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:

https://berkeley.zoom.us/j/95575804119

Title: Treating Small Denominators without KAM

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.

The APDE seminar on Monday, 1/31, will be given by Yilin Wang (MIT & MSRI) both in-person (891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Loewner-Kufarev energy and foliation by Weil-Petersson quasicircles

Abstract: Loewner-Kufarev chain encodes a monotone family of univalent functions (injective holomorphic functions on the unit disk) into a driving measure supported on the circle. It is a powerful tool to study the evolution of domains. Example of application includes Schramm-Loewner evolution (SLE), Hasting-Levitov growth model, semigroups of univalent functions, etc. 

We introduce the Loewner-Kufarev energy on the driving measure. We show that when the energy is finite, the boundary of the evolving codomains forms a foliation of Weil-Petersson quasicircles (a class of non-smooth Jordan curves with more than 20 equivalent definitions). Moreover, this energy is dual to the leaves’ Loewner energy, which is a fundamental quantity for the Kahler geometry on the space of Jordan curves. This result gives the first example of a complete characterization of rough curves in terms of the Loewner-Kufarev driving measure. Our results, although purely deterministic, are inspired by results from the probabilistic SLE duality theorem. This is a joint work with Fredrik Viklund (KTH).

The APDE seminar on Monday, 11/22, will be given by Dominique Maldague (MIT) both in-person (740 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Small cap decoupling for the cone in $R^3$

Abstract: Decoupling involves taking a function with complicated Fourier support and measuring its size in terms of projections onto easier-to-understand pieces of the Fourier support. A simple example is periodic solutions to the Schrodinger equation in one spatial and one time dimension, which have Fourier series expansions with frequency points on the parabola $(n,n^2)$. The exponential sum (Fourier series) itself is difficult to understand, but each summand is very simple. I will explain the basic statement of decoupling and the tools that go into the most current high/low frequency approach to its proof, focusing on upcoming work in collaboration with Larry Guth concerning the cone in $R^3$. Our work further sharpens the refined $L^4$ square function estimate for the truncated cone $C^2=\{ (x,y,z) \in R^3: x^2+y^2 = z^2, 1/2 \leq |z| \leq 2 \}$ from the local smoothing paper of L. Guth, H. Wang, and R. Zhang. A corollary is sharp $(\ell^p,L^p)$ small cap decoupling estimates for the cone $C^2$, for the sharp range of exponents p. The base case of the “induction-on-scales” argument is the corresponding sharpened, refined $L^4$ square function inequality for the parabola, which leads to a new proof of canonical (Bourgain-Demeter) and small cap (Demeter-Guth-Wang) decoupling for the parabola.

The APDE seminar on Monday, 11/15, will be given by Changkeun Oh (University of Wisconsin-Madison) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Decoupling inequalities for quadratic forms and beyond

Abstract: In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.

The APDE seminar on Monday, 11/8, will be given by Michael Hitrik (UCLA) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Semiclassical asymptotics for Bergman projections: from smooth
to analytic

Abstract: In this talk, we shall be concerned with the semiclassical
asymptotics for Bergman kernels in exponentially weighted spaces of
holomorphic functions. We shall discuss a direct approach to the
construction of asymptotic Bergman projections, developed with A.
Deleporte and J. Sj\”ostrand in the case of real analytic weights, and
with M. Stone in the case of smooth weights. The direct approach
avoids the use of the Kuranishi trick and allows us, in particular, to
give a simple proof of a recent result due to O. Rouby, J.
Sj\”ostrand, S. Vu Ngoc, and to A. Deleporte, stating that, in the
analytic case, the Bergman projection can be described up to an
exponentially small error.

The APDE seminar on Monday, 11/1, will be given by Sung-Jin Oh (UC Berkeley) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: A tale of two tails

Abstract: In this talk, I will introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, the method also applies to dynamical spacetimes. In this case, I will explain how the late-time tails are in general different(!) from the stationary case in the presence of dynamical and/or nonlinear perturbations of problem. This is joint work with Jonathan Luk (Stanford).

The APDE seminar on Monday, 10/25, will be given by Junyan Zhang (Johns Hopkins University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ()

Title: Anisotropic regularity of the free-boundary problem in ideal compressible MHD

Abstract: We consider the free-boundary compressible ideal MHD system under the Rayleigh-Taylor sign condition. The local well-posedness was recently proved by Trakhinin and Wang by using Nash-Moser iteration. We prove the a priori estimate without loss of regularity in the anisotropic Sobolev space. Our proof is based on the combination of the “modified” Alinhac good unknown method, the full utilization of the structure of MHD system and the anisotropy of the function space. This is the joint work with Professor Hans Lindblad.

The APDE seminar on Monday, 10/18, will be given by our own Michael Christ online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: On quadrilinear implicitly oscillatory integrals

Abstract: Multilinear oscillatory integrals arise
in various contexts in harmonic analysis,
in partial differential equations, in ergodic theory,
and in additive combinatorics.  We discuss the majorization of integrals
$\int \prod_{j} (f_j\circ\varphi_j)$ of finite products
by negative order norms of the factors,
where integration is over a ball in Euclidean space and $\varphi_j$ are smooth
mappings to a space of strictly lower dimension. The talk focuses
on the quadrilinear case, after work on the trilinear case of Bourgain (1988),
of Joly, M\’etivier, and Rauch (1995), and of the speaker (2019).
Sublevel set inequalities, which quantify the nonsolvability of certain systems
of linear equations, are a central element of the analysis.