| Applied
and Interdisciplinary Research |
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In addition to research in
abstract mathematics,
the UW Math Department engages in a wide variety of applied and
interdisciplinary research. Many of these projects are carried
out in collaboration with other Mathematical Science Departments
at UW, as well as with science departments, research labs, and private
corporations. All of these projects present many opportunities for
graduate student involvement.
Here are brief descriptions of some
of the current projects in which Math Department members are involved.
Cryptography
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.
Immersed interface problems
In many physical problems, such as fluid
flow with bubbles or porous media flow with discontinuous permeabilities,
there is an immersed "interface" or boundary across which some property
changes abruptly. Grants from the National Science Foundation and
the Department of Energy support an interdepartmental effort to develop
numerical methods for partial differential equations in complicated
geometry, or with interfaces across which the solution is not smooth.
Professors Ken
Bube (Math), Randy
LeVeque (Applied Math), and Loyce
Adams (Applied Math) are the primary faculty involved.
Inverse problems
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.
Optimization
The Departments of Mathematics
and Applied
Mathematics have a joint program in theoretical, numerical, and
applied optimization. The core faculty are Professors Jim
Burke, Terry
Rockafellar (Emeritus), and Paul
Tseng. Current research activities of this group include collaborative
activities in the Departments of Statistics
(numerical methods for robust statistics), Bioengineering
(population analysis and biomedical imaging), Computer
Science and Engineering (Markovian decision processes), Finance
(portfolio optimization), and the School
of Aquatic and Fishery Sciences (hydro-power optimization). Most
of these projects have graduate student involvement.
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.
Computational
Molecular Biology
Working with the Fangman/Brewer
genetics lab at the University of Washington, Professor David
Collingwood studies the duplication of genetic material during
cell division. While it has been known for almost 40 years that
different parts of a chromosome replicate at different times during
the cell cycle; the recent invention of genomic DNA microarrays
has allowed experimenters for the first time to simultaneously investigate
origin activation and replication kinetics in a genome-wide fashion.
In a pilot study, Professor Collingwood, in collaboration with the
Fangman/Brewer lab (University of Washington), Elizabeth Winzeler
of the Ronald Davis lab (Stanford) and Lisa Wodicka and David Lockhart
(Affymetrix Corporation), has demonstrated the utility of these
microarrays in determining the genome-wide dynamics of chromosome
replication in the yeast Saccharomyces cerevisiae.
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