Georgia Institute of Technology College of Sciences

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.

Many evolution equations arising from interacting particle systems, particularly the nonlinear Fokker–Planck (McKean–Vlasov) equation, are Wasserstein gradient flows. This talk looks at two complementary questions.
First, when the medium is spatially inhomogeneous and oscillates, does the $\varepsilon$-gradient flow homogenize as $\varepsilon$ goes to 0? We show evolutionary Gamma-convergence of the flow in energy-dissipation-inequality form to an effective gradient flow, and identify the resulting effective Wasserstein metric. Notably, it is strictly larger than the "naive" Gromov–Hausdorff limit of the $\varepsilon$-metrics themselves, showing that the gradient-flow structure encodes genuinely more information than the metric space alone.
Second, given only noisy observations of the flow, can we learn the energy — the interaction kernel, confining potential, and diffusion — that generates it? We introduce self-test loss functions, a derivative-free, high-dimension-friendly estimator built from the weak form of the equation. We give identification analysis when the resulting inverse problem is well- or ill-posed.

Motivated by applications and the classical theory of A-discriminants of Gel'fand, Kapranov, and Zelevinsky, we develop the basic theory of discriminants of multivariate reciprocal polynomials. These reciprocal discriminants parameterize reciprocal polynomials that define a singular hypersurface. We show that a reciprocal discriminant has a rich combinatorial structure. It is a reducible hypersurface whose components are dual varieties to Chebyshev subvarieties which correspond to certain layers in a toric arrangement associated to the exponents of the reciprocal polynomials. The layers which contribute correspond to certain matroidal decompositions of the exponents.

This is joint work with Trevor Karn and Simon Telen.

We study the graphon variational problem that governs lower tail large deviations of the triangle count in a sparse Erdős--Rényi random graph G(n,p). We deduce that the replica-symmetric and symmetry-breaking regimes are separated by a single critical parameter q_c and that the phase transition is first-order. I’ll also discuss tools to transfer these results to the conditioned random graph, showing for example that the edge density is a discontinuous order parameter.