Program

Ancient solutions play a fundamental role in the singularity analysis of geometric flows, often appearing as blow-up limits and revealing asymptotic geometric structures. In this talk, we study ancient solutions to the curve shortening flow on a class of noncompact warped product surfaces with rotational symmetry. Such surfaces include many important examples of rotationally symmetric manifolds and provide a natural setting for studying geometric flows. Motivated by the problem of classifying ancient solutions in curved ambient spaces, we present evidence supporting the conjecture that, under suitable geometric assumptions, slices are the only ancient solutions.

In this talk, we discuss a gap theorem for closed Einstein manifolds with positive Ricci curvature, motivated by Colding's sharp gradient estimate for Green functions and by the recent rigidity theorem of Lee-Park in the Ricci-flat setting. We first recall Manea's sharp gradient estimate for a Green function in positive Ricci curvature and introduce an integral gap quantity. We show that the smallness of this gap forces the first eigenvalue pinching for the Laplacian. Based on this observation, we prove the gap theorems in dimension three and four. If time permits, we will briefly discuss the higher dimensional extensions.

In this talk, we study the ancient pancake solution to the mean curvature flow constructed by Bourni-Langford-Tinaglia and discuss its geometric and asymptotic properties. After reviewing several known classification results for ancient solutions, we present a uniqueness theorem for the pancake solution under a low entropy condition. This is joint work with Kyeongsu Choi.

Compared to the mean curvature flow, relatively few examples are known for Gauss curvature flow. In this talk, we present new compact ancient solutions to the Gauss curvature flow in $\mathbb{R}^3$ with $O(2) \times \mathbb{Z}_2$-symmetry, constructed in the viscosity setting and characterized by bounded width. First, we construct a pancake-like compact ancient solution with flat sides to the Gauss curvature flow contained in a slab. Second, for the $\alpha$-Gauss curvature flow with $\alpha > 1/2$, we construct sausage-like compact ancient solutions asymptotic to a round cylinder as $t \to -\infty$.

In this talk, we discuss an extinction-time estimate for mean-convex mean curvature flow whose time slices are outward-minimizing. The classical argument combines the first variation formula, Hölder’s inequality, and Minkowski’s inequality to obtain a sharp upper bound for the extinction time, with equality attained by shrinking round spheres. The main difficulty is to extend this argument beyond the smooth setting, where singularities may occur before extinction. To address this, we use elliptic regularization of the arrival-time equation and interpret the regularized solutions as translating graphs. This allows us to pass geometric inequalities for smooth approximating level sets to the weak level-set flow. We explain how relaxed Minkowski inequalities, convergence of level-set areas and mean curvature measures, and the no-mass-drop property of mean-convex Brakke flows combine to recover the sharp extinction-time bound in the weak setting.

In this talk, we study complete stable minimal hypersurfaces in Riemannian manifolds. Motivated by a classical result of Fischer--Colbrie and Schoen on the conformal type of complete stable minimal surfaces in $3$-manifolds with nonnegative scalar curvature, we investigate the conformal structure of stable minimal hypersurfaces in higher dimensions. We also establish stability results for complete noncompact two-sided minimal hypersurfaces immersed in warped product manifolds under suitable assumptions on the angle function and the warping function. This is joint work with Keomkyo Seo.

Curvature flows have been a major field of study in geometric analysis due to their various applications in differential geometry and topology. While many curvature flows are already well-understood on closed manifolds, their behaviors remain largely unexplored on manifold with boundary. In this talk, I will give a brief survey of the mean curvature flow and Ricci flow on manifolds with boundary, with an emphasis on free boundary mean curvature flow and Ricci flow with a constant curvature boundary.

We study convergence to equilibrium for the fully parabolic Keller-Segel system in a bounded domain with no-flux/Neumann boundary conditions. 

We prove that any global solution whose cell density satisfies a uniform-in-time $L^p$ bound, with $p>\max(1,d/2)$, converges smoothly to a stationary solution. 

Once such boundedness is available, convergence to an equilibrium follows even when the stationary set may be nonunique. 

The main analytic ingredient is a Łojasiewicz-Simon inequality for the Keller-Segel free energy, formulated within the geometry of a coupled Wasserstein-$L^2$  gradient flow.

In this series talk, we aim to discuss when Schubert varieties are stationary with respect to the first variation of the area functional. Schubert varieties are natural generalizations of determinantal varieties. As a preliminary talk for the subsequent talk, we will first review several aspects of determinantal varieties. The determinantal variety $\Sigma_{pq}$ is defined as the set of all $p \times q$ real matrices, with $p \ge q$, whose ranks are strictly smaller than $q$. It is known that $\Sigma_{pq}$ is a minimal cone in $\mathbb{R}^{pq}$. We will review some important examples and several different approaches to prove the minimality of determinantal varieties.

We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly oscillating corrector, which captures the singular behavior of solutions near periodically distributed holes of critical size. We then prove the uniqueness of a critical value that encodes the coupled effects of oscillations in both the coefficients and the obstacles.

We prove a splitting theorem for Lagrangian mean curvature flow. As an application, we classify self-expanders emerging from a pair of Lagrangian n-planes intersecting along a subspace.

Matrix Schubert varieties are algebraic varieties that provide a natural generalization of determinantal varieties. In contrast to the fact that determinantal varieties give rise to minimal submanifolds, the minimality of Schubert varieties depends on certain combinatorial properties. In this talk, we will discuss Schubert varieties and the combinatorial conditions they must satisfy in order to be minimal submanifolds. This talk is based on joint work with Sangwoo Park and Eungbeom Yeon.

It is known that, in three dimensions, a flat‑flow solution of the volume‑preserving mean curvature flow that converges to a single ball converges exponentially fast in Hausdorff distance. In this talk, we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every $C^k$‑norm. In the proof we develop a version of Brakke’s $\epsilon$-regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movements scheme. The same result holds in the planar case. This is based on joint work with Vedansh Arya and Vesa Julin.