The HADES seminar on Tuesday, November 26th, will be at 2:00pm in Room 740.

Speaker: Ciprian Demeter (Indiana University Bloomington)

Abstract: Partition the unit square into $N^2$ tiny squares with side length $1/N$. Can a selection of roughly N such tiny squares be made, in such a way that each line intersects at most O(1) of them? Related questions will also be discussed.

The HADES seminar on Tuesday, October 15th, will be at 2:00pm in Room 740.

Speaker: Andrea Nuetzi

Abstract: We construct a chain homotopy for the de Rham complex of relative differential forms on compact manifolds with boundary. This chain homotopy has desirable support propagation properties, and satisfies estimates relative to weighted Sobolev norms, where the weights measure decay near the boundary. The estimates are optimal given the homogeneity properties of the de Rham differential under boundary dilation. When applied to the radial compactification of Euclidean space, this construction yields, in particular, a right inverse of the divergence operator that preserves support on large balls around the origin, and satisfies estimates that measure decay near infinity. I will explain how this right inverse can then be used to also obtain a right inverse of the divergence operator on symmetric traceless matrices in three dimensions.

The HADES seminar on Tuesday, October 1st, will be at 2:00pm in Room 740.

Speaker: Tristan Humbert

Abstract: Topological entropy is a measure of the complexity of a
dynamical system. The variational principle states that topological
entropy is the supremum over all invariant probability measures of the
metric entropies. For an Anosov flow, the supremum is uniquely attained
at a measure called the measure of maximal entropy (or Bowen-Margulis
measure).

An important example of Anosov flow is given by the geodesic flow on a
negatively curved closed manifold. For these systems, another important
invariant measure is given by the Liouville measure : the smooth volume
associated to the metric.

A natural question, first raised by Katok is to characterize for which
negatively curved metrics the two measures introduced above coincide.
The Katok’s entropy conjecture states that it is the case if and only if
g is a locally symmetric metric. The conjecture was proven by Katok for
surfaces but remains open in higher dimensions.

In this talk, I will explain how one can combine microlocal techniques
introduced by Guillarmou-Lefeuvre for the study of the marked length
spectrum with geometrical methods of Flaminio to obtain Katok’s entropy
conjecture in neighborhoods of real and complex hyperbolic metrics (in
all dimensions).

The HADES seminar on Tuesday, September 24th, will be at 2:00pm in Room 740.

Speaker: Semyon Dyatlov

Abstract: How fast can the Fourier transform of a nontrivial compactly supported function decay at infinity? Certainly it can be faster than any polynomial (any smooth compactly supported function has such Fourier decay), but it cannot decay exponentially (as this would imply real analyticity of the function). The Beurling–Malliavin Theorem gives a partial answer to this question in dimension 1: if $\omega :\mathbb{R}\to [0,\mathrm{\infty })$ is a Lipschitz continuous weight such that ${\int }_{\mathbb{R}}\frac{\omega (x)dx}{1+x^2}<\mathrm{\infty }$, then there exists a nontrivial function $u$ with $|\hat{u}|\le e^{-\omega }$. This theorem has been used by Bourgain and Dyatlov in the proof of Fractal Uncertainty Principle (FUP) in dimension 1.

I will first give a proof of a weaker version of the Beurling–Malliavin Theorem which is still sufficient for the application to FUP. Then I will discuss the generalization of this theorem to higher dimensions, which has recently been used by Alex Cohen in his proof of higher-dimensional FUP.

The HADES seminar on Tuesday, September 17th, will be at 2:00pm in Room 740.

Speaker: Zhongkai Tao

Abstract: Let be a probability measure on . The Fourier transform of the measure, defined by has been very useful in dynamical systems. A central question is the Fourier decay, that is, the uniform decay rate of . This was studied by Erdős and Salem almost a century ago. While polynomial Fourier decay, i.e.  for some , is expected in many situations, it is only recently that people can prove polynomial Fourier decay for nontrivial measures coming from dynamical systems, c.f. the works of Bourgain, Dyatlov, Li, Sahlsten, Shmerkin, Orponen, de Saxcé, Khalil, Baker, Algom, Rodriguez Hertz, Wang, etc. I will try to explain the key ideas in some recent developments: sum-product estimates, additive combinatorics and the use of dynamical systems.

The HADES seminar on Tuesday, September 10th, will be at 2:00pm in Room 740.

Speaker: Katie Marsden

Abstract: This talk will concern the three dimensional half-wave maps equation (HWM), a nonlocal geometric equation with close links to the better-known wave maps equation. In high dimensions, n≥4, HWM is known to admit global solutions for suitably small initial data. The extension of these results to three dimensions presents significant new difficulties due to the loss of a key Strichartz estimate. In this talk I will introduce the half-wave maps equation and discuss a global wellposedness result for the three dimensional problem under an additional assumption of angular regularity on the initial data. The proof combines techniques from the study of wave maps with new microlocal arguments involving commuting vector fields and improved Strichartz estimates.