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

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

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.

## Dynamical Constraints on RG Flows and Cosmology

I will discuss time-dependent probes of the renormalization group, and derive new constraints that govern the spread of local operators in holographic theories. The same methods lead to sum rules for inflationary correlators, relating observables, like the speed of sound during inflation, to properties of the UV theory.

## Extremal set theory I

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.

## High-Energy Conformal Bootstrap and Tauberian Theory

We analyze modular invariance drawing inspiration from tauberian

theorems. Given a modular invariant partition function with a positive

spectral density, we derive lower and upper bounds on the number of

operators within a given energy interval. They are most revealing at high

energies. In this limit we rigorously derive the Cardy formula for the

microcanonical entropy together with optimal error estimates for various

widths of the averaging energy shell. Finally, we identify a new universal

## "Black Metaphors: How Modern Racism Emerged from Medieval Race-Thinking - and Why It Matters"

## Privacy via ill-posedness

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

## Pseudoholomorphic curves with boundary: Can you count them? Can you really?

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with various constraints on boundary and interior marked points. The presence of boundary poses an obstacle to invariance. In a joint work with J. Solomon (2016-2017), we defined genus zero OGW invariants under cohomological conditions.

## Extremal set theory II

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.