Author Archives: jzhao

Overdamped quasi-normal modes for Schwarzschild black holes

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The HADES seminar on Tuesday, April 7th, will be at 3:30pm in Room 740.

Speaker: Tyler Guo

Abstract: Quasi-normal modes (QNM) of black holes are supposed to describe the ringdown of decaying gravitational waves. This ringdown is dominated by the modes closest to the real axis but the overdamped modes (deeper in the complex) remain of interest to physicists and mathematicians.

In this talk, I will give a rigorous definition of QNM as scattering resonances. I will then explain how, in the Schwarzschild case, the problem of finding QNM can be reduced to a study of eigenvalues of a family of non-self-adjoint 1D operators using the method of complex scaling. To explain the structure of the resulting spectral problems I will consider the simpler self-adjoint case and give a proof of the emergence of Bohr–Sommerfeld quantisation rules. If time permits, I will discuss the modifications needed to treat the non-self-adjoint case.

Late-time tails of the Einstein vacuum equation near Minkowski spacetime in wave gauges

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The HADES seminar on Tuesday, March 31th, will be at 3:30pm in Room 740.

Speaker: Yuchen Mao

Abstract: It has been expected that wave gauges lead to weak time decay of solutions of the Einstein vacuum equation near Minkowski spacetime, even though Lindblad-Rodnianski used the standard wave gauge to prove global nonlinear stability of Minkowski spacetime in their renowned work. Lindblad further observed the weak $\tau^{-1}$ decay by identifying the source of the time asymptotics. In this talk, I will introduce a new result that proves i) the late time tail; i.e. the leading term in the time asymptotic expansion, is indeed $\tau^{-1}$ in the standard wave gauge, and ii) by slightly modifying the wave gauge condition, one can achieve decay better than $\tau^{-1}$, hence better than expected before. The proof adapts the iteration scheme developed in Luk-Oh to the weak null system of wave equations under wave gauges in order to identify the tail, and chooses the wave gauge so that the zeroth spherical harmonic in the semilinear weak null structure that causes the weak decay is canceled.

Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

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

Speaker: Abishek Rajan

Abstract: We consider the space of smooth gradient expanding Ricci soliton structures on S^1\times \mathbb R^3 and S^2\times \mathbb R^2 which are invariant under the action of SO(3)\times SO(2). In the case of each topology, there exists a 2-parameter family of cohomogeneity one solitons asymptotic to cones over the link S^2\times S^1, as constructed by Nienhaus-Wink and Buzano-Dancer-Gallaugher-Wang. Analogous to work of Bamler and Chen, we define a notion of expander degree for these cohomogeneity one solitons through a properness result. We then proceed to calculate this cohomogeneity one expander degree in the cases of the specific topologies

Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation

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The HADES seminar on Tuesday, February 10th, will be at 3:30PM on Zoom.

Speaker: Yongming Li

Abstract: In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach, using the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations.

Global strong well-posedness of the CAO-problem introduced by Lions, Temam and Wang

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

Speaker: Felix Brandt

Abstract: The CAO-problem concerns a system of two fluids described by two primitive equations coupled by nonlinear interface conditions. Lions, Temam and Wang proved in their pioneering work the existence of a weak solution to the CAO-system. Its uniqueness remained an open problem.

In this talk, we show that this coupled CAO-system is globally strongly well-posed for large data in critical Besov spaces. The approach presented relies on an optimal data result for the boundary terms in the linearized system in terms of time-space Triebel-Lizorkin spaces. Boundary terms are controlled by paraproduct methods.

This talk is based on joint work with Tim Binz, Matthias Hieber and Tarek Zöchling.

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.