# 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
SpeakerRobert 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
SpeakerHanming 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 n$\ge$3, 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 $\tau$-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 2003Fall 2002Spring 2002Fall 2001, Spring 2001, Winter 2001, Fall 2000, 1999-2000, 1998-9, 1997-8    or  1996-7.