Seminars and Colloquia by Series

Unstable Manifolds of Stratified Euler Equations

Series
PDE Seminar
Time
Tuesday, November 3, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yanbo Wang – Georgia Tech –

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifold of $(\rho_0,v_0)$. The proof is based on the Lyapunov--Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Kerr Black Hole Uniqueness by Gilbert Weinstein

Series
PDE Seminar
Time
Monday, October 12, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Piedmont room (John Lewis Student Center)
Speaker
Gilbert Weinstein – Ariel University

The no-hair theorem states that the only asymptotically flat stationary solutions of the Einstein vacuum equations are members of the Kerr family, parametrized by mass and angular momentum. Hawking’s rigidity theorem shows that under the hypothesis of analyticity, any such solution is axially symmetric. For axially symmetric solutions with a single connected event horizon, Robinson (1975) proved that the solution must be Kerr. However, the case of multiple black holes has remained largely open. We settle this longstanding problem and show that stationary, axially symmetric multiple black holes cannot be in equilibrium by studying the associated harmonic maps with prescribed singularities. This is joint work with Qing Han, Marcus Khuri, and Jingang Xiong.

TBA

Series
PDE Seminar
Time
Tuesday, October 6, 2026 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Marcelo Disconzi – Vanderbilt University

Asymptotic stability of the degree-one vortex in the 2D abelian Yang-Mills-Higgs model under equivariant perturbations

Series
PDE Seminar
Time
Tuesday, September 29, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonas Lührmann – University of Cologne –

The abelian Yang-Mills-Higgs model on (1 + 2)-dimensional Minkowski space is a classical relativistic field theory, where a scalar complex field is coupled to an electromagnetic field. It admits topological soliton solutions called vortices. We present a proof of the asymptotic stability of the degree-one vortex under equivariant perturbations in the self-dual case.

Blowup for Navier-Stokes: Context, Relevance and History

Series
PDE Seminar
Time
Tuesday, September 15, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Blumenthal – Georgia Tech –

 

On September 8, OpenAI announced a Lean-certified resolution of the Navier-Stokes Millennium Problem by furnishing a classical solution to the forced 3d Navier-Stokes equations which blows up at the origin in finite time. The purpose of this talk will be to provide (i) historical context and relevance for the Millennium Problem; and (ii) give an overall view of the techniques developed towards its resolution from the last ten years or so. I will avoid technical details, and endeavor to make the talk accessible to those outside PDE. While I will briefly address the ongoing priority dispute and allegations of misconduct by OpenAI, this will not be the focus of the talk. 

Radiative damping and dispersive decay estimates for SSH models

Series
PDE Seminar
Time
Tuesday, September 1, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Remy Kassem – Georgia Tech –

We study the effect of time-periodic forcing on the edge state of the semi-infinite Su–Schrieffer–Heeger (SSH) model, a 1D tight-binding model. Numerical simulations and an asymptotic expansion demonstrate that if the frequency of forcing is in resonance with the continuous spectrum of the unforced Hamiltonian, then on a time scale proportional to the inverse square of the forcing amplitude, the edge state decays in amplitude due to the radiation of its energy into the bulk. A proof is work in progress, and makes use of a new dispersive decay estimate for the time-evolution induced by the Hamiltonian. 

$C^{1, \alpha}$ isometric embeddings for contact manifolds

Series
PDE Seminar
Time
Tuesday, April 28, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Sandra Ried – Georgia Institute of Technology

Isometric embeddings between a domain manifold and a target manifold are differentiable maps f such that the pullback of the target metric h coincides with the metric g in the domain manifold. This problem can also be formulated as a non-linear PDE via $\nabla f^{\top} h \nabla f = g$. In the case of contact manifolds, it is additionally required that the embedding preserves a certain restriction on the tangent bundle.

We prove that the Nash iteration scheme can be quantified in order to construct infinitely many $C^{1,\alpha}$-isometric embeddings for contact manifolds. In this way, we extend an existing result regarding non-uniqueness for $C^1$ regularity. The strategy of the proof follows a paper by Conti, De Lellis and Szekelyhidi Jr. on the Riemannian case, which is built on the Nash-Kuiper scheme. The main difficulty in our case is to keep the additional linear constraint coming from the contact setting along the iteration procedure.

In the larger program of a quantitative analysis of isometric embeddings between sub-Riemannian manifolds, our result can be seen as an important first step. Another aspect is the flexibility of this convex integration method: the geometric constraint coming from the contact condition is just one special case of a (potentially large) class of admissible constraints, under which this scheme can still be applied.

Self-Similar Smoothing of A Fluid Boundary Corner

Series
PDE Seminar
Time
Tuesday, April 21, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Neel Patel – University of Maine

The Hele-Shaw problem describes the dynamics of the boundary of a single fluid in porous media. For the nonzero surface tension case, we provide the first proof (to the best of our knowledge) of the existence of solutions that initially have a corner. The main challenge is the analysis of a nonlocal equation whose linearization has coefficients that grow at infinity.

Dynamics of the Nonlinear Schrödinger Equation with an Inverse-Square Potential

Series
PDE Seminar
Time
Tuesday, April 21, 2026 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xiaoyi Zhang – University of Iowa –

Please Note: Special time and special room

I will discuss our recent works on the nonlinear Schrodinger equation with an inverse square potential. The primary results include the asymptotic properties of solutions with energy below or equal to the energy of the ground state, as well as the uniqueness of the ground state for the inter-critical problems.

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