Berkeley Harmonic Analysis and Differential equations student seminar
Fall
2017




Here is a list of links to material and conferences/workshops on PDE’s and Harmonic Analysis.



September 12th
3:40 - 5:00
891 Evans Hall
Kevin O'Neill (UCB)
A Sharp Schrodinger Maximal Estimate in R^2. In this talk, I will present a recent paper of Xiumin Du, Larry Guth, and Xiaochun Li, which proves almost everywhere convergence of solutions to the Schrodinger equation in R^2 for initial data in H^s (s>1/3). I will give an extended introduction to the method of polynomial partitioning, which is used in the proof of their main theorem. A new result which arises during the proof is a bilinear local refinement of the Strichartz inequality which is made possible by the l^2-decoupling theorem of Bourgain and Demeter.

September 19th
3:40 - 5:00
891 Evans Hall
Cristian Gavurs (UC Berkeley)
A wave packet parametrix for wave equations. The aim of this talk is to present the construction of a parametrix for the wave equation with variable coefficients due to Hart Smith. The idea is to write approximate solutions as linear combinations of wave packets by decomposing the initial data using a frame of functions (concentrated in space and frequency) which are then transported across the bicharacteristic flow. Even though this construction and proof are fairly elementary, it works under minimal assumptions on the regularity of the coefficients (two bounded derivatives). The talk is based on the paper https://eudml.org/doc/75304 .


September 26th
3:40 - 5:00
891 Evans Hall
A. Martina Neuman (UC Berkeley)
Decoupling for surfaces in R^4. We present a sharp decoupling result of surfaces in R. The techniques brought forth will demonstrate that the natural decoupling scale to work with non-degenerate surfaces or hypersurfaces in any dimensions is N^(-1/2).
October 3rd
3:40 - 5:00
891 Evans Hall
No HADES

October 10th
3:40 - 5:00
891 Evans Hall
Marina Iliopoulou (UC Berkeley)
Algebraic structure underlying Kakeya-type problems. Kakeya-type questions ask how much tubes that point to different directions overlap. Such problems are central in harmonic analysis, due to their connection with restriction theory, geometric measure theory, PDE and number theory. Over the years there was some indication that Kakeya-type problems have an underlying algebraic structure, but it was only a decade ago that a tool was introduced in the area to reveal such structure, leading to important advances in the field. This tool is the polynomial method. During this talk we will explain the method and see applications to Kakeya-type problems.
October 17th
3:40 - 5:00
891 Evans Hall
Albert Ai (UC Berkeley)
Strichartz estimates for the gravity water waves. In this talk we will introduce the gravity water waves equations, which describe the motion of a fluid influenced by gravity, under a free interface with a vacuum. We will discuss various formulations of the problem, and in particular a paradifferential reduction due to Alazard, Burq, and Zuily. From this formulation we can exhibit the dispersive properties of the water waves system by establishing Strichartz estimates.
October 24th
3:40 - 5:00
891 Evans Hall
Grace Liu (UC Berkeley)
Scattering and modified scattering for the NLS equation. The NLS with nonlinear term |u|^p u has the property that when 1 < p < 2/n, there is no low energy scattering, when 2/n < p < 4/n, there is low energy scattering. When p=2/n, we expect there will be modified scattering. The talk will be based on the work of Hayashi and Naumkin, they proved that the NLS with critical nonlinearity has low energy scattering when n=1,2,3.
October 31st
3:40 - 5:00
891 Evans Hall
Peter Hintz (UC Berkeley)
Resonances and wave decay on Euclidean and hyperbolic spaces. I will introduce the notion of resonances in potential and obstacle scattering in Euclidean and hyperbolic spaces and explain their relation to the local energy decay of solutions of the wave equation. This talk is based on joint work with Maciej Zworski.
November 7th
3:40 - 5:00
891 Evans Hall
Justin Brereton (UC Berkeley)
Stokes' theorem and construction of an invariant measure on an L^2 ball. An invariant measure is useful in proving almost sure well-posedness of a PDE on a compact set. Thomann and Tzvetkov constructed an invariant measure for the derivative nonlinear Schrodinger (DNLS) equation on the torus. In this talk we construct a DNLS-invariant measure on the set of functions with L^2(T) norm equal to a fixed value m, by an argument that is similar to Stokes' theorem, but in infinitely many dimensions.
November 14th
3:40 - 5:00
891 Evans Hall
Casey Jao (UC Berkeley)
Global quasilinear waves in 1+3 dimensions. I will present a classical result, obtained independently by Christodoulou and Klainerman, on the global existence of solutions to certain second order quasilinear wave equations with small data.
November 21st
3:40 - 5:00
891 Evans Hall
NO HADES

November 28th
3:40 - 5:00
891 Evans Hall
Mohandas Pillai (UC Berkeley)