# 2015 Lafayette-Lehigh Geometry-Topology Seminar

The Lafayette-Lehigh Geometry-Topology Seminar will be held Saturday March 21, 2015 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 9:30 AM, across the hall in Pardee 218 (the Saalfrank Common Room). We will also have a light lunch there between the morning and afternoon talks.

** Schedule ** ( Seminar Flier )

10:00-11:00: **Fernando Schwartz ** (University of Tennessee, Knoxville)

** Geometric inequalities for hypersurfaces **

*Abstract*: The classical inequalities of Polya-Szego and Alexandrov-Fenchel provide bounds for the total mean curvature of convex surfaces in Euclidean space in terms of their capacity and area, respectively. In this talk I will present my joint work with A. Freire in which we generalize these inequalities to arbitrary dimensions using Huisken and Ilmanenâ€™s inverse mean curvature flow, and will show some

applications in mathematical relativity.

11:15-12:15: **Sam Taylor ** (Yale University)

** Stable subgroups: Hyperbolic behavior in mapping class groups and right-angled Artin groups **

*Abstract*: In his seminal manuscript, * Hyperbolic groups*, Gromov introduced his now famous notation of large scale negative curvature for finitely generated groups, which we now call (Gromov) hyperbolic. These groups possess a remarkable number of combinatorial, geometric, and algorithmic properties which make them extremely well-suited to the tools of geometric group theory. However, even though hyperbolic groups are quite abundant, many naturally occurring, geometrically significant groups are not hyperbolic. A key insight into studying many such groups is that despite the fact that they are not globally hyperbolic, some still possess a rich class of hyperbolically behaving subgroups.

In this talk, I will introduce stable subgroups, a notion which is meant to capture the idea of being “hyperbolically behaving.” Besides giving a basic introduction to stable subgroups and explaining the intuition behind stability, we’ll also see how these subgroups turn out to be of particular interest in several well-studied situations, e.g., within mapping class groups and right-angled Artin groups.

12:30-1:30: ** Lunch. **

1:30-2:30: **Moon Duchin ** (Tufts University)

** Rational growth in groups **

* Abstract *: Growth functions in finitely-generated groups are measuring something geometric: the volume growth in the metric models of the group. These have been studied at least since Schwartz and Milnor in the 1950s, and they produce all kinds of puzzling behavior. For some groups and presentations, the growth satisfies a simple recurrence relation that makes it particularly easy to compute. Groups with flat geometry and groups with negatively-curved geometry always have this feature. I’ll explain this and describe some new work (joint with Mike Shapiro) that adds a new group to this list for the first time since the 1980s: the Heisenberg group of 3×3 upper-triangular matrices, whose geometry is in neither of those nice classes.

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