Stanford University
Department of Mathematics

# Geometry Seminar Spring 2014

Organizers: Richard Bamler (rbamler@math.*) and Yi Wang (wangyi@math.*)

Time: Wednesday at 4 PM

Location: 383N

(*=stanford.edu)

## Next Seminar

4 June

Speaker: Jonathan Pfaff (Bonn)

Title: Analytic torsion on locally symmetric spaces

Abstract: We will review the definition of analytic torsion on closed manifolds and its relation to the Reidemeister torsion. Then we describe results about the asymptotic behaviour of analytic torsion on compact locally symmetric spaces. Finally, we discuss the extension of these results to the non-compact, finite-volume case.

## Spring Quarter

2 April

Speaker: Setsuro Fujiie (Ritsumeikan University) (Analysis and PDE Seminar)

Title: Semiclassical distribution of resonances created by homoclinic trajectories

Abstract: We consider the Schrödinger operator $$−h^2 \Delta + V (x)$$ in the multidimensional Euclidean space with a semiclassical parameter h and a smooth potential decaying at infinity. Assuming that the underlying classical mechanics on the energy surface of a fixed positive energy has a trapped set consisting of homoclinic trajectories, we describe the precise asymptotic distribution of resonances in a complex neighborhood of this energy. The method is based on the microlocal study of solutions near the trapped set, especially near the hyperbolic fixed point. This is a joint work with Jean-Francois Bony, Thierry Ramond and Maher Zerzeri.

9 April

Speaker: David Maxwell (Fairbanks)

Title: Initial Data in General Relativity Described by Expansion and Conformal Deformation

Abstract: Initial data for the vacuum Cauchy problem in general relativity satisfy a system of nonlinear PDEs known as the Einstein constraint equations. These equations are underdetermined, and it has been a long-standing problem to naturally parameterized the solution space. In particular, although the set of constant mean curvature solutions is fully understood, the far-from-CMC regime is not. In this talk we describe how the two most popular competing strategies for constructing non-CMC solutions (the conformal method and the conformal thin-sandwich method) are in fact the same, and we present some examples illustrating deficiencies these methods have in constructing far-from-CMC solutions. From this analysis, we propose adjustments to the conformal method that have the potential to better describe far-from-CMC initial data.

16 April

Speaker: TBA

Title: TBA

Abstract: TBA

23 April

Speaker: Yi Wang (Stanford)

Title: Isoperimetric inequality and Q-curvature

Abstract: A well-known question in differential geometry is to prove the isoperimetric inequality under intrinsic curvature conditions. In dimension 2, the isoperimetric inequality is controlled by the integral of the positive part of the Gaussian curvature. In my recent work, I prove that on simply connected conformally flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson's Q-curvature. The isoperimetric inequality is valid if the integral of the Q-curvature is below a sharp threshold. Moreover, the isoperimetric constant depends only on the integrals of the Q-curvature. The proof relies on the theory of A_p weights in harmonic analysis.

30 April

Speaker: TBA

Title: TBA

Abstract: TBA

9 May, Friday (Note special day and time: 11-11:50am. Regular location: 383N) Poster

Speaker: Sun-Yung Alice Chang (Princeton)

Title: Boundary value problems on conformal compact Einstein manifolds

Abstract: Given a class of conformally compact Einstein manifolds with boundary, we are interested to study the boundary behavior of their compactified metrics. I will report some joint work with Yuxin Ge, Paul Yang on a compactness result in a 3+1 setting, which is a study of some 4th order elliptic system with some matching 3rd order boundary conditions.

14 May

Speaker: Richard Bamler (Stanford)

Title: There are finitely many surgeries in Perelman's Ricci flow

Abstract: Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $$t \to \infty$$.
In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $$C t^{-1}$$ for large $$t$$. Using this curvature bound it is possible to give a more precise picture of the long-time behavior of the flow.

19 May
MONDAY
(Note special seminar)
4pm, room 380-U

Speaker: Lu Wang (Johns Hopkins)

Title: Rigidity of asymptotically conical shrinking Ricci solitons

Abstract: Shrinking Ricci solitons generalize positive Einstein manifolds and play a central role in the analysis of singularities of the Ricci flow. At present, essentially all known complete noncompact examples are either locally reducible as products or possess conical structures at infinity. I will describe recent joint work with Brett Kotschwar in which we investigate the rigidity of such conical structures and show that, if two shrinking solitons are asymptotic to the same cone along some end of each, then the solitons must actually be isometric on some neighborhoods of infinity of these ends. As an application, we prove that the only complete shrinking soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton. The main tools are the PDE-ODE technique and the Carleman type estimate in the geometric setting.

28 May

Speaker: Peter Hintz (Stanford)

Title: Nonlinear wave equations on de Sitter and Kerr-de Sitter spaces

Abstract: I will discuss the global small data solvability of semilinear and quasilinear wave equations on geometric classes of spaces, which include asymptotically de Sitter and Kerr-de Sitter spacetimes. We obtain our results by showing the global invertibility of the underlying linear operator, which in the quasilinear setting has non-smooth coefficients, on suitable $$L^2$$-based function spaces, which possess appropriate algebra or more complicated multiplicative properties. The linear framework is based on b-analysis, introduced in this context by Vasy to describe the asymptotic behavior of solutions of linear equations. Joint work with Andras Vasy.

4 June

Speaker: Jonathan Pfaff (Bonn)

Title: Analytic torsion on locally symmetric spaces

Abstract: We will review the definition of analytic torsion on closed manifolds and its relation to the Reidemeister torsion. Then we describe results about the asymptotic behaviour of analytic torsion on compact locally symmetric spaces. Finally, we discuss the extension of these results to the non-compact, finite-volume case.

## Past Quarters

For the Winter 2014 Schedule go here

For the Fall 2013 Schedule go here

For the Spring 2013 Schedule go here

For the Winter 2013 Schedule go here

For the Fall 2012 Schedule go here

For the Spring 2012 Schedule go here

For the Winter 2012 Schedule go here

For the Fall 2011 Schedule go here

For the Spring 2011 Schedule go here

For the Winter 2011 Schedule go here

For the Fall 2010 Schedule go here

For the Spring 2010 Schedule go here

For the Winter 2010 Schedule go here

For the Fall 2009 Schedule go here