Monotone twist maps and Dowker-type theorems
Classical Dowker-type theorems (Dowker 1944, Molnar 1955, Eggleston 1957) give geometric inequalities on extremal areas/length of in-/circumscribed polygons about ovals. We link this to convexity of Mather‘s beta function of associated billiard systems. This gives a unified (and much simpler) proof of these inequalities. Furthermore, we derive new inequalities stemming from e.g. wire or magnetic billiards. This is joint work with Sergei Tabachnikov.
Classification of Homogeneous Lorentz 3-manifolds
We show that every (simply connected) homogeneous Lorentzian manifold is either symmetric, or isometric to a Lie group with a left invariant metric, or belongs to a one-parameter family of examples constructed as quotients of some extensions of the Heisenberg group. If time permits we discuss the geometry and the causal structure of these latter examples.
The Vlasov-Poisson system as dynamics on the dual of a Banach Lie algebra
Amongst other differential equations, the Vlasov-Poisson system can be described as the flow of equations of motion (in the Poisson bracket formulation) on the dual of a Banach Lie algebra with a canonical Poisson structure called the Lie-Poisson bracket. I present this construction and compare it to the case where the Lie algebra is finite-dimensional. I describe an idea that relates the Vlasov-Poisson system with a “conventional” Hamiltonian system (i.e. the Hamiltonian flow on a symplectic space) and try and transfer this idea to the aforementioned setting.
The Gromov width of Bott-Samelson varieties
Schubert varieties are subvarieties of flag varieties which are in general singular. They have natural desigularizations called Bott-Samelson varieties. In this talk, we review some general facts about Bott-Samelson varieties and explain how to compute their Gromov widths. The Gromov width of a symplectic manifold is the size of the largest ball equipped with the standard symplectic form that can be symplectically embedded in that manifold. This is a joint work with Stéphanie Cupit-Foutou.
Barcodes for Hamiltonian homeomorphisms of surfaces
In this talk, we will study the Floer Homology barcodes from a dynamical point of view. Our motivation comes from recent results in symplectic topology using barcodes to obtain dynamical results. We will give the ideas of new constructions of barcodes for Hamiltonian homeomorphisms of surfaces using Le Calvez's transverse foliation theory. The strategy consists in copying the construction of the Floer and Morse Homologies using dynamical tools like Le Calvez's foliations.
Bases of Representations from Nakajima quiver varieties
In this talk I will explain how bases of representations can be obtained by studying irreducible components of Nakajima quiver varieties. I will explain in detail how quiver varieties enter the picture and discuss the link to Kashiwara's crystal theory. If time permits, I will try to explain the ideas and difficulties in the affine setting also.
3-Torus FlowVis Tool is online!
The goals of the tool are to allow exploration of vector fields on the 3-torus, as well as easy data communication and publication. For easy communication, the entire state of the tool can be shared as a link. For publication, it is possible to embed an image with a link containing the state in LaTeX, which enables clicking the image in the resulting PDF and being redirected to the tool with identical settings for exploration.