Dominique Maldague (MIT)

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 (sjoh@math.berkeley.edu).

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.

Changkeun Oh (University of Wisconsin-Madison)

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 (sjoh@math.berkeley.edu).

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.

Michael Hitrik (UCLA)

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 (sjoh@math.berkeley.edu).

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.

Sung-Jin Oh (UC Berkeley)

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 (sjoh@math.berkeley.edu).

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).

Junyan Zhang (Johns Hopkins University)

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 (sjoh@math.berkeley.edu)

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.

Michael Christ (UC Berkeley)

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 (sjoh@math.berkeley.edu).

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.

Jacek Jendrej (Université Sorbonne Paris Nord)

The APDE seminar on Monday, 10/11, will be given by Jacek Jendrej (Université Sorbonne Paris Nord) online via Zoom from 9.10am to 10.00am PST (note the time change). To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu)

Title: Soliton resolution for energy-critical equivariant wave maps

Abstract: We consider wave maps R^(1+2) -> S^2, under the assumption of equivariant symmetry. We prove that every solution of finite energy resolves, as time passes, into a superposition of harmonic maps (solitons) and radiation. It was proved in works of Côte, and Jia and Kenig, that such a decomposition holds along a sequence of times. We show that the resolution holds continuously in time via a “no-return lemma” based on the virial identity. The proof combines a modulation analysis of solutions near a multi-soliton configuration with the concentration-compactness method. Joint work with Andrew Lawrie from MIT.

Sanchit Chaturvedi (Stanford)

The APDE seminar on Monday, 10/4, will be given by Sanchit Chaturvedi (Stanford University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu)

Title: Stability of vacuum for the non-cut-off Boltzmann equation with moderately soft potentials.

Abstract: The vector field method developed by Klainerman has been widely successful in the study of wave equations and general relativity. Recently, the vector field approach has been adapted to understand the dispersion due to the transport operator in both collisionless and collisional kinetic models. As a proof of concept, I will discuss the stability of vacuum for Boltzmann equation with moderately soft potentials. The nonlocality of the Boltzmann operator poses a lot of difficulty and forces us to use a purely energy based approach. This is in contrast to the paper by Luk (Stability of vacuum for the Landau equation with moderately soft potentials) on Landau equation in a similar setting where a maximum principle is both proved and needed.

Steve Zelditch (Northwestern University)

The APDE seminar on Monday, 9/27, will be given by Steve Zelditch (Northwestern University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu)

Title: Riemannian 3-manifolds whose eigenfunctions have just
two nodal domains.

Abstract: In 1977, Hans Lewy published an article constructing a
series  $\phi_N$  of spherical harmonics of degree $N$ whose nodal (zero) sets cut the 2-sphere into just 2 nodal domains. It is an ingenious construction. Recently, Junehyuk Jung and I showed that there is a simple and canonical way to construct an infinite dimensional family of Riemannian metrics on certain 3 manifolds,  all of whose eigenfunctions have this property (except for a certain trivial sequence).  The construction generalizes to all odd  dimensions.  This unexpected behavior of 3D eigenfunctions is heuristically related to numerical computations of nodal sets of 3D spherical harmonics, where only one connected component is visible (the `giant component’). Sarnak conjectured that its genus is maximal among degree N polynomials . Jung and I showed that our result holds for random “equivariant” 3D spherical harmonics and showed that in a certain range the genus has maximal order of growth $N^3$.  The purpose of my talk is to explain these phenomena of 3D nodal sets, which have no analogue for the much more studied 2D case

Kenji Nakanishi (Kyoto University)

The APDE seminar on Monday, 9/20, will be given by Kenji Nakanishi (Kyoto University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu)

Title: Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equation

Abstract: This is joint work with Kenjiro Ishizuka (Kyoto). We study global behavior of solutions for the nonlinear Klein-Gordon equation with a damping and a focusing nonlinearity on the Euclidean space. Recently, Cote, Martel and Yuan proved the soliton resolution conjecture for this equation completely in the one-dimensional case: every global solution in the energy space is asymptotic to superposition of solitons. Since the solitons are unstable, a natural question is which initial data evolve into each of the asymptotic forms. We consider the simplest setting in general space dimensions: the global behavior of solutions starting near a superposition of two ground states.

The main result is a complete classification of those solutions into 5 types of global behavior. Two of them are asymptotic to the positive ground state and the negative one respectively. They form two manifolds of codimension-1 that are joined at their boundary, which is the manifold of solutions asymptotic to superposition of two solitons. The connected union of those three manifolds separates the other solutions into the open set of global decaying solutions and that of blow-up. The manifold of 2-solitons was constructed by Cote, Martel, Yuan and Zhao. To get the classification, the main difficulty is in controlling the direction of instability attached to the two soliton components, because the soliton interactions are not integrable in time, breaking the simple linearized approximation. It is resolved by showing that the non-integrable interactions do not essentially affect the direction of instability, using the reflection symmetry of the equation and the 2-solitons.

I will also talk about a much harder difficulty in the 3-soliton case,
which may be called soliton merger.