Abstract:
Aldous introduced the notion of scale-invariant random spatial network (SIRSN) as a random process of paths between any two points in $R^n$ that would be invariant under translation, rotation and scaling. Moreover, the paths of a SIRSN must have a length comparable to the Euclidean one, and there must be “main roads”: the intersection of all the infinitely many paths (except small balls near their starting points) with a finite rectangle must have finite total length. Intuitively, all the paths use the same roads. He suggested that the improper Poisson line process would generate one, but neither he nor Kendall could prove it.
The improper Poisson line process is obtained by dropping lines uniformly at random in $R^n$, with a speed limit on each line. The lower the speed, the more lines there are. We then get a random metric on $R^n$; the time it takes to go from one point to another while respecting the speed limits.
We give a proof that the improper Poisson line process is indeed a SIRSN, with some nice use of the pigeonhole principle. At the end of the talk, we will hint at generalisations with flats of higher dimension.
Bio:
Jonas KAHN has studied at École Normale Supérieure in Paris, getting both theoretical physics and mathematics diplomas, before getting two PhDs, in Leiden (Netherlands) and Orsay (France).
He entered CNRS (Centre National de la Recherche Scientifique) before the end of his PhD in 2008, and moved to UESTC (Chengdu, China) as full professor and 1000 talents recipient in 2022.
He is a mathematician with knowledge in physics, interested in both fundamental and applied subjects, encompassing but not restricted to quantum statistics, operator algebras, stochastic geometry,
Markov chains, optimal transport, imaging, learning, compressed sensing.
He has worked with physicists, biologists, computer scientists and neuroscientists on their subjects, publishing in such journals as Physical Review A, Annals of Probability, Annals of Applied Probability, Annals of Stattistics, Foundations of Computational Mathematics, Transactions on Pattern Analysis and Machine Intelligence or SIAM journal on Imaging Science
