Georgia Institute of Technology College of Sciences

In this talk, we study the problem of minimizing the distance from a data point to a subvariety of the Grassmannian from the perspective of metric algebraic geometry. An important invariant in this line of work is the number of critical points of this optimization problem for general data. When the data is a general point in the ambient space, this number is called the Euclidean distance (ED) degree of the variety. However, we focus mainly on the case when the data is a general point in the Grassmannian, and call the number of critical points the Grassmann distance (GD) degree. As we will see, the GD degree is more subtle than the ED degree. We give formulas for the ED and GD degrees of prominent subvarieties of the Grassmannian, including Schubert and matroid varieties. 

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.

The dual Garside left canonical form is used to solve the conjugacy problem in braid theory, introduced by Xu (3-braids), Kang-Ko-Lee (4-braids) and Birman-Ko-Lee (n-braids). The fractional Dehn twist coefficient (FDTC) is a measurement of the boundary twist of a surface homeomorphism, introduced by Honda-Kazez-Matic. In contact geometry, it is a powerful tool to detect tightness/overtwistedness of a given contact structure. In this talk, I will give applications of the left canonical form to the computation of the FDTC of 4-braids.

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.

In this talk, I will discuss a model four-wave kinetic equation with a power-law dispersion relation. The goal is to identify the sharp decay threshold for local well-posedness in weighted $L^\infty$ spaces.This sharp decay threshold comes from a balance between the high-frequency strength of the collision kernel and the geometry of the resonant manifold. Stronger high-frequency growth requires more decay, while stronger curvature in the dispersion relation provides more geometric gain.We prove local well-posedness above the threshold and ill-posedness below it. The main difficulty is that the gain and loss terms cannot always be treated separately. In the most delicate regimes, the sharp estimate relies on a cancellation mechanism coming from the full nonlinear structure and the resonant geometry.

How does the number of cycles of one length constrain the number of cycles of another length in a planar graph? Rather than maximizing each count separately, we study their possible joint orders of growth as the number of vertices tends to infinity. I will discuss bounds and constructions for short even cycles, together with auxiliary-graph methods that help organize the counting. It is joint work with Greg Blekherman.

TBA