Extremal set theory typically asks for the largest collection of sets satisfying certain constraints. In the first talk of these series, I'll cover some of the classical results and methods in extremal set theory. In particular, I'll cover the recent progress in the Erdos Matching Conjecture, which suggests the largest size of a family of k-subsets of an n-element set with no s pairwise disjoint sets.
School of Mathematics
We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesics and represents the first such appearance of these in sub-Riemannian spectral geometry.
In this talk I will describe joint work with D. Alonso-Orán and A. Córdoba where we extend a result, proved independently by Kiselev-Nazarov-Volberg and Caffarelli-Vasseur, for the critical dissipative SQG equation on a two dimensional sphere. The proof relies on De Giorgi technique following Caffarelli-Vasseur intermingled with a nonlinear maximum principle that appeared later in the approach of Constantin-Vicol. The final result can be paraphrased as follows: if the data is sufficiently smooth initially then it is smooth for all times.