# 2018 Lafayette-Lehigh Geometry-Topology Seminar

The Lafayette-Lehigh Geometry-Topology Seminar will be held Saturday April 14, 2018 at Lafayette College. The Mathematics Department is located in Pardee Hall (building # 37 on the Campus Map). The Directions on the Lafayette Website direct you to the Visitor Parking Deck behind Markle Hall. You may be able to park near Pardee Hall—there are spots adjacent to the building, between Pardee and Watson Courts. Please refer to the the Campus Map.

The talks will be held in Pardee Hall 217. We will gather for coffee and bagels, etc., across the hall in Pardee 218 (the Saalfrank Common Room) starting at 9:15 AM, before the first talks. We will also have a light lunch there between the morning and afternoon talks.

** Schedule **

9:15-9:45 ** Coffee, etc. **

9:45-10:45 **Helen Wong ** (Carleton College/IAS)

** Kauffman bracket skein algebras of a surface **

* Abstract *: The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial for knots in S^3. But the skein algebra is also closely tied to the SL_2(C)-character variety of the surface, which contains Teichmuller space as a real subvariety. We’ll discuss known algebraic properties of the skein algebra, how those properties are used to build representations of the skein algebra, and how the representations could bridge quantum topology and geometric topology.

11:00-12:00 ** Henry Adams ** (Colorado State University)

** Metric reconstruction via optimal transport **

* Abstract *: Given a sampling of points X from a manifold, what information can one recover about the manifold? The Vietoris-Rips simplicial complex of a metric space X at scale r > 0 has as its simplices the finite subsets of X of diameter less than r. A theorem of Latschev states that if X is a sufficiently dense sample from a Riemannian manifold, then the Vietoris-Rips complex of X recovers the topology of the original manifold, so long as the scale r is sufficiently small. For this reason, Vietoris-Rips complexes are a popular tool in applications of topology to data analysis. However, many questions about the behavior of these complexes at larger scales remain open. We describe how the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, …, as the scale r increases. Furthermore, we argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport metric thickening the metric on X. This talk is on joint work with Michal Adamaszek and Florian Frick.

12:15-1:15 ** Lunch **

1:15-2:15 ** Davi Maximo ** (University of Pennsylvania)

** On Morse index estimates for minimal surfaces **

* Abstract *: In this talk we will survey some recent estimates involving the Morse index and the topology of minimal surfaces.

2:30-3:30 ** Christos Mantoulidis ** (MIT)

** Minimal surfaces and the Allen-Cahn equation on 3-manifolds **

* Abstract *: This is joint work with O. Chodosh. The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. Using new curvature estimates and sharp sheet separation estimates for stable Allen-Cahn solutions on 3-manifolds, derived by improving recent work of Wang-Wei, we show: minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index, provided the ambient metric is generic. This confirms, in the Allen-Cahn setting, two conjectures of Marques-Neves: a strengthened multiplicity one conjecture, and the index lower bound conjecture. If combined with recent work of Guaraco and Gaspar-Guaraco, this gives a new proof of Yau’s conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven in dimensions up to 7 by Irie-Marques-Neves) with new geometric conclusions.