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 $C^2=\{(x,y,z)\in R^3:x^2+y^2=z^2,1/2\le |z|\le 2\}$ 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 (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.

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.