Analysis and PDE Seminar Homepage, Fall 2013
Location: 383N
Time: Thursday, 2:153:15pm.
Thursday, October 3:
Maarten de Hoop (Purdue)
``Stability estimates for the inverse boundary value problem
for the Helmholtz equation and iterative reconstruction''
Abstract: We consider timeharmonic waves described by the Helmholtz equation
and view the DirichlettoNeumann map as the data. We establish a
relationship between this map and seismic data, and include
attenuation. We obtain conditional Lipschitz stability for the
associated inverse boundary value problem with partial boundary
data. The stability is obtained for wavespeed models of the form of
finite linear combinations of piecewise constant functions, naturally
including the presence of conormal singularities. The dimension of the
corresponding space is determined by the number of linearly
independent combinations. We give an estimate of the stability
constant in terms of the dimension of this model space. Using
multifrequency data, we then design a multilevel iterative method
with associated conditions expressed in terms of compression for
convergence to the unique solution derived from the mentioned
stability estimates.
Joint research with E. Beretta, L. Qiu and O. Scherzer.
Thursday, October 31:
Austin Ford (Stanford)
``Second microlocalization at coisotropic submanifolds''
Abstract: Solutions to linear hyperbolic equations such as the classical wave
equation often enjoy what Hörmander (and later Melrose) termed
``Lagrangian regularity''. Indeed, the fundamental solution to the
wave equation is a Lagrangian distribution associated to the graph of
geodesic flow on the cotangent bundle. In this talk, we will describe
a generalization of this type of regularity where the Lagrangian
submanifold of the cotangent bundle of the ambient space is replaced
by a coisotropic submanifold, i.e., ``coisotropic regularity''. We
will also describe a ``second microlocal'' pseudodifferential calculus
associated to such a coisotropic, i.e., a calculus which will test for
coisotropic regularity in an analogous fashion to how the usual
pseudodifferential calculus tests for smooth (or Sobolev) regularity
(via a second wavefront set). Time permitting, we'll discuss the
propagation of second microlocal singularities in analogy with
Hörmander's theorem on propagation of singularities in the smooth
regularity setting.
Thursday, November 7:
Ivana Alexandrova (SUNY Albany)
``Resonances in Scattering by Two Magnetic Fields at Large Separation and a New Complex Scaling Method''
Abstract: We study the quantum resonances in magnetic scattering in two dimensions. The scattering system consists of two obstacles by which the
magnetic fields are completely shielded. The trajectories trapped between
the two obstacles are shown to generate the resonances near the positive
real axis, when the distance between the obstacles goes to infinity. The
location is described in terms of the backward amplitudes for scattering by
each obstacle. A difficulty arises from the fact that even if the supports of
the magnetic fields are largely separated from each other, the corresponding vector potentials are not expected to be well separated. To overcome
this, we make use of a gauge transformation and develop a new type of complex scaling method. We can cover the scattering by two solenoids at large
separation as a special case. The obtained result heavily depends on the
magnetic fluxes of the solenoids. This indicates that the Aharonov–Bohm effect influences the location of resonances. This is joint work with Hideo Tamura.
Wednesday, November 13, Evans Hall,
Bay Area Microlocal Analysis seminar in Berkeley (Note location and
day!)

2:103pm, Evans 891: Peter Hintz (Stanford)
``Semilinear wave equations on asymptotically de Sitter, Kerrde
Sitter and Minkowski spacetimes''
Abstract: I will discuss the small data solvability of suitable semilinear
wave and KleinGordon equations on geometric classes of spaces, which
include asymptotically de Sitter, Kerrde Sitter and Minkowski spacetimes.
Our results are obtained by showing the global Fredholm property, and
indeed invertibility, of the underlying linear operator on suitable
$L^2$based function spaces, which also 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. An interesting
feature of the analysis is that resonances, namely poles of the inverse of
the Mellin transformed bnormal operator, play an important role. Joint
work with Andras Vasy.

4:10  5:00pm, Evans 736: Gabriel Durkin and Sergey Knysh (NASA
Ames Research Center)
``Quantum Fisher Information for Noisy Dynamics: A semiclassical
approach''
Abstract:
Parameter estimation of probability distributions is one of the most basic
tasks in information theory, and has been generalized to the quantum regime
since the description of quantum measurement is essentially probabilistic.
The "quantum metrology" prescription is straightforward for closed systems
evolving unitarily, becoming more challenging when the system is coupled
(more realistically) to an external environment. This produces "noisy"
nonunitary dynamics. This noise, or decoherence, degrades the precision in
any parameter estimation  precision that is quantified by Quantum Fisher
Information. For physically relevant noise models including both phase
diffusion and dissipation we investigate the scaling of single parameter
precision with the noise amplitude and "resource" N, the number of system
dimensions. Using a novel operator approach (rather than WKB), we find new
saturable precision bounds in the asymptotic limit of large N in tandem with
those quantum states uniquely capable of reaching these bounds. Convergence
to asymptotic behaviour can set in quickly for modest N<100, and as such our
analysis is relevant to sensing and metrology experiments incorporating
ensembles of 10 to 100 particles; potentially atoms, magnetic spins,
fluxqubits or photons.
Tuesday, December 3 (Note unusual date), 2:153:15pm, 383N:
Dietrich Häfner (Grenoble)
``Boundary values of resolvents of selfadjoint operators in Krein
spaces and applications to the KleinGordon equation''
Abstract :
30 years ago, E. Mourre showed that a local in energy positive
commutator estimate for a selfadjoint operator H entails a limiting
absorption principle for this operator and thus the absence of
singular continuous spectrum. This result had a very deep impact in
scattering theory leading in particular to asymptotic completeness
results for quantum Nparticle systems. A lot of efforts had been made
to weaken the original hypotheses in the work of Mourre. A central
requirement remained however that the hamiltonian H is a selfadjoint
operator on a Hilbert space. Whereas this is a very natural
requirement for the Schrödinger equation, it turns out that it is in
general not fulfilled for the KleinGordon equation when this equation
is coupled to an electric field or associated to a lorentzian metric
which is not stationary (in the sense that there exists no global
timelike Killing vector field). The natural setting in this situation
seems to be the one of a selfadjoint operator on a so called Krein
space. A Krein scalar product is a non degenerate hermitian form which
is not necessarily positive. Krein spaces are generalizations of
Hilbert spaces.
The talk is devoted to the proof of resolvent estimates for the boundary values
of resolvents of selfadjoint operators on a Krein space. Our result generalizes the result of Mourre to the Krein space setting. We will also discuss applications to the KleinGordon operator coupled to an electromagnetic field. This is joint work with Vladimir Georgescu and Chrsitian Gérard.
Thursday, December 12:
Nishanth Gudapati (Albert Einstein Institute)
``Critical SelfGravitating Wave Maps''
Abstract: Wave maps are maps from a Lorentzian manifold to a Riemannian manifold which are critical points of a Lagrangian which
is a natural geometrical generalization of the free wave Lagrangian. Selfgravitating wave maps are those from an
asymptotically flat Lorentzian manifold which evolves according to Einstein's equations of general relativity with
the wave map itself as the source. The energy of wave maps is scale invariant if the domain manifold is 2+1 dimensional,
hence it is referred to as the critical dimension.
Apart from a purely mathematical interest, the motivation to study critical selfgravitating wave maps is that they
occur naturally in 3+1 Einstein's equations of general relativity. Therefore, studying critical selfgravitating wave maps
could be a fruitful way of understanding the ever elusive global behavior of Einstein's equations. A few central
questions concerning the study of critical selfgravitating wave maps are local and global existence, blow up criterion,
profile, compactness and bubbling.
In this talk, after a brief discussion on the background and formulation of the Cauchy problem of critical selfgravitating
wave maps, we shall present a recent proof of the nonconcentration of energy of critical equivariant selfgravitating wave
maps before pointing out potential generalizations and applicable methods therein.
Monday, December 16, 34pm, 383N (Note unusual date and time):
Jared Wunsch (Northwestern)
``The trace formula for diffractive closed geodesics on a conic
manifold''
Abstract: We consider the trace of the wave group on a compact manifold with conic singularities.
The trace of the wave group, which on the one hand equals $\sum
e^{it\lambda_j}$ where $\lambda_j^2$ are the eigenvalues of the
Laplacian, is on the other hand a distribution in $t,$ singular at
lengths of closed geodesics. Those closed geodesics that interact
with the cone points generically do so `diffractively,' carrying
singularities into regions of phase space inaccessible to ordinary geodesic flow. We describe a formula for the leading order singularity of the wave trace at lengths of closed diffractive geodesics, generalizing the formula due to DuistermaatGuillemin in
the smooth setting and to Hillairet in the setting of flat surfaces
with conic singularities. This is joint work with G. Austin Ford.