Abstracts

On the Multiplicity One Conjecture for Mean Curvature Flows of Surfaces

Abstract:

The Mean Curvature Flow describes the evolution of a family of embedded surfaces in Euclidean space that move in the direction of the mean curvature vector. It is the gradient flow of the area functional and a natural analog of the heat equation for an evolving surface. Initially, this flow tends to smooth out geometries over brief time-intervals. However, due to its inherent non-linearity, the Mean Curvature Flow equation frequently leads to the formation of singularities. The analysis of such singularities is a central goal in the field.

A long-standing conjecture addressing this goal has been the Multiplicity One Conjecture. Roughly speaking, the conjecture asserts that singularities along the flow cannot form by an "accumulation of several parallel sheets”. In recent joint work with Bruce Kleiner, we resolved this conjecture for surfaces in \(\mathbb{R}^3\). This had several applications. First, combining our work with previous results, we obtain that the problem of evolving embedded 2-spheres via the Mean Curvature Flow equation is well-posed within a natural class of singular solutions. Second, we remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze to show that the Mean Curvature Flow starting from any generically chosen embedded surface only incurs cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of general Mean Curvature Flows that flow through singularities.

October 24, 2025

The Square Root Method in the Algebraization of Dirichlet Series

Abstract:

Founded upon an interplay of integral coefficients and large analytic continuations, the algebraization of formal functions is an old theme that dates back to Emile Borel’s reading of a passage from Hadamard’s Essai sur l’étude des fonctions données par leur développement de Taylor. The entire corpus of arithmetic algebraization works has so far aligned to the title of Hadamard’s essay: the case that the formal function was initially given as a power series. While this line of work connects directly to some applications—old and new—to function field arithmetic, it is by far less obvious if and how to perform an analogous Diophantine analysis in the global realm: the case that the formal function is initially given by a general (non-sparse) Dirichlet series. In this symposium talk, we explain how to devise an algebraization method in order to characterize automorphic forms as the only analytic continuation mechanism to a sufficiently large domain of the complex plane, for pairs of Dirichlet series with almost integral coefficients and connected by a functional equation of the standard \(GL(2)\) type. We will be guided by blueprint theorems of a Drinfeld—Vladut kind, except now concerning the “non almost doubling" of zeros in functional field arithmetic. In the number-theoretic realm, the key new principle proposes how to use the global Euler product in the context of the original Deuring—Heilbronn phenomenon. This ultimately leads to an effectivization of the first historical proof of the finiteness of the class number one quadratic imaginary fields. 

October 24, 2025

Higher Expansion, Representation Stability and Minimal Submanifolds

Abstract:

A set of Riemannian manifolds is a \(k\)-expander family if their \(k\)-waists are bounded below by a positive multiple of their volume, i.e. any map to \(k\)-dimensional space has a fiber of volume comparable to the ambient space. Constructing bounded-geometry examples for \(k > 1\) was a difficult problem posed by Gromov, with the first solutions coming from cosystolic expander CW-complexes - complexes satisfying a system of isoperimetric inequalities in mod 2 cohomology. It has been recently observed that non-abelian generalizations of cosystolic expansion can be used to prove representation stability, where approximate homomorphisms between groups are shown to be close to true ones. In my talk, I will present a different approach to (non-abelian) higher expansion using minimal submanifolds and the geometry of locally symmetric spaces. Based on a joint work with Ben Lowe.

October 24, 2025

Limit Measures for Topologically and Geometrically Random Surfaces

Abstract:

An immersed surface \(S\) in a Riemannian manifold \(M\) induces a probability measure on the space \(G_2(M)\) of two-planes in the tangent bundle of \(M\). If \(M\) is a hyperbolic 3-manifold and \(S_n\) is a sequence of surfaces with principal curvatures going to zero, then any weak* limit of their induced measures is a convex combination of the Liouville (equidistributed) measure on \(G_2(M)\), and measures that come from immersed totally geodesic surfaces in \(M\). We consider two ways of generating a “random” nearly geodesic surface in \(M\), one by bounding the genus, and the other by bounding the area. We show that limits of the measures in the former case must come exclusively from the totally geodesic surfaces (if there are any in \(M\)), while limits in the latter case must have some portion that is equidistributed.

This is joint work with V. Markovic and I. Smilga.

October 24, 2025