# 2013 Lafayette-Lehigh Geometry-Topology Seminar

The Lafayette-Lehigh Geometry-Topology Seminar will be held Saturday March 23 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., around 10:00 AM, across the hall in Pardee 218 (the Saalfrank Common Room).

** Schedule **

10:30-11:30: **Vincent Bonini ** (California Polytechnic State University)

** On horospherically convex hypersurfaces in H^{n+1} and complete conformal metrics on subdomains of S^n **

* Abstract *: We will develop a global correspondence between properly immersed horospherically convex hypersurfaces in hyperbolic space and complete metrics in the conformal class of the round metric on subdomains of sphere. If time permits, we will also discuss some results concerning the injectivity of the hyperbolic Gauss map and when an immersed horospherically convex hypersurface can be unfolded along the normal flow into an embedded hypersurface. Our global correspondence theorem allows us to compare elliptic problems associated with Weingarten hypersurfaces in hyperbolic space to those of conformal metrics on domains of the sphere, with the hope of providing a unified framework for the two subjects that will shed light on further investigation and research.

11:45-12:45: **Cynthia Curtis ** (The College of New Jersey)

** Knots, Dehn surgery, and invariants from character varieties **

*Abstract*: It has long been known that any three-manifold may be obtained by Dehn surgery on a link in the three-sphere. Recent study on the properties of 3-manifolds obtained by Dehn surgery on a given knot has often involved knot invariants coming from the character variety of the knot complement. We discuss three such invariants: the Culler-Shalen semi-norm, the SL(2,C) Casson invariant, and the A-polynomial. We share recent work with Hans Boden on the SL(2,C) Casson invariant which has led to new computational insights regarding the A-polynomial.

1:00-2:00: Lunch Break: **Light lunch ** in Pardee 218.

2:00-3:00: **Andrew Cooper ** (University of Pennsylvania)

** Singularities of the Lagrangian Mean Curvature Flow **

*Abstract*: The mean curvature flow is the downward gradient of the area functional on submanifolds. If the ambient manifold is Kahler-Einstein and the initial submanifold is Lagrangian, then submanifold remains Lagrangian as long as the flow exists.

In complex Euclidean space, starting from compact initial data, the flow must encounter a singularity at some finite time. We will show how to exploit the Lagrangian structure to understand the formation of such singularities, particularly in the type II (“slow-forming” or “scale-breaking”) case.

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