Research
Note:
This section is heavily under construction.
Each page
will contain a short description of the research area,
including links to sites of interest and papers.
Combinatorics and Discrete Geometry
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- 3D Scanning: Professor Thomas Duchamp,
along with members of the
UW Departments of Statistics and
Computer Science and Engineering and
Microsoft Research,
has been working for several years on the mathematics
of 3D scanning -- creating computer models of surfaces from data
generated by laser scanners or range cameras. The goal of 3D scanning is the
inverse of computer aided manufacturing: given a physical object, create a
computer model of the object, capturing its shape, color, reflectance, and
other visual properties. Mathematically, this means developing efficient
algorithms for finding equations to represent a surface in space and its
physical properties from a sample of hundreds of thousands of points
scattered around the surface. The main mathematical tools are
subdivision
surfaces and wavelet analysis. In 1996, a group of UW faculty members formed
a company called Manifold Graphics, Inc., to develop commercial applications
of these ideas.
- Ocean Dynamics: Professor Robin Graham, in collaboration with
Frank Henyey, an ocean physicist at the UW Applied Physics Lab, studies
mathematical aspects of Hamiltonian theories of fluid flow for modeling the
dynamics of the ocean. The mathematical structure of the Hamiltonian
framework is similar to that used in electromagnetism and theories of
elementary particles in that it is a "gauge" theory. The variables used to
express the theory are referred to as "potentials" and are not the
"physical" variables that an experimentalist measures. Nonetheless, just as
for the electromagnetic potential, the potentials can be very useful in
understanding and working with the physical quantities.
Ergodic Theory and Symbolic Dynamics
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- Garth Warner (PhD Michigan, 1966)
- Tatiana Toro (PhD Stanford, 1992)
The classical theory of differential equations is concerned with "forward
problems": given a differential equation, find a solution. But in many
real-world situations, one has to go in the reverse direction: given some
information about the solutions to a differential equation, find the unknown
coefficients in the governing equation. The Mathematics Department has a
strong group working on various aspects of inverse problems. Professors
Gunther Uhlmann and John Sylvester
study various kinds of
tomography, which is a powerful method for probing the world around
us by directing energy in the form of waves or electric currents at an
object and observing the energy after it has interacted with the object.
Professor Ken Bube
works on inverse problems related to seismic exploration
of the earth. Professors Jim Morrow and Ed Curtis study how to determine the
structure of an electrical network from measurements of voltage and currents
at its boundary terminals.
For more information, see the home page of the
Inverse Problems Group.
- Garth Warner (PhD Michigan, 1966)
- John Westwater (PhD Cambridge, 1967)
-
Algebra (from Eric Weisstein's World of Mathematics)
- Cryptography, the analysis and construction of
secret codes, has long been important to governments, and has become big
business since the explosion of commerce on the Internet. In 1985,
Professor Neal Koblitz, a UW number theorist, and Victor Miller, then at IBM,
discovered a new system of cryptography that is much more secure than
previously known systems. Their system is based on elliptic curves,
which are curves defined by certain polynomial equations that play a central
role in abstract number theory and algebraic geometry. Professor Koblitz
continues to make major contributions to the theory of elliptic curve
cryptography, and serves as an advisor to Certicom Corporation, the leading
commercial provider of elliptic curve cryptographic systems.
- Neutron transport: Professor Anne Greenbaum
consults with groups
at Lawrence Livermore National Laboratory on problems involving neutron
transport. One of the problems involves numerical simulation of a nuclear
oil well logging procedure in which a neutron source is placed in a borehole
and measured radiation returning to a detector is used to deduce
characteristics of the surrounding material.
- Numerical methods and software for conservation laws: Hyperbolic
systems of partial differential equations are used in a number of
applications to model phenomena governed by conservation laws, such as soil
liquefaction, planetary nebulae, gravitation, and shallow water waves.
Professor Randy LeVeque
and his collaborators have studied the numerical
analysis of such systems extensively, and have developed a software package
called CLAWPACK (Conservation LAWs PACKage), designed to find numerical
solutions for a wide variety of problems involving such systems.
Optimization and Variational Analysis
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Partial Differential Equations
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Representation Theory of Lie Groups and Lie Algebras
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