Seminar Geometrie / Graduate Lectures

Julian Weigert (MPI-MIS Leipzig)

Geometry of polytopal adjoints
Abstract: Adjoints are special polynomials associated to semialgebraic sets. They appear in several different flavours throughout mathematics and physics, particularly in the emerging field of positive geometries. In this talk we focus on the simplest case, where the semialgebraic set is a convex polytope. Instead of looking at the polytope directly, we study the geometry of the hypersurface that it defines. The main result that will be presented is a new interpolation technique for computing the adjoint hypersurface and for predicting some of its singularities. This is joint work in progress with Clemens Brüser.

Joris Koefler (MPI-MIS Leipzig)

Taking the amplituhedron to the limit
Abstract: The amplituhedron is a semi-algebraic set given as the image of the positive Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters to infinity. We derive its algebraic boundary in terms of Chow varieties and explain its geometric interpretation from the point of view of Positive Geometry in terms of secants of the rational normal curve. This discussion shows that the limit amplituhedron is a positive geometry with a unique differential form.

27.05.2025 (Di)
WIL C/104
13:30

Verteidigung: Kenan Vrabac (TU Dresden)

Amenability of finite energy path and loop groups
Abstract: Verteidigung der Master-Arbeit

Vukasin Stojisavljevic (CRM/UdeM - CNRS/Sorbonne)

Nodal topology and topological data analysis
Abstract: Classical Courant's nodal domain theorem provides an upper bound on the number of nodal domains of a Laplace-Beltrami eigenfunction in terms of its eigenvalue. It has long been known that direct generalizations of this theorem to linear combinations of eigenfunctions fail in general. By introducing the coarse count of nodal domains, based on topological data analysis, we will show a "coarse generalization" of the Courant's theorem. More precisely, we combine the theory of persistence modules and barcodes with the theory of Sobolev spaces in order to study oscillations of functions on different scales. As another application of the same method, we will prove a version of the classical Bézout's theorem for linear combinations of Laplace-Beltrami eigenfunctions. The talk is based on a joint work with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

15.04.2025 (Di)
WIL C/103
13:30

kein Seminar