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.

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 $c_{d}$.

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).

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.