The HADES seminar on Tuesday, May 6th, will be at 3:30pm in Room 740.
Speaker: Xiaoyu Huang
Abstract:Assume that for the 3D incompressible Euler equation, the initial vorticity is concentrated in an
The HADES seminar on Tuesday, April 29th, will be at 3:30pm in Room 740.
Speaker: Maciej Zworski
Abstract: Chern numbers are analytically constructed topological invariants of complex line bundles over surfaces and a special case of Chern classes. I will introduce this concept in a simple case of the Bloch sphere and show how related objects such as the Berry curvature (introduced by Barry Simon) appear in the adiabatic theorem (which I will state and explain). I will then move to tori and the line bundle of Bloch eigenfunctions. Its topology will then be related to decay of eigenfunctions as discussed in many physics and maths papers including the recent paper by Simon Becker, Zhongkai Tao and Mengxuan Yang (with some of the authors present to answer any hard questions).
The HADES seminar on Tuesday, April 22nd, will be at 3:30pm in Room 740.
Speaker: Robert Schippa
Abstract: We revisit the classical Córdoba-Fefferman square function estimate and give applications to moment inequalities for exponential sums. Next, we extend the CF estimate to higher dimensional manifolds with tangent spaces satisfying a transversality condition. Finally, we show square function estimates for the conical extension by the High-Low method, which in the scalar case is due to Guth-Wang-Zhang.
The HADES seminar on Tuesday, April 1st, will be at 3:30pm in Room 740.
Speaker: Ryan Martinez
Abstract: The method of normal forms was introduced to PDEs by Shatah, who used it to
study the long time behavior of semilinear Klein Gordon equations and the method
has been widely used in the context of semilinear problems. The modified energy
method of Hunter, Ifrim, Tataru, and Wong extends the idea of normal forms to
quasilinear problems.
In this talk, we will discuss the method of normal forms, the related method
of modified energy, and my recent work which extends these in a novel way. The aim is
to give a selection of nonlinear PDEs which demonstrate in detail how these methods
are used, why they work, and what gains they achieve.
The HADES seminar on Tuesday, March 11th, will be at 3:30pm in Room 740.
Speaker: Hanna Kim
Abstract: We study problems involving the optimization of eigenvalues in various boundary conditions. The Steiner symmetrization was the important key to solving the classical isoperimetric inequality, where the solution is the ball. Based on this problem, analogous problems were introduced in spectral problems with Dirichlet, Neumann and Robin boundaries and so on. I will discuss recent results on showing maximization of third Robin eigenvalue for negative parameters. This work is based on joint work with R. Laugesen.
The HADES seminar on Tuesday, March 18th, will be at 3:30pm in Room 740.
Speaker: Sung-Jin Oh
Abstract: The electron magnetohydrodynamics equation (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, (E-MHD) is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. In this talk, I’ll discuss a recent proof, joint with In-Jee Jeong, of the local wellposedness of the Cauchy problems for (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity.
More specifically, my goal is to explain the main new ideas introduced in this work and the related work of Pineau-Taylor on quasilinear ultrahyperbolic Schrodinger equations, which also have nondegenerate but nonelliptic principal terms. Both works significantly improve upon the classical work of Kenig-Ponce-Rolvung-Vega on such PDEs, in the sense that the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola-Metcalfe-Tataru in the case of an elliptic principal term.
The HADES seminar on Tuesday, March 4th, will be at 3:30pm in Room 740.
Speaker: Yuchen Mao
Abstract: Solving an underdetermined PDE such as a divergence equation plays a central role in problems like general relativistic gluing. Starting from divergence equations on Euclidean spaces, I will introduce a method of constructing integral solution operators for a wide class of underdetermined differential operators with prescribed support properties. By duality, this will also produce integral representation formulas for overdetermined differential operators. The method extends various ideas from Bogovskii, Oh-Tataru, and Reshetnyak. The construction is based on an assumption called the recovery on curves condition (RC) imposed on the operators. I will also give an algebraic sufficient condition of RC that is easier to verify, which is called the finite-codimensional cokernel condition (FC). At the end, I will show some examples that satisfy FC on space forms and derive their integral formulas in the flat case. This is joint work with Philip Isett, Sung-Jin Oh, and Zhongkai Tao.
The HADES seminar on Tuesday, February 25th, will be at 3:30pm in Room 740.
Speaker: Jason Zhao
Abstract: It has been observed both experimentally and mathematically that solutions to linear dispersive equations, such as the Schrodinger and Airy equations, posed on the torus exhibit dramatically different behaviors between rational and irrational times. For example, the evolution of piecewise constant data remains so at rational times, while it becomes continuous and fractalised at irrational times. A natural question to ask is whether this Talbot effect, as it broadly known, persists under non-linear dispersive flows. Focusing on the KdV equation, we will present two perspectives which follow in the spirit of the seminal works of Bourgain (1993) and Babin-Ilyin-Titi (2011): the first is the non-linear smoothing effect observed by Erdogan-Tzirakis (2013), and the second is the numerical work of Hofmanova-Schratz (2017) and Rousset-Schratz (2022}.
The HADES seminar on Tuesday, February 18th, will be at 3:30pm in Room 740.
Speaker: Zhongkai Tao
Abstract: The Strichartz estimate is an important estimate in proving the well-posedness of dispersive PDEs. People believed that the lossless Strichartz estimate could not hold on manifolds with trapping (for example, the local smoothing estimate always comes with a logarithmic loss in the presence of trapping). Surprisingly, in 2009, Burq, Guillarmou, and Hassell proved a lossless Strichartz estimate for manifolds with trapping under the “pressure condition”. I will talk about their result as well as our recent work with Xiaoqi Huang, Christopher Sogge, and Zhexing Zhang, which goes beyond the pressure condition using the fractal uncertainty principle and a logarithmic short-time Strichartz estimate.
The HADES seminar on Tuesday, February 4th, will be at 3:00pm in Room 740 (UNUSUAL TIME).
Speaker: Mengxuan Yang
Abstract: Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles, the resulting material is superconducting. In 2011, Bistritzer and MacDonald proposed a model that is experimentally very accurate in predicting magic angles. In this talk, I will introduce some recent mathematical progress on the Bistritzer–MacDonald Hamiltonian, including the generic existence of Dirac cones and the mathematical characterization of magic angles. I will also discuss topological aspects of this model, as well as some new mathematical discoveries in twisted multilayer graphene.