Wednesday, Mar. 8, 1610-1700, Ayers 112

Title: A discrete curve shortening flow for polygons.

Abstract: In relatively recent news, Nature published an article on Community Detection on Networks using Ricci Flow in 2019 by C. C. Ni et al, where they discuss the application of a discrete version of the Ricci flow on graph networks (e.g. social networks) to detect cliques (highly connected subcommunities). Notions of curvature of discrete structures has been studied by Ollivier (2008), Forman (2003) et al. The study of curvature of nonsmooth spaces goes back even further. Discrete curvature has been of particular interest lately due to the study of the geometry of information. On the other hand, the geometric analysis of smooth structures is a highly established area, with the resolution of the Poincare Conjecture by Perelman (2002) using the Ricci flow being its crowning achievement. We look at a conceptually much simpler flow defined on polygons in the plane, and a discrete analog of a theorem of Grayson in curve shortening flow. Grayson (1987) proved that under the usual curve shortening flow, embedded curves in the plane contract to a point, and become more and more circular as they do so. Chow and Glickenstein (2007, Semidiscrete Geometric Flows of Polygons) define a discrete analogue of this flow for n-sided polygons in \mathbb{R}^p (p not necessarily equal to 2). They show that this flow contracts polygons to a point, and up to an affine transformation, the polygons converge to a planar, convex, regular polygon. This result does not use any high-level mathematics, and is completely accessible to anyone with an understanding of linear algebra and ODEs.

Wednesday, Apr. 5, 1610-1800, Ayers 112

Title: Geometric view of elliptic PDEs, Part I .

Abstract: In this first part, we will review the correspondence developed by Espinar-Gálvez-Mira between elliptic hypersurfaces in the Hyperbolic space of dimension n+1 and elliptic conformal metrics to the standard n-dimensional sphere. We will review the most interesting results, such as the new proof of Toponogov Th, and possible new directions we can apply to this correspondence.

Wednesday, Apr. 12, 1610-1800, Ayers 112

Title: Geometric view of elliptic PDEs, Part II.

Abstract: In this second part, we will review the theory of overdetermined elliptic problems in space forms and its relation with the theory of minimal and constant mean curvature hypersurfaces in space forms. We will discuss this field's techniques, classification, existence results, and open problems. If time permits, we will also relate Part I and Part II..

Wednesday, Apr. 19, 1610-1700, Ayers 112

Title: The preservation of product structures under the Ricci flow.

Abstract: In this talk, we show that if $g(t)$ is a solution to the Ricci flow satisfying a certain non-uniform curvature bound and $g(0)$ splits as a product, then the solution splits as a product for all time. The problem is framed as one of uniqueness for a related system to which we apply a maximum principle, in which the persistence of the product structure is encoded in time-dependent projections. We will also discuss the application of this method to other problems.

Wednesday, Apr. 26, 1610-1700, Ayers 112

Title: Plateau's problem via the theory of phase transitions.

Abstract: Plateau's problem asks whether every boundary curve in 3-space is spanned by an area minimizing surface. Various interpretations of this problem have been solved using eg. geometric measure theory. Froehlich and Struwe proposed another approach, in which the desired surface is produced using smooth sections of a twisted line bundle over the complement of the boundary curve. The idea is to consider sections which minimize an analogue of the Allen--Cahn functional (a classical model for phase transition phenomena) and show that these concentrate energy around a solution of Plateau's problem. After some background on the link between phase transition models and minimal surfaces, I will describe new work with Marco Guaraco in which we produce smooth solutions of Plateau's problem using the approach described above.

Wednesday, May 3, 1610-1700, Ayers 112

Title: Compact curve shortening flow solutions out of non-compact curves.

Abstract: We construct a "slingshot" solution to cuurve shortening flow—a compact, embedded solution that evolves out of a non-compact curve and exists for a finite time.

Wednesday, May 10, 1610-1700, Ayers 112

Title: Near horizon geometries and quasi-Einstein metrics.

Abstract: Extremal black holes in general relativity admit a well-defined Near Horizon Geometry that describes the geometry of the space time near its event horizon. Einstein’s equations on the Lorentzian space time imply that that the Riemannian metric on Near Horizon Geometry satisfies an equation involving the Ricci curvature and a smooth vector field X called the Quasi Einstein equation. Comparison geometry and rigidity phenomena on Quasi Einstein metrics have been previously studied in Riemannian geometry, partly for their connection to Ricci flow and optimal transport. In this talk I’ll discuss adaptations of work on Quasi Einstein Riemannian geometry that give obstructions and rigidity for near horizon geometries as well as interesting examples of Quasi Einstein metrics coming from the near horizon literature. As this line of inquiry is relatively new, I’ll also discuss open questions.

Wednesday, May 15, 1730-1830, Ayers 113

Title: Translators in extrinsic fully nonlinear curvature flows.

Abstract: In this talk I will talk about solutions extrinsic curvature flows such that the deformation occurs through translations in a fixed unit direction. In particular, I will emphasize in properties related to maximum principles applied to geometric quantities of these solutions: uniqueness and non existence results. Finally, if time permits, I will discuss a collaborative work with Sathya Rengaswami on the asymtotics behavior of bowl-type solutions and applications.