Random tensors and fractional NLS

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The HADES seminar on Tuesday, September 30th, will be at 3:30pm in Room 740.

Speaker: Rui Liang

Abstract:In this talk, we will consider the Schrödinger equation with cubic nonlinearity on the circle, with initial data distributed according to the Gibbs measure.  We will discuss the challenges and strategies involved in establishing the Poincaré recurrence property with respect to the Gibbs measure in the full dispersive range. This work, using the theory of the random averaging operator developed by Deng-Nahmod-Yue ’19, addresses an open question proposed by Sun-Tzvetkov ’21. We will also explain why the Gibbs dynamics for the full dispersive range is sharp in some sense. Finally, we will see how the theory of random tensors works for extending this work to multi-dimensional settings.

Integral formulas for under/overdetermined differential operators

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The HADES seminar on Tuesday, September 23rd, will be at 3:30pm in Room 740.

Speaker: Sung-Jin Oh

Abstract: In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (IHÉS) that introduces a new versatile approach to constructing integral solution operators (i.e., right-inverses up to finite rank operators) for a broad class of underdetermined operators, including the divergence operator, linearized scalar curvature operator, and the linearized Einstein constraint operator. They are optimally regularizing and, more interestingly, have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). My goal is to (1) describe our approach, (2) demonstrate how it generalizes the well-known construction of Bogovskii, which has proved very useful in fluid dynamics, and (3) explain how it connects underdetermined PDEs with the rich literature on the dual problem on overdetermined differential operators.

Instability, chaos, and nonlinear energy transfer

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The HADES seminar on Wednesday, September 17th, will be at 4:00pm in Room 732.

Speaker: Jacob Bedrossian

Abstract: In this talk we survey several recent results regarding nonlinear dynamics of stochastic differential equations. First, we discuss joint results with Alex Blumenthal, Keagan Callis, and Kyle Liss regarding the existence of stationary measures to SDEs with degenerate damping. This requires the nonlinearity to consistently pump energy from the forced modes to the damped modes. We determine sufficient conditions on the nonlinearity for this and then prove that “generic” examples of fluid-like SDEs satisfy these conditions. Second, we discuss joint results with Alex Blumenthal and Sam Punshon-Smith regarding estimating lower bounds of Lyapunov exponents and using this to prove non-uniqueness of stationary measures for SDEs with almost-surely invariant subspaces. In particular, we prove for L96 with every 3rd mode stochastically forced that for strong forcing there is exactly 2 stationary measures — the trivial one supported only on the forced modes and a second mode which is absolutely continuous with respect to Lebesgue measure and so determines the long term dynamics of almost every initial condition.

Nonuniqueness of solutions to the Euler equations with integrable vorticity

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The HADES seminar on Thursday, September 11th, will be at 3:30pm in Room 736.

Speaker: Anuj Kumar

Abstract: Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in $L^p (p > 1)$.  A celebrated open question is whether the uniqueness result can be generalized to solutions with $L^p$ vorticity. In this talk, we resolve this question in negative for some $p > 1$. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Bruè and Maria Colombo.

Vortex Filament Conjecture for Incompressible Euler Flow

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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 $\epsilon$-tube around a smooth curve in $\mathbb R^3$. The Vortex Filament Conjecture suggests that one can construct solutions in which the vorticity remains concentrated around a filament that evolves according to the binormal curvature flow, for a significant amount of time. In this talk, I will discuss recent developments on the vortex filament conjecture.

What are Chern numbers and why do they matter in condensed matter physics?

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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).

Square function estimates and applications

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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.

Normal Forms, the Modified Energy Method, and an Extension

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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.

Isoperimetric inequalities on different boundary problems

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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.

Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field

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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.