# 2019 Lafayette-Lehigh Geometry-Topology Seminar

The Lafayette-Lehigh Geometry-Topology Seminar will be held Saturday March 30, 2019 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) before the first talks. We will also have a light lunch there between the morning and afternoon talks.

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

9:45-10:45 ** Otis Chodosh ** (Princeton University)

* The topology and Morse index of minimal surfaces *

*Abstract*: Minimal surfaces are critical points of the area functional. I’ll discuss some recent existence and classification results about minimal surfaces, based on some recent works with C. Mantoulidis and D. Maximo.

11:00-12:00 ** Peter McGrath ** (University of Pennsylvania)

** Area Bounds for Free Boundary Minimal Submanifolds **

* Abstract *: Fraser-Schoen and Brendle proved that the area of a k-dimensional free boundary submanifold of the unit n-ball is bounded from below by the area of the k-dimensional unit ball. I will discuss (joint work with Brian Freidin) some recent generalizations of these works in positively curved ambient manifolds.

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

1:15-2:15 ** Jenya Sapir ** (Binghamton University)

** Tessellations from long geodesics on surfaces **

* Abstract *: I will talk about joint work of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of a typical tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.

2:30-3:30 ** Damin Wu ** (University of Connecticut/IAS)

** Invariant metrics and the Greene-Wu conjectures. **

* Abstract *:An invariant metric on a complex manifold is in some sense a generalization of the Poincare metric on the unit disk. The classical invariant metrics are the Bergman metric, Caratheodory-Reiffen metric, the Kobayashi-Royden metric, and the complete Kahler-Einstein metric of negative scalar curvature. In 1979, R. E. Greene and H. Wu conjectured that on a simply-connected complete Kahler manifold of negatively pinched sectional curvature, the Bergman metric and the Kobayashi-Royden metric are uniformly equivalent to the background Kahler metric. In this talk, we shall briefly discuss the background; then present a proof of these conjectures, as well as a result on the Kahler-Einstein metric. It is based on the joint work with S. T. Yau.