Analysis and PDE Seminar Homepage, Winter-Spring 2012
Location: Room 383N.
Time: Thursdays, 2:30-3:30pm, or 2:15-4:05pm if a double header.
Thursday, January 19:
Laurent Saloff-Coste (Cornell)
``The heat kernel on manifolds with ends''
Two-sided heat kernel estimates are a very efficient tools to capture
fundamental geometric and analytic properties of the underlying
Most known two-sided heat kernel estimates apply to spaces that have
rather "homogeneous" properties. In this talk, I will discuss
detailed heat kernel
estimates for a natural and rather concrete class of manifolds which
are definitely not homogeneous (joint work with Alexander Grigoryan).
Note: there will be another talk by the speaker in a department
colloquium on Tuesday, January 17, at 2:30pm in 383N,
`Random walks from a geometric perspective'
In this talk, I will discuss random walk as a way to explore
generated groups. The focus will be on the relations between various
behaviors of random walk
and the properties of the group from the view point of algebra and/or
I will attempt to give a broad view of the subject from Harry
Kesten's 1958 Ph.D. Thesis to the present day.
Wednesday, January 25, 4pm, 383N, Geometry seminar slot:
Richard Melrose (MIT)
``Smooth gluing and extremal Kaehler metrics''
Abstract: As part of the general question raised by Calabi of the existence
of extremal metrics in a given Kaehler class, several authors have
considered the possibility of `lifting' a constant scalar curvature
Kähler (or more generally an extremal) metric under the blow-up of a
finite collection of points (in particular see Arezzo and Pacard, Arezzo,
Pacard and Singer and Székelyhidi). I will discuss joint work with
Michael Singer in which we show how to carry out these `gluing
constructions' systematically in terms of smooth analysis at the boundary
of a manifold with corners and as a consequence obtain more detailed
information on the degeneration of these families of metrics.
Thursday, February 9:
Benoit Kloeckner (Grenoble)
``Geometry of Wasserstein spaces: Hadamard spaces''
Jérôme Bertrand, Université Toulouse 3)
Abstract: Optimal transport enables one to define a metric on the
of sufficiently concentrated probability measures of a metric space
the squared distance between two measures is defined by the minimal
cost of a transport plan betwen them, given that the cost for moving
unit of mass is the square of the distance it is moved by.
The goal of the talk is to study some geometric properties of this
metric space of measures, called the Wasserstein space of X, when X
non-positively curved and locally compact. We shall focus on the
following result: if X is CAT(-1), then the Euclidean plane cannot
isometrically embedded in the Wasserstein space of X, while there is
wealth of bi-Lipschitz embeddings.
Thursday, February 23, Room 200-034, 2:30-3:30pm (note unusual location):
Simon Marshall (Northwestern)
``L^p bounds for eigenfunctions on locally symmetric spaces''
Abstract: There is a classical theorem of Sogge which provides bounds
for the L^p norms of a Laplace eigenfunction on a compact Riemannian
manifold, which are sharp on the sphere and for spectral clusters. I
will present a generalization of this theorem to eigenfunctions of
the full ring of invariant differential operators on a locally
symmetric space, as well as a theorem on the restriction of
eigenfunctions to maximal flat subspaces. Time permitting, I will
discuss ways in which these bounds can be improved using inputs from
Thursday, March 8:
Jan Swoboda (Bonn)
``The Yang-Mills gradient flow and its variants''
Abstract: We review our construction of a Morse homology theory for the
Yang-Mills gradient flow in two dimensions and its relation to
Weber's heat flow homology. We discuss compactness and
Morse-Smale transversality for the perturbed flow, which
invokes a novel L^2 local slice theorem due to Mrowka-Wehrheim.
Finally we show how a modified Yang-Mills functional leads to
an "elliptic Yang-Mills flow" for which a Floer type homology
theory is currently under construction.
(Last part in joint work with R. Janner).
Wednesday, March 14, 4pm, 383N (geometry seminar slot):
András Vasy (Stanford)
Scattering on hyperbolic and Lorentzian spaces
In this talk I describe a new approach to analysis on (Riemannian)
asymptotically hyperbolic spaces. This approach connects them via an
extension across the boundary to asymptotically de Sitter
(Lorentzian) spaces, as well as to a family of operators arising from
an asymptotically Minkowski-type space.
Although the problems to be analyzed are no longer elliptic, we now
have microlocal tools to handle such problems in a Fredholm
framework, stable under perturbations.
This talk will emphasize the geometric aspects of the connections
between these spaces, briefly touching on the underlying analysis.
Similar tools also apply for analysis on black hole backgrounds.
Friday, March 16, Room 736, Evans Hall, Berkeley, Bay Area Microlocal
Nick Haber (Stanford)
Propagation of singularities around a Lagrangian submanifold
of radial points
In this talk we consider the wavefront set of a solution u to Pu = f,
where P is a pseudodifferential operator with real-valued homogeneous
principal symbol p. We assume that the Hamilton vector field of p has
a certain configuration of 'radial points,' that is, points where the
vector field points radially outward in the cotangent fiber.
Hormander's propagation of singularities theorem gives no information
at such radial points. Nevertheless, we are able to give additional
statements about the regularity of u. In addition, we discuss a
further regularity result in this radial setting, in the sense of how
close u is to being a Lagrangian distribution. All work presented is
joint with Andras Vasy.
Rafe Mazzeo (Stanford)
Spectral geometry on the Riemann moduli space
Abstract: I will describe joint work with Ji, Müller and Vasy
concerning the analytic properties of the Laplacian for
the Weil-Petersson metric on the Riemann moduli space.
I also hope to describe new work with Swoboda concerning
fine regularities of the Weil-Petersson metric.
Thursday, April 5:
Simeon Reich (Technion)
Problems and Results in Nonlinear Analysis
Abstract: We intend to present an update regarding several open problems
in Nonlinear Analysis which have been of recent research interest.
These problems concern, for example, the generic method,
infinite products of operators, as well as the asymptotic
properties of discrete and continuous semigroups of holomorphic
Semyon Dyatlov (Berkeley)
Semiclassical limits of plane waves
Abstract: On a complete noncompact Riemannian manifold which is either
Euclidean or hyperbolic near infinity, we study microlocal
convergence of distorted plane waves E(z,\xi) as z \to \infty.
Here z is the spectral parameter and \xi indicates the direction
of the wave at infinity. The functions E(z,\xi) are generalized
eigenfunctions of the Laplacian, they are also known as Eisenstein
functions in the hyperbolic setting. We show that if the trapped set
has zero Liouville measure, then plane waves converge to a limiting
measure, if we average in \xi and in z\in [R,R+1]. The rate of
convergence is estimated in terms of the maximal expansion rate and
classical escape rate of the geodesic flow, giving a negative power
of z when the flow is Axiom A. As an application, we obtain
expansions of local traces and of the scattering phase with fractal
remainders. Joint work with Colin Guillarmou.
Daniel Tataru (Berkeley)
Price's law for electromagnetic waves on Schwarzschild/Kerr
Abstract: I will describe recent work, joint with Jason Metcalfe, Jacob
Sterbenz and Mihai Tohaneanu, on pointwise decay estimates for solutions
to the Maxwell system on black hole asymptotically flat relativistic
backgrounds. This is related to the nonlinear black hole stability
problem for Einstein's equations.
Friday, May 18, at 2pm (special day and time), in 383N:
Neil Trudinger (ANU)
``On the local theory of prescribed Jacobian equations''
Thursday, May 24:
Clara Aldana (Max Planck Institute)
``Compactness of isospectral sets of open surfaces''
Abstract: I will start mentioning the inverse spectral problem
for closed surfaces and the compactness result by Osgood,
Phillips and Sarnark in this setting. Then I will talk about
how to generalize this problem to complete surfaces with finite
topology, going from isoresonant surfaces to what we call
relatively isospectral surfaces. The last part part of the talk
is joint work with Pierre Albin and Frederic Rochon.
Thursday, May 31:
Dean Baskin (Northwestern)
``Asymptotics of radiation fields in asymptotically Minkowski
Abstract: Radiation fields are (appropriately rescaled) limits of solutions
of wave equations along light rays. In this talk I will describe a class of
(non-static) asymptotically Minkowski space times for which the radiation
field is defined and indicate how methods of Vasy can be used to express the
asymptotics in terms of the resonances of a related Riemannian problem on an
asymptotically hyperbolic manifold. In particular, even on Minkowski space,
these methods give a new understanding of the Klainerman-Sobolev estimates.
This is joint work with Andras Vasy and Jared Wunsch.