Analysis and PDE Seminar: semester plan

January

• January 12, 2023: Stefan Le Coz, Toulouse
  • Title: Ground states on nonlinear quantum graphs
     
  • Abstract: The nonlinear Schrödinger equation is an ubiquitous model in physics, with numerous applications in areas as divers as Bose-Einstein condensation or nonlinear optics. In many physical situations, the underlying space is essentially one dimensional and can be modeled as a metric graph, i.e. a collection of  vertices and edges with finiter or infinite lengths. The mathematical study of this type of model is very recent and is gaining an incredible momentum. In this talk, we will review some of the results concerning the existence of nonlinear Schrödinger ground states on graphs and present a numerical approach for their computation. This is a joint work with Christophe Besse and Romain Duboscq.
• January 26, 2023: Frédéric Valet, Bergen
  • Title: Collision of two solitary waves for the Zakharov-Kuznetsov equation
     
  • Abstract: The Zakharov-Kuznetsov (ZK) equation in dimension 2 is a generalization in plasma physics of the one- dimensional Korteweg de Vries equation (KdV). Both equations admit solitary waves, that are solutions moving in one direction at a constant velocity, vanishing at infinity in space. When two solitary waves collide, two phe- nomena can occur: either the structure of two solitary waves is conserved without any loss of energy and change of sizes (elastic collision), or the structure is lost or modified (inelastic collision). As a completely integrable equation, KdV only admits elastic collisions. The goal of this talk is to explain the collision phenomenon for two solitary waves having almost the same size for ZK, and to describe the inelasticity of the collision. The talk is based on current works with Didier Pilod.

March

• March 16, 2023: Abdon Moutinho, Paris Nord
  • Title: On the collision problem of two kinks with low speed.
     
  • Abstract: We will talk about our results on the elasticity of the collision of two kinks with low speed v>0 for the nonlinear wave equation of dimension 1+1 known as the phi^6 model. We will show that the collision of the two solitons is "almost" elastic and that, after the collision, the size of the energy norm of the remainder and the size of the defect of the speed of each soliton can be, for any k>0, of the order of any monomial v^{k} if v is small enough.
     
• March 23, 2023: Gianmarco Vega-Molino, Bergen
  • Title: Horizontal Holonomy of H-type Foliations
     
  • Abstract: In this talk I will present the notion of sub-Riemannian holonomy for H-type foliations, a class of sub-Riemannian manifolds jointly introduced with Fabrice Baudoin, Erlend Grong, and Luca Rizzi. Arising as certain sub-Riemannian geometries transverse to Riemannian foliations, they are ideally suited to study the intersection of "extrinsic" and "intrinsic" approaches to sub-Riemannian geometry. Riemannian holonomy is the study of endomorphisms of the tangent spaces induced by the parallel transport of affine connections; in a sub-Riemannian setting one can study an analogue via adapted connections.

    The talk will include an introduction to these structures suitable for persons not familiar with sub-Riemannian geometry, and we will come to a theorem describing the horizontal holonomy of these spaces.

Easter break

April

• April, 20, 2023: Didier Pilod, Bergen
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• March 30, 2023: Erlend Grong, Bergen
  • Title: Sub-Riemannian geometry, most probable paths and transformations.
     
  • Abstract: Doing statistics on a Riemannian manifold becomes very complicated for the reason that we lack pluss to define such things as mean and variance. Using the Riemannian distance, we can define a mean know as the Fréchet mean, but this gives no concept of asymmetry, also known as anisotropy. We introduce an alternative definition of mean called the diffusion mean, which is able to both give a mean and the analogue of a covariance matrix for a dataset on a Riemannian manifolds.Surprisingly, computing this mean and covariance is related to sub-Riemannian geometry. We describe how sub-Riemannian geometry can be applied in this setting, and mention some finite dimensional and infinite-dimensional applications.