# Geometry & Analysis Seminar Spring 2017

McHenry Room 4130

For further information please contact Professor Longzhi Lin or call 831-459-2400

**April 6, 2017
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**No Seminar**

**April 13, 2017**** **

**No Seminar**

**April 20, 2017**

**No Seminar**

**April 27, 2017**

**Alvaro del Pino, ICMAT, Madrid, Spain**

**Title: Spaces of Distributions**

**Abstract: A (tangent) distribution is a subbundle of the tangent space. A classical question in differential topology asks to describe the space of all possible distributions of a given rank on a given manifold. This classical question has a classical answer: obstruction classes provide a nice framework in which actual computations can be performed.**

**One can then study the topology of those subspaces of distributions defined by some geometrically meaningful condition. There are two particular instances in which answers can be provided: foliations and (even-)contact structures. In both cases there is a well--developed theory and active research. At the same time, very little is known about other spaces of distributions.**

**In the talk I will review the h-principle philosophy and explain the role it has played in foliation theory and contact topology. This will allow me to motivate some of the recent developments in the study of Engel structures (which is a class of distributions particular to dimension 4).**

**May 4, 2017**

**Xuwen Zhu, Stanford University**

**Title: The configuration space of constant scalar curvature metrics with conical singularities**

**Abstract: For a compact Riemann surface M, we would like to understand the space of constant curvature metrics with prescribed conical singularities. When all the cone angles are less than 2π, the existence and uniqueness is known due to the works of Luo-Tian, McOwen and Troyanov; the recent work of Mazzeo-Weiss gives the deformation theory and constructs the moduli space of such metrics. However when some or all of the cone angles are bigger than 2π, at least when the curvature is positive, the analysis is much more complicated, which is suggested by the recent breakthrough of Mondello-Panov by synthetic geometry method that there are global constraints on the space of cone angles.**

**We discover that one key ingredient of the obstructed deformation is related to allow some of the points to split into clusters with smaller cone angles. We construct a resolution of the configuration space of conical metrics, and prove a new regularity result that the family of hyperbolic or flat metrics with conic singularities has a nice compactification as the cone points coalesce, and moreover, the fibrewise family of constant curvature metrics is polyhomogeneous on this compactification. And we hope to apply this new construction to describe the moduli space of spherical conic metrics with all possible cone angles. This is joint work in progress with Rafe Mazzeo.**

**May 11, 2017**

**Matt Grace, UC Santa Cruz**

**Title: Lagrangian correspondences and continuously extending the mean index**

**Abstract: Lagrangian correspondences may be used to identify the usual symplectic group Sp(2n) with a dense open submanifold of the Lagrangian Grassmannian for the twisted symplectic product of a 2n dimensional symplectic vector space. The mean index is a real valued function defined on the path space over Sp (2n) based at the identity. In this talk we discuss previous results and potential obstructions regarding continuous extensions of the mean index to paths based at the diagonal Lagrangian in the aforementioned Lagrangian Grassmannian. Time permitting we may explore the topological structure of the embedding of Sp (2n) in the Lagrangian Grassmannian. **

**May 18, 2017**

**Peter Hintz, UC Berkeley**

**Title: Reconstruction of Lorentzian manifolds from boundary light observation sets**

**Abstract: In joint work with Gunther Uhlmann, we consider the problem of reconstructing the topological, differentiable, and conformal structure of subsets S of Lorentzian manifolds M with boundary from the collection of boundary light observation sets: these are the intersections of light cones emanating from points in S with a fixed subset of the boundary of M; here, light rays get reflected according to Snell's law upon hitting the boundary. This can be viewed as a simple model of wave propagation in the interior of the Earth. We solve this inverse problem under a convexity assumption on the boundary of M.**

**May 25, 2017**

**Vincent Bonini, Cal Poly, San Luis Obispo**

**Title: Nonnegatively Curved Hypersurfaces in Hyperbolic Space**

**Abstract: The Asymptotic Boundary Theorem of Alexander and Currier shows that a complete, noncompact hypersurface embedded in hyperbolic space with nonnegative sectional curvature has at most two points in its boundary at infinity. The presence of two points in the boundary at infinity is a rigidity condition that forces the hypersurface to be equidistant about a geodesic. Moreover, hypersurfaces with single point boundaries can be classified in terms of their asymptotics. In this talk we discuss some recent progress on the generalization of the Asymptotic Boundary Theorem to the setting of hypersurfaces in hyperbolic space with nonnegative Ricci curvature.**

**June 1, 2017**

**Seongtag Kim, Princeton University**

**Title: Rigidity of conformal scalar curvature on Riemannian manifolds**

**Abstract: Let (M,G) be an n-dimensional complete Riemannian manifold and Ω be a domain in (M,G) with smooth boundary ∂Ω. Let R[G] be the scalar curvature of and H[G] be the mean curvature of ∂Ω. In this talk, I consider the following problem: Given a compact smooth domain Ω with ∂Ω, can one find a conformal metric g whose scalar curvature R[g] ≥ R[G] on Ω and the mean curvature H[g] ≥ H[G] on ∂Ω with G=g on ∂Ω?**

**Hang and Wang proved that g should be on the hemisphere in the standard sphere. Recently, Qing and Yuan extend the scalar curvature rigidity for conformal deformations of metrics to the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature. In this talk, I present the rigidity results of this problem on the domains in a general Riemannian manifold.**

**June 8, 2017**

**Alice Chang, Princeton University**

**Title: On a conformal characterization of CP ^{2}**

**Abstract: I will discuss some joint work with M. Gursky and P.Yang where we characterized the 4 sphere and CP ^{2} in turns of some conformally invariant conditions and some recent perturbation result extending the characterization to a neighborhood of CP^{2} (work in progress with Siyi Zhang).**

**June 15, 2017**

**No Seminar **