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Program Oxford days in Grenoble, 2019
In the context of the partnership of University of Oxford, and University Grenoble Alpes, we welcome six general audience talks in Geometry, Topology, Analysis, Probability and Statistics.
Location : Grenoble, Amphi tour IRMA (click for a map) Program THURSDAY 10th of October 13h45 - Welcome 14-14h55. Panos Papasoglu "Uryson width and volume" 15h05-16h. Marc Lackenby "The triangulation complexity of 3-manifolds" break 16h45-17h40. Jason Lotay "Minimal Lagrangians and where to find them"
FRIDAY 11th of October 9h15 - 10h10. Melanie Rupflin. "Minimal surfaces and geometric flows" 10h20 - 11h15. Christina Goldschmidt "The critical random transposition random walk" break 11h45-12h40. Julien Berestycki. Brownian bees in the infinite swarm limit
================================================== Panos Papasoglu: Uryson width and volume Abstract: I will give a brief survey of some problems in curvature free geometry and sketch a new proof of the following: Theorem (Guth). There is some \delta (n)>0 such that if (M^n,g) is a closed aspherical Riemannian manifold and V(R) is the volume of the largest ball of radius R in the universal cover of M, then V(R)\geq \delta(n) R^n for all R. I will also discuss some recent related questions and results.
Marc Lackenby : The triangulation complexity of 3-manifolds.
Abstract: The triangulation complexity of a closed orientable 3-manifold M is the minimal number of tetrahedra in any triangulation of M. It is a natural, but poorly understood, invariant. In my talk, I will explain how it may be computed, to within a bounded factor, for any hyperbolic 3-manifold that fibres over the circle with fibre a given closed orientable surface S. I will show that it is equal, up to a bounded factor, to the translation distance of the action of the monodromy on the mapping class group of S. I will also explain how the methods that we develop can be applied to lens spaces; we determine their triangulation complexity to within a universally bounded factor. All this is joint work with Jessica Purcell.
Jason Lotay: Minimal Lagrangians and where to find them Abstract: A classical problem going back to ancient Greece is to find the shortest curve in the plane enclosing a given area: the isoperimetric problem. A similar question is whether given a curve on a surface it can be deformed to a shortest one. Whilst the solutions to these classical problems are well-known, natural generalisations in higher dimensions are mostly unsolved. I will explain how this leads us to the study of minimal Lagrangians and the question of how to find them, which will take us to the interface between symplectic topology, Riemannian geometry and analysis of nonlinear PDEs, with links to theoretical physics. Melanie Rupflin: Minimal surfaces and geometric flows
Abstract: The classical Plateau problems has been one of the most influential problems in the development of modern analysis. Posed initially by Lagrange, it asks whether a closed curve in Euclidean space always spans a surfaces with minimal possible area, a question that was answered positively by Douglas and Rado around 1930.
In this talk I want to consider some aspects of the classical Plateau Problem and its generalisations and discuss furthermore how one can "flow" to such minimal surfaces by following a suitably defined gradient flow of the Dirichlet energy, i.e. of the integral of gradient squared.
Christina Goldschmidt: The critical random transposition random walk
Abstract: Create continuous-time random walk on the symmetric group by successively composing independent transpositions chosen uniformly at random from among the possibilities, at rate n/2. The uniform distribution is stationary for this Markov chain. A well-known result of Schramm states that this process undergoes a phase transition: if t < 1, the cycles of the permutation at time t are O(log n) in size, whereas for t > 1, a positive proportion of the numbers {1,2,\ldots,n} are contained in giant cycles, whose relative sizes are distributed approximately as Poisson-Dirichlet(0,1) (so that although the whole random walk is far from having reached stationarity, it has mixed on part of the space). In this talk, I will characterise the behaviour of the critical random transposition random walk, and shed light on the emergence of the Poisson-Dirichlet distribution. This is joint work with Dominic Yeo.
Julien Berestycki. Brownian bees in the infinite swarm limit Abstract: The N-Branching brownian motion is a particle system in which independent particles move on the real-line as Brownian motions, branch at rate one and the total population size is kept constant equal to N by removing the leftmost particle at each branching event. This model, which was introduced by Brunet and Derrida to study certain noisy reaction-diffusion equations, can also be seen as a model for the evolution of a population under selection. Its hydrodynamic limit was shown recently to exist and satisfies a free boundary problem. I will present an analogous result in higher dimension (the so-called Brownian bees model) for which we can prove the existence of the hydrodynamic result, the global existence of the solution to the associated free boundary problem and describe the large time behaviour of the particle system and of its deterministic limit.
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