Swansea - Grenoble colloquium in mathematics

Grenoble, September 11-12, 2017

All talks will take place in seminar room 4, ground floor of Institut Fourier

Monday, September 11:

9:30 Opening

10:00 Jeffrey Giansiracusa (Swansea and IF)

11:00 Coffee and tea

11:30 Edwin Beggs (Swansea)

12:30 Buffet lunch (2nd floor)

14:00 Jean Fasel (IF)

15:00 Grigory Garkusha (Swansea)

16:00 Coffee and tea

16:30 Round table

Tuesday, September 12:

9:00 Dmitri Finkelshtein (Swansea)

10:00 Coffee and tea

10:30 Elaine Crooks (Swansea)

11:30 Biagio Lucini (Swansea)

12:30 Buffet lunch (2nd floor)

14:00 Raphaël Rossignol (IF)

15:00 Pierre Saramito (LJK)

Organizers:

Edwin Beggs, Elaine Crooks, Biagio Lucini (Swansea)

Jean Fasel, Thierry Gallay, Louis Funar (Grenoble)

Titles and abstracts

Edwin Beggs (Swansea)

Title: Applications of noncommutative and complex geometry

Abstract: I will talk about noncommutative differential geometry from the point of view of forms, vector fields and connections. The philosophy is that most of the methods of classical differential geometry should extend to noncommutative geometry. Noncommutative models of calculus provide a novel way to treat conserved quantities in numerical analysis, but also raise interesting topological questions. However, it turns out that for studying many examples of relevance for Physics we are also forced into nonassociative geometry. I will talk about the importance of measurement in theories of quantum gravity. I will also talk about the application of complex analysis to the inverse scattering method in integrable systems in 1+1

dimensions, and the possibility (no more than that at the moment) of extending this to solitons in higher dimensions.

Elaine Crooks (Swansea)

Title: Travelling waves in anisotropic smectic C* liquid crystals

Abstract: We consider minimality conditions for the speed of monotone travelling waves in a model of a sample of smectic C* liquid crystal subject to a constant electric field, dealing with both isotropic and anisotropic cases. Such conditions are important in understanding switching properties of a liquid crystal, and our focus is on understanding how the presence of anisotropy can affect the speed and nature of switching. Through a study of travelling-wave solutions of a quasilinear parabolic equation, we obtain an estimate of the influence of anisotropy on the minimal speed, sufficient conditions for linear and non-linear minimal speed selection mechanisms to hold in different parameter regimes, and a characterisation of the boundary separating the linear and non-linear regimes in parameter space. This is joint work with Michael Grinfeld and Geoff Mackay (Strathclyde).

Jean Fasel (IF, Grenoble)

Title: Correspondences between correspondences

Abstract: In this talk, we will briefly survey the construction of Voevodsky’s category of motives. We will then explain the need for a more general version of this category, and sketch a few possible constructions: framed correspondences (after Voevodsky-Garkusha-Panin) and finite Chow-Witt correspondences. We will then explain the relationships between those, and (time permitting) give a few computations.

Dmitri Finkelshtein (Swansea)

Title: Long-time behavior for monostable equations with nonlocal diffusion

Abstract: We consider a class of monostable nonlinear equations with nonlocal diffusion and local or nonlocal reaction in R^d. If the nonlocal diffusion is given through an anisotropic probability kernel decaying at least exponentially fast in a direction and if the initial condition has the similar property, then the solution propagates in this direction at most linearly in time. For a particular equation arising in population ecology, we study then traveling waves and describe the front of propagation. In contrast, if either the kernel or the initial condition (for the general equation) have appropriately regular heavy tails, then an acceleration of the front propagation for solutions is observed and studied. For the one-dimensional case we show also that the propagation to the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may yield different orders of the acceleration.

Grigory Garkusha (Swansea)

Title: Reconstructing rational stable motivic homotopy theory

Abstract: Using a recent computation of the rational minus part of SH(k) by Ananyevskiy-Levine-Panin, a theorem of Cisinski-Déglise and a version of the Röndigs-Řstvćr theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered from finite Milnor-Witt correspondences in the sense of Calmčs-Fasel.

Jeffrey Herschel Giansiracusa (Swansea and Grenoble)

Tropical geometry as a scheme theory

Tropical geometry has become a powerful toolset for tackling problems in algebraic geometry, combinatorics, and number theory. The basic objects have traditionally been considered as certain polyhedral sets and heuristically thought of as algebraic objects defined over the real numbers with the max-plus semiring structure. I will explain how to realize this within an extension of scheme theory and describe the particular form of the equations of tropical varieties in terms of matroids.

Biagio Lucini (Swansea)

Title: Efficient numerical simulations and strong metastabilities: from the 2D Potts to Lattice Gauge Theories

Abstract: Thanks to the availability of exact results, two dimensional spin systems are often taken as a reference frame to test novel Monte Carlo techniques. In this context, the Potts model at large number of states can be used to assess the ability of a numerical method to overcome strong metastabilities. In this talk, I will discuss a recently introduced Monte Carlo method (known in the literature as the LLR method) that has been proved to be very efficient at dealing with metastabilities arising at first order transition points and I will compare numerical results with the exact results on a q=20 Potts model. The striking outcome of the study is that the numerical determinations of thermodynamic observables (including the order-disorder interface tension at criticality) displays an impressive degree of accuracy with respect to the exact value (down to a level of a few parts in a million). This confirms the reliability and the efficiency of the LLR method, suggesting its applicability also in scenarios in which exact results are not available. Indeed I will show that the method enables us to go well beyond the state of the art in a four-dimensional compact U(1) model at criticality.

Raphaël Rossignol (IF)

Title: Scaling limits of dynamical percolation on critical Erdös-Rényi random graphs

Abstract: An Erdös-Rényi random graph with parameters p and n is a graph on n vertices where each possible edge is present with probability p, independently from the others. When n is large, there is a particular scaling of p where one assists to the appearance of a giant connected component. Inside this "critical" window, focusing on the largest components, Addario-Berry, Broutin and Goldschmidt have shown in 2012 that one may renormalize the components in order to obtain convergence in distribution, when the number of vertices n goes to infinity, to a collection of random graphs that we shall call the CER (for Continuous Erdös-Rényi random graph). The aim of this talk is to explain what is the limit, as n goes to infinity, of a special resampling procedure of the Erdös-Rényi random graph (dynamical percolation).

Pierre Saramito (LJK)

Title: Some computations with level-sets, surface PDEs and non-Newtonian fluid mechanics

Abstract: We consider three problems from applied mathematics and numerical computations, motivated by interactions with physicians and biologists. The first problem bases on the mathematical Canham-Helfrich model for red blood cells, membranes and vesicles, and its numerical resolution involves the level-set method and the eikonal equation. The second problem deals with surface PDEs, for the development of biological tissues by a continuum modeling. A concrete application to the embryogenesis is presented: it bases on a Stokes-like problem on arbitrarily complex closed surfaces. The third problem considers a continuum model for liquid foams: it involves complex fluid mechanics, combining viscous, plastic and elastic effects together, in an elastoviscoplastic fluid model. All these three problems are presented with few words on the domain of application and examples of numerical computations and comparisons with physical observations.