Gunther
Uhlmann has been working on an array
of inverse problems since the 1980s. Some
are related to partial differential
equations, like the EIT
problem; others arise from differential
geometry, like the boundary rigidity
problem. His theoretical work on cloaking
also makes connection to material science.
In 2011 the AMS awarded Gunther the Bôcher
Prize.

RTG co-Principal Investigators

Robin
Graham is a geometric analyst, one
of whose main interests is conformal
geometry. He has worked on scattering
theory and Dirichlet-to-Neumann maps in
the setting of asymptotically hyperbolic
manifolds, with connections to the AdS/CFT
correspondence in string theory. Among
other topics, he is presently studying the
inverse problem of boundary rigidity for
asymptotically hyperbolic metrics, in
collaboration with Uhlmann
and P.
Stefanov.

Jim
Morrow has been the director since
1988 of the REU
program at UW in Discrete Inverse
Problems. Since 1994 he has been director
of Mathday
at UW, a program which brings over
1,200 high school students to campus for a
day of mathematical discovery. He also is
co-director of SIMUW
(Summer Institute for Mathematics at the
University of Washington).

Hart
Smith works on the application of
harmonic analysis to partial differential
equations. The current focus of his
research involves using wave packet
methods to study the behavior of
travelling waves and eigenfunctions on
compact manifolds. He pioneered the use of
curvelets to study wave propagation in
rough media. In recent collaboration with
Uhlmann
and de
Hoop, these curvelet techniques were
applied to the development of
computational schemes for the seismic
imaging problem.

Tatiana
Toro works at the interface of
partial differential equations and
geometric measure theory. Her recent work
concerns free boundary regularity problems
with rough data, and has settled several
questions concerning the relationship
between the geometry of a set and the
regularity of the measures it supports.
Tatiana spoke at ICM
2010 in Hyderabad, India.

RTG Postdoctoral Fellows

Jonas
Azzam works in the fields of
geometric measure theory and analysis on
metric spaces. His research uses
multi-scale analysis and techniques and
methodology from harmonic analysis to
quantitatively study the geometry of
subsets of Euclidean space and general
metric spaces. Recently, his work has
focused on quantitative rectifiability and
Lipschitz analysis.

Brent
Werness studies the
Schramm-Loewner Evolutions, a topic which
incorporates ideas from probability theory
and complex analysis to study the geometry
of curves in models from two-dimensional
statistical physics. Recently, he has
approached these problems using techniques
from rough path theory, and
quasi-conformal geometry.

RTG Graduate Fellows

Matthew
Badger completed his Ph.D. under Tatiana Toro. While
supported by the RTG grant he studied
geometric measure theory and its
application to PDE, specifically harmonic
measure on non-tangentially accessible
domains in higher dimensions. A focus of
his research was to prove analogues in
space of results in the plane by using
geometric measure theory to replace
complex analysis. He now has a position at
Stony
Brook University.

Joel
Barnes is a third-year graduate
student studying probability theory with Krzysztof Burdzy. His
current research concerns obtaining
solutions to a free-boundary parabolic PDE
as the hydrodynamic limit of an
interacting particle system. The system is
connected to an alternate model of
exclusion dynamics with a different
stationary distribution than the
traditional simple exclusion process.

Gregory
Drugan is a fourth year graduate
student in the math department. He is
interested in differential geometry and
partial differential equationsand is
currently working on problems involving
geometric flows. His thesis advisor is Yu Yuan.

Sean
Holman works in the area of inverse
problems. The general objective in such
problems is to determine parameters of a
medium based on measurements made around
the boundary. In his graduate work he
considered the problem of recovering
anisotropic media based on polarization
measurements of light rays passing through
them. In winter 2010 he began a position
at Purdue
University, working on geophysical
inverse problems.

Mark
Hubenthal is interested in numerous
topics related to inverse problems
including different imaging methods,
invisibility, inverse transport, and
scattering. He is currently working on
photo-acoustic tomography, which is
closely related to the inverse Schrodinger
problem with internal measurements. One
particular question of interest concerns
uniqueness and stability while restricted
to using only illuminations supported on
part of the boundary of the medium. Mark's
Ph.D. advisor is Gunther
Uhlmann.

Stephen
Lewis is a third year graduate
student studying geometric measure theory
and PDE with Tatiana
Toro.

Stephen
McKeown is a second year Ph.D.
student. He is interested in pure
differential geometry and its interactions
with theoretical physics.

Lee
Patrolia is a PhD student working on
inverse problems and PDE. She is currently
working on aspects of photo- and thermo-
acoustic tomography and inverse problems
for the linear transport equation. In
particular, she is investigating an
inverse problem for the stationary linear
transport equation on a Riemannian
manifold. Her thesis advisor is Gunther
Uhlmann.

Justin
Tittelfitz is currently in his
fourth year of the graduate program. He is
primarily interested in inverse problems
involving the elastic wave equation,
especially those with applications to
medical and seismic imaging. One
particular goal is the recovery of an
elastic wave's initial displacement via
boundary measurements. Justin's Ph.D
advisor is Gunther
Uhlmann.

Joshua
Tokle works on the analytic side of
probability theory. Under his advisor, Zhen-Qing Chen, he
is working to establish two-sided
estimates for the heat kernels of
symmetric alpha-stable processes and
censored stable processes in domains with
some boundary regularity conditions.

James
Vargo works on inverse problems
related to boundary rigidity. In these
problems the aim is to reconstruct a
Riemannian metric in the interior of a
manifold from measurements of the
geodesics taken at the boundary. This
problem is an abstract model for travel
time tomography in which one tries to
reconstruct wave speeds inside a medium
from travel times between boundary points.
Jim now has a position at Texas
A&M University.

Additional members of IPDE group at UW

Ken
Bube's research is at the interface
between numerical analysis, partial
differential equations, and geophysical
imaging. In ongoing projects with
geophysical researchers in industry, he is
working on numerical algorithms for
ill-posed problems, including seismic
traveltime tomography and multiple
removal. His work involves theoretical
analysis of problems with non-unique
solutions, coupled with numerical
algorithms to regularize these problems.

Anne
Greenbaum's research involves
numerical linear algebra and the numerical
solution of partial differential
equations. Particular interests include
nonnormal matrices, iterative methods for
linear systems, and integral equations.

Jack
Lee is a geometric analyst whose
research in recent years has centered on
Riemannian geometry and general
relativity. Lee is also the author of
several widely used graduate texts in
differential geometry.

Randall
LeVeque developes and uses
high-resolution finite volume methods to
solve hyperbolic partial differential
equations arising in fluid dynamics and
wave propagation problems, including
further development of the CLAWPACK
software (www.clawpack.org).
There are applications in geology ---
tsunamis, volcanoes, landslides --- and in
biomedicine: shock wave therapy and
lithotripsy.

Tu
Nguyen's current interest is inverse
problems, in particular the partial data
inverse problem. He has worked on unique
continuation, local well-posedness of
dispersive equations, and Lagrangian mean
curvature flows.

Daniel
Pollack's research centers on the
applications of partial differential
equations and analysis to problems in
geometry and physics. In recent years his
work has concerned solutions of the
Einstein field equations of mathematical
general relativity. For an introduction to
these topics, and a list of open problems,
see his co-written Mathematical
general relativity: a sampler.

John
Sylvester researches the development
of multi-dimensional mathematical tools
for remote sensing applications. A primary
emphasis is to draw precise and useful
conclusions from data that is both limited
and noisy. The mathematical problems
underlying these remote sensing problems
are the Inverse Scattering and Inverse
Source problems.

Yu
Yuan works in partial differential
equations and differential geometry. His
recent research focuses on fully nonlinear
equations such as special Lagrangian
equations (which include Monge-Ampere
equations).