CUSO Graduate Colloquium

Colloquium schedule

Titles and Abstracts

Wednesday

Ioan Manolescu (University of Fribourg): An invitation to percolation.
Percolation was introduced in 1957 as a surprisingly simple probabilistic model for liquids diffusing through porous materials. It is one of a large variety of statistical mechanics models: large systems of particles governed by local interactions that combine to produce surprising large-scale behaviours. In this talk we will discuss percolation -- and its more general variant called FK-percolation -- on the d-dimensional lattice Z^d. We will touch on topics such as the sharpness of the phase transition, its continuity or discontinuity and finally, will mention some recent progress on the critical phase of the two dimensional case.

Nicola Paddeu (University of Fribourg): Normal curves in metabelian nilpotent groups.
We consider nilpotent Lie groups for which the commutator subgroup is abelian. We equip them with subRiemannian metrics and we study length-minimizing curves. We focus on normal geodesics: length-minimizing curves that are solutions of a Hamiltonian type differential equation. We use a symplectic reduction procedure to prove a correspondence between normal geodesics and polynomial Hamiltonians in some Euclidean space. From this correspondence we get several results on the global behaviour of normal curves.

Anna Katharina Bot (University of Basel): A smooth complex rational affine surface with uncountably many nonisomorphic real forms.
A real form of a complex algebraic variety X is a real algebraic variety whose complexification is isomorphic to X. Many families of complex varieties have a finite number of nonisomorphic real forms, but up until recently no example with infinitely many had been found. In 2018, Lesieutre constructed a projective variety of dimension six with infinitely many nonisomorphic real forms, and last year, Dinh, Oguiso and Yu described projective rational surfaces with infinitely many as well. In this talk, I’ll present the first example of a rational affine surface having uncountably many nonisomorphic real forms.

Thursday

Kamil Khettabi (University of Geneva): Deriving macroscopic properties of a physical system from a microscopic description
I will explain how one can study macroscopic properties of a diluted gas from its microscopic description. For instance, one can show that at sufficiently small temperature and well chosen density, the gas forms a macroscopic bubble with a deterministic shape, even though randomness is occuring in the mathematical description of such phenomenon. I will then explain what happens when one takes into consideration the gravitationnal field.

Luke Higgins (University of Fribourg): Milnor spheres, and metrics of nonnegative sectional curvature on them.
In a paper from 1956, John Milnor presents a family of manifolds which are homeomorphic to a sphere but which carry non-standard smooth structures. This talk will present Milnor’s construction of such manifolds – so-called Milnor spheres – followed by a survey of some interesting results regarding them.

Marco Inversi (University of Basel): Energy conservation for fluid flows in an Onsager critical class.
The incompressible Euler equations govern the evolution of an ideal fluid. It is well known that the total kinetic energy is preserved along the time evolution of a regular fluid flow. However, when the motion is very rough, there is theoretical and experimental evidence of formation of chaotic structures that support the dissipation of kinetic energy. Mathematically, this problem translates into finding the critical regularity for weak solutions to the incompressible Euler equations to have conservation or dissipation of kinetic energy (Onsager’s conjecture). Currently, the Onsager conjecture is almost solved. It has been been proved that energy is conserved in any subcritical class and there are examples of solutions in any supercritical class violating the energy conservation. In a joint paper with Luigi De Rosa, we gave the first proof of energy conservation for weak solutions to the incompressible Euler system in a critical space, both in absence and presence of physical boundary. This is the first energy conservation result that holds in the incompressible case and fails in the compressible setting.

Letizia Issini (University of Geneva): Lamplighter and divergence in groups.
In this talk, we will present some approaches used in geometric group theory, such as the study of quasi-isometry invariants of finitely generated groups. You may have heard about some of these, like growth and amenability. We will be talking about another one, divergence. The lamplighter group is an interesting example of a finitely generated group, and starting from it we will describe the algebraic structure of a wreath product. Finally, we will sketch the proof that wreath products of groups have linear divergence.

Simon Santschi (University of Bern): Interpolation in Substructural Logics with Exchange.
The full Lambek calculus with exchange (FLe) is a natural generalization of intuitionistic logic (IL) and various other logics in the sense that they can be obtained by adding extra axioms/rules to FLe. A classical result of Maksimova states that there are exactly 8 axiomatic extensions of IL with the deductive interpolation property, an important property for logics. The picture for axiomatic extensions of FLe is much less clear. Indeed, a characterization is only known in very specific cases. From the current literature we know that there are at least countably infinitely many axiomatic extensions of FLe with the deductive interpolation property. A natural question is whether there are uncountably many such axiomatic extension. In this talk I will give a positive answer to this question and show that there are continuum-many axiomatic extensions of FLe with the deductive interpolation property. The method to show this is algebraic in nature and uses the algebraizability of FLe and its axiomatic extensions. In fact, we show that there exist continuum-many classes of algebraic models that have the amalgamation property, which is the algebraic counterpart of the deductive interpolation property. These classes are constructed using certain classes of abelian groups, and the proof of the amalgamation property relies on the classical result from abelian group theory that every abelian group essentially embeds into an injective abelian group. This is joint work with Wesley Fussner.

Friday

Léonard Tschanz (University of Neuchâtel): The Steklov problem on hypersurface of revolution: definitions, results and conjecture
The goal of this presentation is to get familiar with the Steklov problem on hypersurfaces of revolution of the Euclidean space. We will define what are the mathematical objects that want to study, and present some recent results that hold about them as well as the general ideas of the proofs. In the process, an interesting question will naturally be raised; we will formulate a conjecture about it, conjecture that we will defend with numerical experiments.

Ulrik Hansen (University of Fribourg): The Many Connections of Statistical Mechanics.
From a physical point of view, statistical mechanics pertain to the description of macroscopic thermodynamic theory through random microscopic models. Sometimes, the same macroscopic phenomenon admits description through multiple mathematically viable microscopic models, with the most famous being the relation between random walks and electrical networks. As such, one should not be surprised to see that long, deep connections permeate such models. In this talk, we shall briefly introduce Bernoulli percolation, which is the simplest model of a random subgraph, along with the fundamental questions one asks in its study. Then, we shall go on to show that many natural modifications of the model all turn out to be related in one way or another: Either by direct coupling or in spirit. Thus, one may take the same questions that one studies for Bernoulli percolation, ask them for these other models, and conclude that the models have the kindness to help answering for each other. Based on joint work with Boris Kjær and Frederik Ravn Klausen.