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 JeanFrancois 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 longstanding problem to naturally parameterized the solution space. In particular, although the set of constant mean curvature solutions is fully understood, the farfromCMC regime is not. In this talk we describe how the two most popular competing strategies for constructing nonCMC solutions (the conformal method and the conformal thinsandwich method) are in fact the same, and we present some examples illustrating deficiencies these methods have in constructing farfromCMC solutions. From this analysis, we propose adjustments to the conformal method that have the potential to better describe farfromCMC initial data.

16 April

Speaker: TBA
Title: TBA
Abstract: TBA

23 April

Speaker: Yi Wang (Stanford)
Title: Isoperimetric inequality and Qcurvature
Abstract: A wellknown 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 Qcurvature. The isoperimetric inequality is valid if the integral of the
Qcurvature is below a sharp threshold. Moreover, the isoperimetric constant depends only on the integrals of the Qcurvature. 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: 1111:50am. Regular location: 383N) Poster

Speaker: SunYung 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 longtime behavior of the flow.

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

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 PDEODE technique and the Carleman type estimate in the geometric setting.

28 May

Speaker: Peter Hintz (Stanford)
Title: Nonlinear wave equations on de Sitter and Kerrde 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 Kerrde Sitter spacetimes. We obtain our results by showing the global invertibility of the underlying linear operator, which in the quasilinear setting has nonsmooth coefficients, on suitable \(L^2\)based function spaces, which possess appropriate algebra or more complicated multiplicative properties. The linear framework is based on banalysis, 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 noncompact, finitevolume
case.
