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 $\omega:\mathbb R\to [0,\infty)$ is a Lipschitz continuous weight such that $\int_{\mathbb R}\frac{\omega(x)\,dx}{1+x^2} <\infty$, then there exists a nontrivial function $u$ with $|\hat u|\leq e^{-\omega}$. 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 
%09=%09%5Cint%09e%5E%7Bi%09x%09%5Ccdot%09%5Cxi%09%7D%09d%5Cmu(x) "\hat{\mu}(\xi) = \int e^{i x \cdot \xi } d\mu(x)")
 "\hat{\mu}(\xi)")
%7C%09%5Cleq%09C%7C%5Cxi%7C%5E%7B-%5Cbeta%7D "|\hat{\mu}(\xi)| \leq C|\xi|^{-\beta}")
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.
The HADES seminar on Tuesday, May 7th, will be at 3:30pm in Room 748.
Speaker: Pranava Jayanti
Abstract: We introduce the physics of superfluidity, including two mathematical models. We begin with a micro-scale description of the interacting dynamics between the superfluid and normal fluid phases of Helium-4 at length scales much smaller than the inter-vortex spacing. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of solutions in $\mathbb{T}^n$ (for n=2,3) for a power-type nonlinearity, beginning from small initial data. Depending upon the strength of the nonlinear self-interactions, we obtain solutions that are global or almost-global in time. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while deriving time-independent a priori estimates. We compare two different approaches: purely energy based, versus a combination of energy estimates and maximal regularity. The results are from recent collaborations with Juhi Jang and Igor Kukavica.
We will also briefly discuss some results pertaining to a macro-scale model known as the HVBK equations, some of which is joint work with Konstantina Trivisa.
The HADES seminar on Tuesday, April 30th, will be at 3:30pm in Room 939.
Speaker: Ning Tang
Abstract: The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity. In this expository talk I will present a result by Jan Sbierski on the C^0 inextendibility of the maximal analytic Schwarzschild spacetime. I will start with a proof for the continuous inextendibility of Minkowski spacetime, followed by a comparison between this and the continuous inextendibility of Schwarzschild exterior. Then I will sketch the proof of continuous inextendibility of Schwarzschild interior.
The HADES seminar on Tuesday, April 23rd, will be at 3:30pm in Room 939.
Speaker: Andrew Scharf
Abstract: Multilayered poroelastic structures are found in many biological tissues, such as cartilage and the cornea, and find use in the design of bioartificial organs and other bioengineering applications. Motivated by these applications, we analyze the interaction of a free fluid flow modeled by the time-dependent Stokes equation and a multilayered poroelastic structure consisting of a thick Biot layer and a thin, linear, poroelastic membrane separating it from the Stokes flow. The resulting equations are linearly coupled across the thin structure domain through physical coupling conditions such as the Beavers-Joseph-Saffman condition. I will discuss previous work in which weak solutions were shown to exist by constructing approximate solutions using Rothe’s method. While a number of partitioned numerical schemes have been developed for the interaction of Stokes flow with a thick Biot structure, the existence of an additional thin poroelastic plate in the model presents new challenges related to finite element analysis on multiscale domains. As an important step toward an efficient numerical scheme for this model, we develop a novel, fully discrete partitioned method for the multilayered poroelastic structure problem based on the fixed strain Biot splitting method. This work is carried out jointly with Sunčica Čanić and Jeffrey Kuan at the University of California, Berkeley and Martina Bukač at the University of Notre Dame.
The HADES seminar on Tuesday, March 12th, will be at 2:00pm in Room 748. (NOTE THE UNUSUAL SPACE AND TIME)
Speaker: Federico Pasqualotto
Abstract: In this talk, I will review recent results concerning the singularity formation problem for 3d incompressible fluids. In particular, I will focus on the role of angular regularity and explain why higher angular regularity makes blow-up constructions harder. I will finally outline recent work in collaboration with Tarek Elgindi for the 3d Euler equations on R^3, in which we construct the first singularity scenario entirely smooth in the angular variable.