Colloquia, Spring 2023
Klas Modin (Chalmers and University of Gothenburg)
Title: A Brief History of Geometric Mechanics
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.
Alain Valette (University of Neuchâtel)
Title: The Wasserstein distance on metric trees
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.
Jeff Steif (Chalmers and University of Gothenburg)
Title: Boolean Functions, Critical Percolation and noise sensitivity
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.
Steen Markvorsen (Technical University of Denmark)
Title: A view towards some applications of Riemann-Finsler geometry
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.
Josephine Sullivan (KTH)
Title: Are All the Linear Regions of a ReLU Network Created Equal?
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.
Colloquia, Autumn 2022
Tom Britton (University of Stockholm)
Title: Mathematical models for epidemics (including Covid-19)
In the talk I will first describe the basics for mathematical epidemic models, and indicate the many extensions that exist. Then I will briefly describe three problems I (and many others) have worked on during Covid-19: Herd immunity, Optimizing interventions in time and magnitude, and Estimating the effects of various behaviour changes during the epidemic.
Gustaf Söderlind (University of Lund, emeritus)
Title: Logarithmic norms and Quadratic forms
The logarithmic norm was introduced in 1958 for matrices, for the purpose of estimating growth rates in initial value problems. Since then, the concept has been extended to spectral theory, nonlinear maps, differential operators and function spaces. There are applications to operator equations in general, including evolution equations as well as boundary value problems and PDEs. A broad introduction to the topic will be given, demonstrating the recurring pattern found in various applications.
Jan Philip Solovej (University of Copenhagen)
Title: The dilute limit of quantum gases
I will introduce the mathematics of quantum many-body problems. In particular, I will discuss systems of many interacting particles that obey either Bose or Fermi statistics. I will discuss the notion of the thermodynamic limit, where the size of the system tends to infinity while the particle density is fixed. The quantity of main interest is the ground state energy density (energy per volume) as a function of particle density. I will review some results in the limits of large and small particle density in 1, 2, and 3 dimensions. The main focus will be on small density, i.e., the dilute limit. This has been an area of very active research in the past few years and an old conjecture from the 1950s relating the issue to the theory of superfluidity has been resolved. There are however still many open problems.
Gustavo Jasso (University of Lund)
Title: Brackets, trees and the Borromean rings
I will describe some of the combinatorics that emerge from the familar associativity equation
(ab)c = a(bc),
and explain how these encode fundamental algebraic structures that are now ubiquitous in Algebra, Geometry and Topology.
Fredrik Kahl (Chalmers University of Technology)
Title: Deep Learning for Geometry and Geometry for Deep Learning