Geometry & Analysis Seminar Winter 2015

Friday January 9, 2015 - ROOM 1240 MCHENRY LIBRARY

Maunderings in enumerative geometry

Gary Kennedy, Ohio State University

The OED defines "maundering" as "rambling talk" or "drivel." I will ramble on about some examples and techniques of enumerative algebraic geometry. The examples are (1) counting lines on surfaces, (2) counting rational curves in the plane, (3) counting rational curves on hypersurfaces. Regarding techniques, I will describe a tower of parameter spaces for curvilinear data first introduced by Semple and later exploited by enumerative geometers. Independently, a group of differential geometers rediscovered the tower. Alex Castro was the first to make the connection between the two constructions. Finally, using plane curves, I will look at the burgeoning field of tropical geometry (named in honor of Imre Simon) and explain how it gives a new perspective on questions of enumeration.

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Friday January 16, 2015 - ROOM 1240 MCHENRY LIBRARY

Geometry of gradient Ricci solitons

Chenxu He, University of Oklahoma

Ricci solitons are self-similar solutions to Hamilton’s Ricci flow and they play a particular role in the singularity analysis of Ricci flow. They are also natural generalization of Einstein manifolds. We present a few results in the study of geometry of gradient Ricci solitons, including the stability problem for the shrinking Ricci solitons with respect to Perelman’s shrinker entropy and show the classification on symmetric spaces of compact type, and the deformation of steady gradient Ricci solitons and show that the infinitesimal deformation is trivial in low dimension. This is a joint work with Huai-Dong Cao.

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Friday January 23, 2015 - ROOM 1240 MCHENRY LIBRARY

Teichmuller Theory and Minimal Surfaces in Hyperbolic 3-manifolds

Zheng Huang, Graduate Center and College of Staten Island, CUNY

The asymptotic Plateau problem studies the existence and uniqueness questions on minimal hypersurface in hyperbolic $n$-space asymptotic to a given boundary on the sphere at infinity. The existence and non-uniqueness for 3-D case were solved by Anderson in early 1980s. We describe some recent generalization and applications. Much of this is based on joint work with B. Wang.

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Thursday January 29, 2015 4:00pm-5:00pm ROOM 4130 MCHENRY LIBRARY

Coarse Ricci curvature and the manifold learning problem

Antonio Ache, Princeton University

We use the framework exploited by Bakry and Emery for the study of logarithmic Sobolev inequalities to consider the statistical problem of estimating the Ricci curvature (and more generally the Bakry-Emery tensor of a manifold with density) of an embedded submanifold of Euclidean space from a point cloud drawn from the submanifold. Our method leads to a definition of Coarse Ricci curvature alternative to the one proposed by Y. Ollivier. This is joint work with Micah Warren.

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Thursday February 5, 2015 - 4:00-5:00pm ROOM 4130 MCHENRY LIBRARY

The Curvature Estimate Of Higher Codimensional Mean Curvature Flow

Mao-Pei Tsui, University of Toledo

Curvature estimate has always been an essential and difficult problem in the study of geometric evolution equations. K. Ecker and G. Huisken have derived a priori estimate for the curvature (second fundamental form) when they study the mean curvature flow of the graph of a function in Euclidean space.
In this talk, I will explain that a similar curvature estimate also exists for higher codimensional mean curvature flow under certain natural conditions. This is joint work with Knut Smoczyk and Mu-Tao Wang.

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Friday February 13, 2015 - ROOM 4130 MCHENRY LIBRARY

Free boundary minimal annuli in convex three-manifolds

Davi Maximo, Stanford University

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature - including strictly convex domains of the Euclidean space R^3. (This is joint work with I. Nunes and G.Smith)

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Friday February 20, 2015 - ROOM 4130 MCHENRY LIBRARY

Surfaces with constant mean curvature in a Riemannian manifold of dimension $3$

Paul Laurain, Université Paris Diderot

The surfaces with constant mean curvature (cmc) in a spacelike hypersurface are geometrically and physically very interesting, as shown in [Huisken-Yau-96] or in the beautiful thesis of H.L. Bray.

However, the purpose of this talk is not to develop the physical properties of cmc but to see on an example what are the analytical difficulties encountered when studying these surfaces.

In fact, we will show how to study the cmc in terms of partial differential equations in order to derive geometric properties. We emphasize in particular the key difficulties generated by the conformal invariance of the problem as the phenomena of concentration and we will show how the structure of the equation helps us to overcome them.

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Friday March 6, 2015 ROOM 4130 MCHENRY LIBRARY

On type-preserving representations of the four-punctured sphere group

Tian Yang, Stanford University

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Friday March 13, 2015 ROOM 4130 MCHENRY LIBRARY

Conformal scalar invariants of surfaces in general conformal 3-manifolds

Jingyang Zhong, University of California, Santa Cruz

In this talk we are interested in establishing a fundamental theorem for surfaces in conformal 3-sphere and conformal 3-manifolds in general. To do so we construct so-called associated surface by extending the use of ambient spaces of Fefferman and Graham, build up relation between conformal geometry of original surface and (pseudo)-Riemannian geometry of its associated surface. We are also looking to study general theory of conformal scalar invariants under this setting.

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