# 2011 Lafayette-Lehigh Geometry-Topology Seminar

The Lafayette-Lehigh Geometry-Topology Seminar will be held Saturday March 26 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, along the Quad. 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 9:30 AM across the hall in Pardee 218 (the Saalfrank Common Room).

** Schedule **

10:15-11:15: **Jeff Jauregui** (University of Pennsylvania)

**Nonnegative scalar curvature on compact manifolds with boundary**

*Abstract*: We begin with a discussion of some physical motivation for studying Riemannian 3-manifolds with nonnegative scalar curvature. The positive mass theorem of Schoen–Yau and Witten, along with a generalization due to Shi–Tam, are essentially statements on the boundary behavior of such manifolds. We consider the problem of asking whether some 2-dimensional geometric data can be realized as the boundary of a 3-manifold of nonnegative scalar curvature, and, time permitting, will give some applications to general relativity.

11:30-12:30: **Lan-Hsuan Huang** (Columbia University)

**Hypersurfaces with non-negative scalar curvature**

*Abstract*: For hypersurfaces in Euclidean space, many classical questions concern the relations between the intrinsic curvature and the extrinsic curvature. For example, if the sectional curvature of the hypersurface is assumed non-negative, a seminal work of R. Sacksteder says that the induced second fundamental form must be semi-positive definite.

In a joint work with Damin Wu, we study hypersurfaces satisfying a much weaker condition on the intrinsic curvature. We prove that closed hypersurfaces with non-negative scalar curvature must be mean convex; namely, the mean curvature is non-negative. The proof relies on a new identity which relates the scalar curvature and mean curvature of the hypersurface to the mean curvature of the level sets. Moreover, using a similar argument, we prove the positive mass theorem for asymptotically flat hypersurfaces with non-negative scalar curvature in all dimensions.

12:30-1:30: **Light lunch ** in Pardee 218.

1:30-2:30: **Bob Gilman** (Stevens Institute of Technology)

** Word Hyperbolic Groups **

* Abstract *: Usually the curvature of a geometric object is a local property; but Alexandrov curvature applies to metric spaces which may not have any local structure, such as the Cayley diagram of a finitely generated group. In a seminal paper published in 1989, Mikhail Gromov defined such a group to be *word hyperbolic* if its Cayley diagram is negatively curved in the sense of Alexandrov. We will survey developments since then and mention some recent results.

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