Classification of Low-Dimensional Solitons in (Generalized) Ricci Flow

Michael Womack

April 9th, 2026 (Thursday) at 3pm - RH 440R

Abstract

The Ricci flow is a geometric evolution equation which evolves the metric on a Riemannian manifold in a manner analogous to a heat equation. In the 1980's, Richard Hamilton used the Ricci flow to classify 3-manifold with positive curvature, and developed an approach to solving the Thurston Geometrization Conjecture. A key difficulty was that the Ricci flow forms singularities, and Perelman's seminal works in 2003 provided the tools to complete Hamilton's program. The singularities formed by the Ricci flow are modeled by so-called \emph{Ricci solitons}, so understanding the behavior of these solitons is crucial to understanding the Ricci flow. This fact remains true for extensions of the Ricci flow, such as Generalized Ricci flow, which have applications in complex geometry and mathematical physics.

We will give a brief introduction to how solitons arise as singularity models in the Ricci flow, some history of the classification of low dimensional Ricci solitons, and then contrast with a surprising new result classifying the Ricci-Yang-Mills solitons on surfaces. The new result is based on forthcoming work by the speaker.

About the Speaker

Michael Womack is a 5th year PhD student advised by Jeff Streets. His primary research interests are in generalized Ricci flow and discrete Ricci flow, and outside of math he enjoys powerlifting, running and music.

Uniqueness in the Plateau problem for calibrated area-minimizers

Bryan Dimler

January 27th, 2026 (Tuesday) at 4pm - ISEB 1200

Abstract

The Plateau problem is one of the cornerstone problems in geometry, dating back to Joseph-Louis Lagrange in 1760. It can be stated for surfaces in R^3 as follows: 

Given a simple closed curve C in R^3, does there exist a surface M in R^3 with least surface area having boundary C?

It is named after Joseph Plateau, who studied the problem experimentally using soap film wire frame experiments in the 1800s. The first existence results were proved by Douglas and Rado in the early 1930s. Later, Federer, Fleming, and Almgren proved existence in the general hypersurface and high codimension settings by developing the theory of area-minimizing integral currents. However, aside from the generic uniqueness results of Frank Morgan in the 1970s and 1980s, uniqueness of solutions has been elusive. 

In this talk, we discuss exciting recent progress on uniqueness in the Plateau problem. Namely, we prove that all calibrated area-minimizers with sufficiently regular boundaries uniquely solve the oriented Plateau problem with their boundary data. This is based on a joint work with Chen-Kuan Lee (Notre Dame).

About the Speaker

Bryan is a 6th year PhD student studying under Connor Mooney and Richard Schoen.