School of Mathematics

Privacy via ill-posedness

Anna Gilbert
University of Michigan; Member, School of Mathematics
November 4, 2019

In this work, we exploit the ill-posedness of linear inverse
problems to design algoithms to release differentially private data or
measurements of the physical system. We discuss the spectral
requirements on a matrix such that only a small amount of noise is
needed to achieve privacy and contrast this with the poor conditioning
of the system. We then instantiate our framework with several
diffusion operators and explore recovery via l1 constrained
minimisation. Our work indicates that it is possible to produce

Extremal set theory I

Andrey Kupavskii
Member, School of Mathematics
October 29, 2019

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.

Spectrum and abnormals in sub-Riemannian geometry: the 4D quasi-contact case

Nikhil Savale
University of Cologne
October 28, 2019

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.

The Surface Quasigeostrophic equation on the sphere

Angel Martinez Martinez
Member, School of Mathematics
October 28, 2019

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.