The Analysis and PDE seminar will take place Monday Oct 1st in 740 Evans from 4-5pm.
The Analysis and PDE seminar will take place Monday Sept 17 in 740 Evans from 4-5pm.
Title: A proof of the instability of AdS spacetime for the Einstein–massless Vlasov system.
Abstract: The AdS instability conjecture is a conjecture about the initial value problem for the Einstein vacuum equations with a negative cosmological constant. It states that there exist arbitrarily small perturbations to the initial data of the AdS spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes after sufficiently long time. In the recent years, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein–scalar field system.
In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein–massless Vlasov system. The construction of the unstable family of initial data will require working in a low regularity setting, carefully designing a family of initial configurations of localised Vlasov beams and estimating the exchange of energy taking place between interacting beams over long period of times. Time permitting, I will briefly discuss how the main ideas of the proof can be extended to more general matter fields, including the Einstein–scalar field system.
The Analysis and PDE seminar will take place Monday Sept 10 in 740 Evans from 4-5pm.
Title: Quantitative additive energy estimates for regular sets and connections to discretized sum-product theorems
Abstract: We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain’s sum-product theorem.