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 public key 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 and practice of cryptography.

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 core faculty working in optimization in the Department of Mathematics are Professors Jim Burke, Dimitry Drusvyatskiy, Thomas Rothvoss and Rekha Thomas. Their interests span a broad spectrum from continuous to discrete optimization, and from theory to algorithms to applications. The optimization group has strong ties with several other departments on campus such as Electrical Engineering, Aeronautics & Astronautics, Computer Science & Engineering, Statistics and Applied Mathematics with several adjunct faculty from these departments in the Department of Mathematics. In addition, the group has interactions with local industries such as Amazon, Google Inc. and Microsoft Corporation. The weekly optimization seminar is an interdepartmental affair, organized currently by students in the Departments of Mathematics and Electrical Engineering.

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.