Analysis-Geometry Seminar
Meets Fridays 12:15 pm in
509 Lake Hall
We generally go to lunch
after the talk.
This seminar features talks in
the fields of partial differential equations, functional analysis,
differential geometry and topology, and mathematical physics. The
seminar is supplemented by the Graduate
Students
Analysis and Geometry seminar.
The current organizer of the
seminar is Maxim
Braverman.
Upcoming Talks:
- January 16, 2015
Speaker: Robert McOwen (Northeastern University)
Title:
Groups of Asymptotic Diffeomorphisms
Abstract: We consider classes of
diffeomorphisms of Euclidean space with partial asymptotic
expansions at infinity; the remainder term lies in a
weighted Sobolev space whose properties at infinity fit with
the desired application. We show that two such classes of
asymptotic diffeomorphisms form topological groups under
composition. As such, they can be used in the study of fluid
dynamics according to the method of V. Arnold. Specific
applications have been obtained for the Camassa-Holm
equation and the Euler equations.
- January 30, 2015
Speaker: Hanming
Zhou (University
of Washington)
Title:
An inverse kinematic problem with internal sources
Abstract: Given a bounded domain M in
Rn with a conformally Euclidean metric g=ρ dx2,
in this paper we consider the inverse problem of recovering
a semigeodesic neighborhood of a domain Γ⊂ ∂ M and the
conformal factor ρ in the neighborhood from the travel time
data (defined below) and the Cartesian coordinates of Γ. We
develop an explicit reconstruction procedure for this
problem. The key ingredient is
the relation between the reconstruction and a Cauchy problem
of the conformal Killing equation. This is a joint work with
L. Pestov and G. Uhlmann.
- February 13, 2015
Speaker:
Dan Mangoubi
(Hebrew University)
Title:
Harmonic functions with common zeros
Abstract: Let f be a positive
harmonic function in the unit disk D(0,1) of R2.
The classical Harnack inequality gives a universal bound on
f(x)/f(y) in D(0, 1/2). We ask for a
generalization of Harnack inequality to harmonic functions
of variable sign. What can be said about the family of
all harmonic functions sharing a common given zero
set? We give an answer in dimension two. We also survey a
recent solution to our question in dimension three by
Logunov-Malinnikova. Finally, we point out a seemingly
surprising phenomenon in dimension three and give partial
results in this direction.
The talk is partly based on a joint work
in progress with Adi Weller-Weiser and her M.Sc. thesis.
- March 6, 2015
Speaker: Maxim
Braverman (Northeastern
University)
Title:
Berry phase and the phase of the determinant
Abstract: In 1984, Michael Berry
discovered that an isolated eigenstate of an adiabatically
changing periodic Hamiltonian H(t) acquires a phase, called
the Berry phase. I will discuss the Berry phase and show
that it is equal to the phase of the zeta-regularized
determinant of the imaginary-time Schrodinger operator. Some
examples of this phenomenon were known to physicists, but no
general result and now rigorous proofs were available until
now.
- March 13, 2015
Speaker: Gabor Lippner (Northeastern
University)
Title:
Discrete curvature-dimension inequalities
Abstract: Finding the 'right' notion
of curvature in various abstract settings has attracted
considerable interest in the past decade. I will describe
recent developments motivated by the Bakry-Emery calculus in
the context of graphs, focusing on applications to local and
global heat kernel estimates.
- March 20, 2015
Speaker:
Koichi Kaizuka (Gakushuin
University)
Title:
Scattering theory for the Laplacian on symmetric space of
noncompact type and its application
Abstract: We develop the stationary
scattering theory for the Laplacian on symmetric spaces of
noncompact type. Typical examples of symmetric
spaces of noncompact type are hyperbolic spaces and bounded
symmetric domains. We consider asymptotic properties of the
solutions to the Helmholtz equation in the Agmon-Hörmander
space. Our approach is based on detailed analysis for the
Helgason Fourier transform and the elementary spherical
function on symmetric spaces of noncompact type. As an
application of our scattering theory, we prove a conjecture
by R. S. Strichartz (J. Funct. Anal.(1989)) concerning a
characterization of a family of generalized eigenfunctions
of the Laplacian.
- March 27, 2015
Speaker: Ezra
Getzler (Northwestern
University)
Title:
Lie n-categories
Abstract: I discuss the definition of a Lie
n-category and discuss some examples. I will also explain
what it means for two Lie n-categories to be
equivalent. This is joint work with Kai Behrend,
motivated by Rezk's theory of Segal spaces and Joyal's
theory of quasicategories. The goal is to develop an
analogue of Kuranishi's theory of analytic germs of
deformations for complexes of holomorphic vector bundles,
and thus an analytic version of the theory of derived stacks
of Toën and Vezzosi and of Lurie.
- April 3, 2015
Speaker: Ting Zhou (Northeastern
University)
Title:
On uniqueness of an inverse problem for the time-harmonic
Maxwell equations
Abstract: The inverse boundary value
problem for the time-harmonic Maxwell equations is a
nonlinear problem to determine electromagnetic parameters of
the medium, namely the magnetic permeability, the electric
permittivity and the conductivity, on a bounded domain using
the measurements of the electromagnetic fields on the
boundary of the domain. I will present both the boundary
uniqueness and interior uniqueness of the parameters, where
we assume that the unknown parameters are described by
continuously differentiable functions. The key ingredient in
proving the uniqueness is the complex geometrical optics
(CGO) solutions.
(this is a joint work with Dr. Pedro Caro.)
- April 10, 2015
Speaker: Dmitry Jakobson
(McGill University)
Title:
Nodal sets in conformal geometry
Abstract: We study conformal
invariants that arise from nodal sets and negative
eigenvalues of conformally covariant operators, which
include the Yamabe and Paneitz operators. We give several
applications to curvature prescription problems. We
establish a version in conformal geometry of Courant's Nodal
Domain Theorem. We also show that on any manifold of
dimension n3,
there exist many metrics for which our invariants are
nontrivial. We prove that the Yamabe operator can have an
arbitrarily large number of negative eigenvalues on any
manifold of dimension n≥3. We obtain
similar results for some higher order GJMS operators on some
Einstein and Heisenberg manifolds. This is joint work with
Yaiza Canzani, Rod Gover and Raphael Ponge. If time permits,
we shall discuss related results for operators on
graphs.
- April 17, 2015
Speaker: Simone Cecchini (Northeastern
University)
Title:
Von Neumann algebra valued differential operators over
non-compact manifolds
Abstract: We provide criteria for self-adjointness and
-Fredholmness
of first and second order differential operators acting on
sections of infinite dimensional bundles, whose fibers are
modules of finite type over a von Neumann algebra A
endowed with a trace τ.
We extend the Callias-type index to operators acting on
sections of such bundles and show that this index is stable
under compact perturbations. (Joint work with Maxim
Braverman).
- June 4, 2015 at 11 am Note the special day and time
Speaker:
Xiaonan MA (
Université
Paris 7)
Title:
Geometric quantization for proper moment maps: the Vergne
conjecture
Abstract: We establish an analytic interpretation for the
index of certain transversally elliptic symbols on
non-compact manifolds. By using this interpretation, we
establish a geometric quantization formula for a Hamiltonian
action of a compact Lie group acting on anon-compact
symplectic manifold with proper moment map. In
particular, we present a solution to a conjecture of
Michele Vergne in her ICM 2006 plenary lecture.
- June 25, 2015 at 11 am Note the special day and time
Speaker: Jesse Gell-Redman
(Johns
Hopkins University)
Title:
The heat kernel on incomplete cusp edge spaces
Abstract: The Weil-Petersson metric on the Riemann moduli
space is an incomplete Riemannian metric which looks locally
near the singular set like products of families of
Riemannian horns, a.k.a. incomplete cusps. Recent work
by Mazzeo-Swoboda and Melrose-Zhu explores the regularity of
the Weil-Petersson metric at the singular set, e.g. finding
asymptotic expansions for the components of the
metric. We discuss our project to analyze Laplace's
equation and the heat equation on this space. In a
model case -- incomplete cusp edge spaces with a `Witt'
condition -- the Laplace-Beltrami operator is essentially
self-adjoint, and the fundamental solution to the heat
equation is fully regular (has an asymptotic expansion at
the singular locus). We derive asymptotics for the
heat trace and prove a Hodge Theorem.
Joint with Jan Swoboda
- October 9, 2015
Speaker: Long Jin (Harvard)
Title:
TBA
Abstract: TBA
You can also view past talks that we have had: Spring 2007, Fall 2007, Spring 2006, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Winter 2003, Fall 2002, Spring 2002,
Fall 2001, Spring 2001, Winter 2001, Fall 2000, 1999-2000,
1998-9,
1997-8
or
1996-7.
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