The APDE seminar on Monday, 4/25, will be given by Hong Wang (UCLA) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Title: Distance sets spanned by sets of dimension d/2
Abstract: Suppose that E is a subset of $\mathbb{R}^{d}$, its distance set is defined as $\Delta(E):=\{ |x-y|, x, y \in E \}$. Joint with Pablo Shmerkin, we prove that if the packing dimension and Hausdorff dimension of $E$ both equal to $d/2$, then $\dim_{H} \Delta(E) = 1$.
We also prove that if $\dim_{H} E \geq d/2$, then $\dim_{H} \Delta(E) \geq d/2 + c_{d}$ when $d = 2, 3$; and $\underline{dim}_{B} \Delta(E) \geq d/2 + c_{d}$ when $d > 3$ for some explicit constants $c_{d}$.
The APDE seminar on Monday, 4/18, will be given by Tadahiro Oh (University of Edinburgh) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Abstract: In this talk, I will discuss the (non-)construction of the focusing Gibbs measures and the associated dynamical problems. This study was initiated by Lebowitz, Rose, and Speer (1988) and continued by Bourgain (1994), Brydges-Slade (1996), and Carlen-Fröhlich-Lebowitz (2016). In the one-dimensional setting, we consider the mass-critical case, where a critical mass threshold is given by the mass of the ground state on the real line. In this case, I will show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988).
In the three dimensional-setting, I will first discuss the construction of the $\Phi^3_3$-measure with a cubic interaction potential. This problem turns out to be critical, exhibiting a phase transition: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. Then, I will discuss the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic $\Phi^3_3$-model). As for the local theory, I will describe the paracontrolled approach to study stochastic nonlinear wave equations, introduced in my work with Gubinelli and Koch (2018). In the globalization part, I introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.
The first part of the talk is based on a joint work with Philippe Sosoe (Cornell) and Leonardo Tolomeo (Bonn), while the second part is based on a joint work with Mamoru Okamoto (Osaka) and Leonardo Tolomeo (Bonn).
The APDE seminar on Monday, 4/11, will be given by Malo Jézéquel (MIT) both in-person (891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Title: Semiclassical measures for higher dimensional quantum cat maps.
Abstract: Quantum chaos is the study of quantum systems whose associated classical dynamics is chaotic. For instance, a central question concerns the high frequencies behavior of the eigenstates of the Laplace-Beltrami operator on a negatively curved compact Riemannian manifold M. In that case, the associated classical dynamics is the geodesic flow on the unit tangent bundle of M, which is hyperbolic and hence chaotic. Quantum cat maps are a popular toy model for this problem, in which the geodesic flow is replaced by a cat map, i.e. the action on the torus of a matrix with integer coefficients. In this talk, I will introduce quantum cat maps, and then discuss a result on delocalization for the associated eigenstates. It is deduced from a \emph{fractal uncertainty principle}. Similar statements have been obtained in the context of negatively curved surfaces by Dyatlov-Jin and Dyatlov-Jin-Nonnenmacher, and the case of two-dimensional cat maps have been dealt with by Schwartz. The novelty of our result is that we are sometimes able to bypass the restriction to low dimensions. This is a joint work with Semyon Dyatlov.
The APDE seminar on Monday, 4/4, will be given by Philip Gressman (University of Pennsylvania) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Title: Testing conditions for multilinear Radon-Brascamp-Lieb inequalities
Abstract: We will discuss a new necessary and sufficient testing condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves.
The APDE seminar on Monday, 3/28, will be given by Yan Guo (Brown University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Title: Gravitational Collapse for Gaseous Stars
Abstract: In this talk, we will review recent constructions of blowup solutions to the Euler-Poisson and Euler-Einstein systems for describing dynamics of a gaseous star. This is a research program initiated with Mahir Hadzic and Juhi Jang.
The APDE seminar on Monday, 3/14, will be given by Jacob Shapiro (University of Dayton) both in-person (891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Title: Semiclassical resolvent bounds for compactly supported radial potentials
Abstract: We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(|x|) – E$ in dimension $n \ge 2$, where, $h, \, E > 0$ and $V : [0, \infty) \to \mathbb R$ is $L^\infty$ and compactly supported. We show that the weighted resolvent estimate grows no faster than $\exp(Ch^{-1})$, and prove an exterior weighted estimate which grows $\sim h^{-1}$ . The analysis at small angular momenta proceeds by a Carleman estimate and the WKB approximation, while for large angular momenta we use Bessel function asymptotics. This is joint work with Kiril Datchev (Purdue University) and Jeffrey Galkowski (University College London).
The APDE seminar on Monday, 3/6, will be given by Baoping Liu (Peking University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Title: Large time asymptotics for nonlinear Schrödinger equation
Abstract: We consider the Schrödinger equation with a general interaction term, which is localized in space. Under the assumption of radial symmetry, and uniformly boundedness of the solution in $H^1(\mathbb{R}^3)$ norm, we prove it is asymptotic to a free wave and a weakly localized part. We derive further properties of the localized part such as smoothness and boundedness of the dilation operator. This is joint work with A. Soffer.
The APDE seminar on Monday, 2/28, will be given by Sebastian Herr (Bielefeld University) online via Zoom from 9:10am to 10:00am PST (NOTE THE SPECIAL TIME). To participate, email Sung-Jin Oh ().
Title: Global wellposedness for the energy-critical Zakharov system below the ground state
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that it is globally well-posed in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\”odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.
The APDE seminar on Monday, 2/14, will be given by Pierre Germain (NYU) both in-person (891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().
Title: Boundedness of spectral projectors on Riemannian manifolds
Abstract: Given the Laplace(-Beltrami) operator on a Riemannian manifold, consider the spectral projector on (generalized) eigenfunctions whose eigenvalue lies in an interval of size $\delta$ around a central value $L$. We ask the question of optimal $L^2$ to $L^p$ bounds for this operator. Some cases are classical: for the Euclidean space, this is equivalent to the Stein-Tomas theorem; and for general manifolds, bounds due to Sogge are optimal for $\delta > 1$. The case $\delta < 1$ is particularly interesting since it is connected with the global geometry of the manifold. I will present new results for the hyperbolic space (joint with Tristan Leger), and the Euclidean torus (joint with Simon Myerson).
In lieu of the regular APDE seminar on Monday, 2/7, I would like to advertise the talk of Svetlana Jitomirskaya (UCI), which will be given in-person (740 Evans) and online via Zoom from 4:10pm to 5:00pm PST. The Zoom meeting room link is:
Abstract: Small denominator problems appear in various areas of analysis, PDE, and dynamical systems, including spectral theory of quasiperiodic Schrödinger operators, non-linear Schrödinger equations, and non-linear wave equations. These problems have traditionally been approached by KAM-type constructions. We will discuss the new methods, originally developed in the spectral theory of quasiperiodic Schrödinger operators, that are both considerably simpler and lead to results completely unattainable through KAM techniques. For quasiperiodic operators, these methods have enabled precise treatment of various types of resonances and their combinations, leading to .proofs of sharp (arithmetic) spectral transitions, the ten martini problem, and the discovery of universal hierarchical structures of eigenfunctions.