# 2016 Lafayette-Lehigh Geometry-Topology Seminar

The Lafayette-Lehigh Geometry-Topology Seminar will be held Saturday March 5, 2016 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 9:15 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 )

9:45-10:45: **Genevieve Walsh ** (Tufts University)

** Groups, boundaries, quasi-isometries, and commensurability **

* Abstract *: We discuss three distinct notions of equivalence classes of hyperbolic groups. Two groups can be abstractly commensurable, or quasi-isometric. Furthermore, every hyperbolic group comes equipped with a topological object, its boundary. Groups that are abstractly commensurable are quasi-isometric. Groups which are quasi-isometric have homeomorphic boundaries. How far do these implications go in the opposite direction? The known answers are surprisingly diverse. We will also mention some joint work in progress with Luisa Paoluzzi and Peter Haissinsky.

11:00-12:00: ** Xiuxiong Chen ** (Stony Brook)

** Recent progress in Kaehler geometry **

*Abstract*: There is a long standing conjecture which relates the existence of Kaehler Einstein metrics with positive scalar curvature to the stability of underlying polarization, which goes back to Yau in the 1980s. In this lecture, we will give an expository account of its resolution by Chen-Donaldson-Sun in 2012. If time permitted, we will also go over several other new proofs this important result emerging in recent years.

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

1:15-2:15: ** Xin Zhou** (MIT)

**Min-max theory in Gaussian space and the Entropy Conjecture**

* Abstract *: Minimal surfaces are critical points of the area functional. The min-max theory is a variational theory for constructing saddle point type, unstable minimal surfaces. In this talk, we will introduce a min-max theory in a specific space–the Gaussian probability space. Minimal surfaces in Gaussian space are also called self-shrinkers, which model the singularities of a geometric parabolic PDE–the Mean Curvature Flow. Self-shrinkers are unstable with respect to the second variation of area. Any variational construction of self-shrinkers must be of min-max type.

As an application, we will address a conjecture concerning the entropy of closed surfaces. The entropy is a quantity which measures complexity of a surface. Colding-Ilmanen-Minicozzi-White conjectured that the entropy of a closed surface is bounded from below by that of a round sphere. We will give a min-max proof of this conjecture for spheres. In fact, the entropy is in many ways similar to that of the Willmore functional, and our argument is analogous in many ways to that of Marques-Neves on the Willmore problem. This is based on a joint work with Dan Ketover.

2:30-3:30 ** Vidit Nanda ** (University of Pennsylvania)

** Applied & Computational Algebraic Topology **

*Abstract*: Recent applications of algebraic-topological methods to scientific and engineering contexts have produced intriguing results: from the discovery of a new type of breast cancer, to the detection of coverage in mobile sensor networks and the prediction of protein compressibility directly from crystallography data. This talk will survey the underlying methodology of topological data analysis, describe its primary computational challenges, and show how a discrete version of Morse theory renders enormous topological computations tractable. We ask nothing of the audience beyond a basic understanding of linear algebra and a burning desire to chase gradient trajectories between critical points.