FoCM 2014 conference
Workshop A6 - Real Number Complexity
December 13, 17:30 ~ 18:00 - Room A22
The Betti numbers of an intersection of random quadrics
Erik Lundberg
Florida Atlantic University, USA - elundber@fau.edu
Let $X$ be an intersection of $k$ quadrics chosen at random from the so-called Kostlan ensemble. As the number of variables increases, we study the asymptotics of each Betti number of $X$ and show that the expected $i$th Betti number is asymptotically one. In particular, an intersection of quadrics is asymptotically connected on average. In the case of an intersection of $k=2$ quadrics, we give additional detail on the sum of all Betti numbers, providing an asymptotic with two orders of precision. The proofs apply the Agrachev-Lerario spectral sequence from Algebraic Topology combined with results from Random Matrix Theory. The case of three quadrics leads to considering a new model for random curves based on determinantal representations.
Joint work with Antonio Lerario (Lyon 1, France).