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.