The APDE seminar on Monday, 9/12, will be given by Nets Katz (Caltech) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: A proto-inverse Szemer\’edi Trotter theorem

Abstract: The symmetric case of the Szemer\’edi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters) produces a configuration of $N$ point and $N$ lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemer\’edi Trotter is densely related to a successful instance of the recipe. We discuss the relation of this statement to the inverse Szemer\’edi Trotter problem. (joint work in progress with Olivine Sillier.)

The APDE seminar on Monday, 8/29, will be given by Vadim Kaloshin (ISTA) both in-person (in 740 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Marked Length Spectral determination of analytic chaotic billiards

Abstract: We consider billiards obtained by removing from the plane three strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides natural labeling of periodic orbits. Jointly with J. De Simoi and M. Leguil we show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of all obstacles. For obstacles without symmetry assumption, V. Otto recently showed that the Marked Length Spectrum along with information about two obstacles determines the geometry of all remaining obstacle. 

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 ().

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.

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 ().

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).

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.