Dartmouth Geometry Seminar
Fall 2018–Spring 2019
Tuesday 4-5 PM
307 Kemeny Hall
Note: Special meeting times are marked in red.
Date Speaker Title
Sep. 25 Chen-Yun Lin
(Duke University)
An embedding theorem: geometric analysis behind data analysis
Sep. 27 in KH 108, 2:15-2:50 pm Jeff Brock
(Yale University)
Renormalized volume of hyperbolic 3-manifolds I
2:55 - 3:30 pm Ken Bromberg
(University of Utah)
Renormalized volume of hyperbolic 3-manifolds II
3:50 - 4:25 pm Javier Aramayona
(Autonomous University of Madrid)
On the abelianization of mapping class groups
Oct. 9 Peter McGrath
(University of Pennsylvania)
Existence and Uniqueness for Free Boundary Minimal Surfaces
Oct. 23 Baris Coskunzer
(Boston College)
Minimal Surfaces in Hyperbolic 3-Manifolds
Oct. 30 Otis Chodosh
(Princeton University)
A splitting theorem for scalar curvature
Nov. 6 Sema Salur
(University of Rochester)
Calibrated Geometries and Applications

September 25, 2018: Chen-Yun Lin "An embedding theorem: geometric analysis behind data analysis"

Abstract: High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the VDM for its mathematical foundation. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.

September 27, 2018: Jeff Brock "Renormalized volume of hyperbolic 3-manifolds I"
Ken Bromberg "Renormalized volume of hyperbolic 3-manifolds II"
Javier Aramayona "On the abelianization of mapping class groups"

October 9, 2018: Peter McGrath "Existence and Uniqueness for Free Boundary Minimal Surfaces"

Abstract: Let $\mathbb{B}^3$ be the unit ball in \( \mathbb{R}^3\) and consider the family of surfaces contained in \(\mathbb{B}^3 \) with boundary on the unit sphere \( \mathbb{S}^2 \). The critical points of the area functional amongst this class are called Free Boundary Minimal Surfaces. The latter surfaces are physically realized by soap films in equilibrium and have been the subject of intense study. In the 1980s, it was proved that flat equatorial disks are the only free boundary minimal surfaces with the topology of a disk. It is conjectured that a surface called the critical catenoid is the unique (up to ambient rotations) embedded free boundary minimal annulus. I will discuss some recent progress towards resolving this conjecture. I will also discuss some sharp bounds for the areas of free boundary minimal surfaces in positively curved geodesic balls which extend works of Fraser-Schoen and Brendle in the Euclidean setting.

October 23, 2018: Baris Coskunzer "Minimal Surfaces in Hyperbolic 3-Manifolds"

Abstract: In this talk, we will discuss the existence question for closed embedded minimal surfaces in 3-manifolds. After reviewing the classical results on the subject, we will show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic 3-manifolds.

October 30, 2018: Otis Chodosh "A splitting theorem for scalar curvature"

Abstract: This is joint work with Michael Eichmair and Vlad Moraru. I will discuss a scalar curvature generalization of the classical splitting theorem (due to Cheeger--Gromoll) for Ricci curvature.

November 6, 2018: Sema Salur "Calibrated Geometries and Applications"

Abstract: Calibrated submanifolds are distinguished classes of minimal submanifolds and their moduli spaces are expected to play an important role in geometry, low dimensional topology and theoretical physics. Examples of these submanifolds are special Lagrangian 3-folds for Calabi-Yau, associative 3-folds and coassociative 4-folds for G2 and Cayley 4-folds for Spin(7) manifolds. In this talk we first give an introduction to calibrated geometries and a survey of recent research on the deformation theory of calibrated submanifolds inside Ricci-flat manifolds. We then study the deformations of Lagrangian submanifolds and extend the theory to “Lagrangian Type” submanifolds inside G2 manifolds. If time permits, we will also discuss relations between G2 and contact structures.

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