, among other things, Fourier analysis on the hypercube. No background concerning percolation or Fourier analysis will be assumed.

### April 26

#### Speaker

Steen Markvorsen (Technical University of Denmark)

#### Title

A view towards some applications of Riemann-Finsler geometry

#### Abstract

In this talk I will comment on some Finsler-type geometric key phenomena that arise naturally in such otherwise disparate fields and topics as: Seismology; dMRI (diffusion Magnetic Resonance Imaging); Wildfire Modelling; the metric of Colour Space; and (if time permits) Riemann-Finsler Conductivity.

### May 17

#### Speaker

Josephine Sullivan (KTH)

#### Titl

### 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.

Are All the Linear Regions of a ReLU Network Created Equal?

#### Abstract

This presentation will describe recent research described in the paper "Are All Linear Regions Created Equal?" published at AISTATS 2022. The function represented by most ReLU neural networks is piecewise affine. The number of linear regions defined by this function has been used as a proxy for the network's complexity. However, much empirical work suggests, especially in the overparametrized setting, the number of linear regions does not capture the effective non-linearity of the network's learnt function. We propose an efficient algorithm for discovering linear regions and use it to investigate the effectiveness of density in capturing the nonlinearity of trained VGGs and ResNets on CIFAR-10 and CIFAR-100. We contrast the results with a more principled nonlinearity measure based on function variation, highlighting the shortcomings of linear regions density. Furthermore, interestingly, our measure of nonlinearity clearly correlates with model-wise deep double descent, connecting reduced test error with reduced nonlinearity, and increased local similarity of linear regions.