Author Archives: jzhao

A Microlocal Calculus on Filtered Manifolds

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The HADES seminar on Tuesday, December 9nd, will be at 3:30pm in Room 740.

Speaker: Steven Flynn

Abstract: Sub-Riemannian geometries arise naturally in quantum mechanics and control theory, yet fundamental questions about quantum dynamics remain open, suggesting that new microlocal tools are needed to extend classical results to these singular geometries.

I will present a pseudodifferential calculus for filtered manifolds with operator-valued symbols built using representation theory of nilpotent groups. The key innovation is an explicit quantization procedure for noncommutative symbols adapted to the filtration, extending the Van Erp-Yuncken calculus while maintaining essential properties: closure under composition, parametrices, and Sobolev continuity.

This framework enables systematic microlocal analysis on equiregular sub-Riemannian manifolds. This is joint work with Véronique Fischer and Clotilde Fermanian-Kammerer.

Effective non-linear PDEs from statistical many-body dynamics

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The HADES seminar on Tuesday, November 25th, will be at 3:30pm in Room 740.

Speaker: Joe Miller

Abstract: Interacting systems of particles and waves are foundational in many natural phenomena. This talk will outline mathematical approaches for deriving effective, statistical descriptions of such many-body dynamics by connecting them to solutions of nonlinear partial differential equations. Key examples include (i) the Boltzmann equation, which emerges as a limit of interacting hard spheres, (ii) the nonlinear Schrödinger equation, which describes quantum particle dynamics initialized near a Bose-Einstein condensate, (iii) the Vlasov equation, which is an effective model for both non-collisional particles evolving under Newtonian dynamics or as a semiclassical limit of fermionic interactions, and (iv) the kinetic wave equations, which model the statistical behavior of interacting waves. I will discuss my joint work on each of these equations, highlighting how to frame these PDEs as limits of the underlying particle or wave dynamics. Time permitting, I will discuss ongoing work on deriving a Boltzmann mean field game from a jump diffusion process.

Long-Time Dynamics of the 3D Vlasov-Maxwell System with Boundaries

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The HADES seminar on Wednesday, November 12th, will be at 4:00pm in Room 732.

Speaker: Chanwoo Kim

Abstract: We construct global-in-time classical solutions to the nonlinear Vlasov-Maxwell system in a three-dimensional half-space beyond the vacuum scattering regime. We also prove dynamical asymptotic stability under general perturbations in the full three-dimensional nonlinear Vlasov-Maxwell system.

Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^2$

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The HADES seminar on Tuesday, October 28st, will be at 3:30pm in Room 740.

Speaker: Beomjong Kwak

Abstract: In this talk, we present an optimal $L^4$-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb{T}^2$. We first recall the previously known results and counterexamples on the Strichartz estimates on the torus. Then we present our new Strichartz estimate, which has an optimal amount of loss, and the small-data global well-posedness of (mass-critical) the cubic NLS in $H^s,s>0$ as its consequence. An intuition for the relation between them is then provided. Our Strichartz estimate is based on a combinatorial proof. We introduce our key proposition, the Szemerédi-Trotter theorem, and explain the idea of the proof. This is a joint work with Sebastian Herr.

Zero viscosity limit of 1D viscous conservation laws at the point of first shock formation

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The HADES seminar on Wednesday, October 22st, will be at 4:00pm in Room 732.

Speaker: Sanchit Chaturvedi

Abstract: Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small-scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. We provide an asymptotic expansion in viscosity of the viscous solution via the help of matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. Furthermore, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. This is joint work with John Anderson and Cole Graham.

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.

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.

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.

Integral formulas for under/overdetermined linear differential operators

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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.