Daniel Tataru (UC Berkeley)

The APDE seminar on Monday, 5/2, will be given by Daniel Tataru (UC Berkeley) both in-person (in 891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Low regularity solutions for nonlinear waves

Abstract: The sharp local well-posedness result for generic nonlinear wave equations was proved in my work with Smith about 20 years ago. Around the same time, it was conjectured that, for problems satisfying a suitable nonlinear null condition, the local well-posedness threshold can be improved. In this talk, I will describe the first result establishing this conjecture for a good model. This is joint work with Albert Ai and Mihaela Ifrim.

Hong Wang (UCLA)

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 (sjoh@math.berkeley.edu).

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 $c_{d}$.

Tadahiro Oh (University of Edinburgh)

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 (sjoh@math.berkeley.edu).

Title: Gibbs measures, canonical stochastic quantization,
and singular stochastic wave equations

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).

Malo Jézéquel (MIT)

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 (sjoh@math.berkeley.edu).

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.

Philip Gressman (U Penn)

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 (sjoh@math.berkeley.edu).

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.

Yan Guo (Brown University)

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 (sjoh@math.berkeley.edu).

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.

Jacob Shapiro (Dayton)

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 (sjoh@math.berkeley.edu).

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).

Baoping Liu (Peking)

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 (sjoh@math.berkeley.edu).

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.

Sebastian Herr (Bielefeld)

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 (sjoh@math.berkeley.edu).

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.

Pierre Germain (NYU)

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 (sjoh@math.berkeley.edu).

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).