Project Details
Projekt
Kähler Package for the Grassmann Zonoid Algebra
Subject Area
Mathematics
Term
since 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 539677879
We study a probabilistic real intersection theory in compact homogeneous spaces M. The examples of prime interest are the real Grassmannians. The intersection of randomly moved submanifolds of M is captured by a notion of multiplication of certain convex bodies, that we call Grassmann zonoids. They live in the exterior power of a Euclidean vector space V, chosen as the cotangent space V of M. Alternatively, these zonoids can be viewed as positive measures on the Grassmannians of V. (They should be invariant under the action of the isotropy group.) There are close connections to integral theory and the theory of valuations. One goal of the project is to prove a generalization of the Alexandrov-Fenchel inequality for higher order Grassmann zonoids. This would imply that certain expected intersection numbers arise as coefficients of Lorentzian polynomials and are therefore define logconcave sequences. In the same direction, we want to show that the Grassmann zonoid algebra has a homomorphic image that satisfies the properties of a Kähler package. Another goal is to investigate the Grassmann zonoid algebra with the tools of harmonic analysis and representation theory. In particular, we would like to compute the volume of the Schubert zonoids and to understand their multiplication.
DFG Programme
Priority Programmes
Subproject of
SPP 2458:
Combinatorial Synergies
International Connection
Italy
Cooperation Partner
Professor Dr. Antonio Mercondes Lerario
Co-Investigator
Dr. Leo Mathis
