# School of Mathematics

## "What is Geometric Entropy, and Does it Really Increase?"

## Calibrations of Degree Two and Regularity Issues

Calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of calibrated currents. We will review some of these issues, then focusing on the two-dimensional case where we will show a surprising connection with pseudo-holomorphic curves and an infinitesimal regularity result, namely the uniqueness of tangent cones

## Small Height and Infinite Non-Abelian Extensions

## HoTT is a Polyvalent Foundation of Mathemtics

## On the Category of hSets

## An Arithmetic Analogue of Fox's Improved Triangle Removal Lemma

We give an arithmetic version of the recent proof of the improved triangle removal lemma by Fox [Fox11], for the group $F_2^n$.

## Resonances for Normally Hyperbolic Trapped Sets

Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set of trapped geodesics is r-normally hyperbolic for large r and satisfies a pinching condition. Our dynamical assumptions are stable under small smooth perturbations and motivated by applications to black holes. We also establish a high frequency analog of resonance expansions.