New professorships

Last year, thanks to two very generous donations, two new professorships were endowed: the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics, and the Walker Family Endowed Professorship in Mathematics. Inaugural four-year appointments to these positions have now been made: Sándor Kovács is the McKibben and Merner Endowed Professor, and Gunther Uhlmann is the Walker Family Endowed Professor. These two mathematicians describe some of their work on the following pages� �

Sándor Kovács

Sándor Kovács is a Professor in the Department of Mathematics. He joined the UW faculty in 2000 after holding faculty positions at the University of Chicago and the Massachusetts Institute of Technology. At the University of Washington he has directed 4 Ph.D. theses and is currently working with 8 graduate students. Since 2002 he has served as the Academic Director of the Summer Institute for Mathematics at the University of Washington (SIMUW), a summer program for high school students interested in mathematics.

Kovács earned his B.S. degree at Etvös University in Budapest, Hungary in 1990 and his Ph.D. at the University of Utah in 1995.

Kovács's area of expertise is higher dimensional complex algebraic geometry, in particular moduli and classification problems. "Higher" in this context means dimension three or more. Here "three" is actually a bit more than it may sound at first. Complex algebraic geometry considers objects defined by polynomials in several variables with complex numbers as coefficients. Recall that complex numbers are real numbers extended with the imaginary unit i, the square root of -1. The field of complex numbers is usually represented by a plane, that is, a two-dimensional object over the (good old) real numbers. As a result, an object that is 3-dimensional over the complex numbers is "really"' 6-dimensional according to our more traditional concept of dimension. A "complex curve" is actually a surface and a "complex surface" is actually four dimensional.

The ultimate goal of algebraic geometry is to classify all algebraically defined geometric objects. It is expected that such a classification will not produce a finite list, or even a countable one. The desired list will actually be nothing like a list. In fact, it is expected that the classification will produce some geometric objects whose points parametrize our objects, thereby these objects will themselves become the "lists." However, this is actually more useful than having a simple list, because the geometric properties of these parameter spaces, called "moduli spaces," contain a lot of information on how the different objects can be transformed into one another.

Moduli spaces of curves were constructed by David Mumford in the 1960s. At the time it was expected that surfaces wouldn't have to wait long to get their own moduli spaces. It turns out, however, that the case of surfaces presents many more difficulties than originally expected and accordingly the moduli theory of surfaces is still in its infancy today.

Kovács's main research interests revolve around the moduli theory of surfaces and higher dimensional varieties. In a joint work with Brendan Hassett he solved a long standing technical issue regarding the construction of moduli spaces in general.

An important characteristic of higher dimensional algebraic geometry is the inescapable fact that one must work with singular varieties, that is, objects containing points whose neighborhoods are significantly different from the neighborhoods of other points. For instance on a sphere, any point looks locally the same as any other point. This is, however, not true for example on a cone. The neighborhood of the vertex is nothing like the neighborhood of any other point.

Working with higher dimensional objects forces the necessity of understanding singularities. Kovács and some of his students has worked on various singularity questions, such as giving new characterizations of known singularity classes and establishing relationships between different classes.

Another area of Kovács's research builds both on moduli theory and working with singularities. At the 1962 International Congress of Mathematicians Igor Shafarevich made a powerful conjecture that still drives important research today. The original conjecture, roughly speaking, predicted that there are only finitely many families of curves with some numerical invariants fixed parametrized by a fixed curve. This was proved in the geometric case by Yurij Arakelov and Aleksey Parshin in the late 60s and early 70s and in the number field case by Gerd Faltings in the early 80s.

After a decade of dormancy, the geometric case of the conjecture regained interest through the (independent) work of Luca Migliorini and Kovács. These works ignited a flurry of new results in the area. The last 10 years saw far reaching generalizations of the original conjecture. Eckart Viehweg with various collaborators, most notably with Kang Zuo, and Kovács first working alone and recently joining forces with Stefan Kebekus and Max Lieblich produced better and better results. Today our knowledge is on a different scale than it was even after Arakelov's groundbreaking result. The current results handle families of objects of arbitrary dimension over an arbitrary dimensional base. Not surprisingly the problem evolved into several sub-problems and these are usually phrased in terms of moduli spaces. The best known results at this time in various aspects of the problem are due to the working duos of Viehweg and Zuo, Kebekus and Kovács, and Kovács and Lieblich.

These results, as often in mathematics, do not signal the end of the road, but instead lead to new and more interesting and important questions waiting to be answered. Kovács is currently working on these questions collaborating with several mathematicians from around the world, including Princeton University, Rice University, the Universities of Georgia, Pennsylvania, Texas and Utah in the United States, Université Joseph Fourier in France, Universität zu Köln in Germany and Instituto Nacional de Matemática Pura e Aplicada in Brazil. He is also involved in a long-term project that aims at collecting the scattered pieces of the construction of moduli spaces of surfaces and fill in the gaps where necessary. This, once completed, will be a major and important accomplishment.

Kovács was awarded the Rényi Kató Prize by the Bolyai János Mathematical Society in 1990, a Centennial Research Fellowship by the American Mathematical Society for 1998-2000, a CAREER Research Award by the National Science Foundation for 2001-2006 and a Research Fellowship by the Alfred P. Sloan Foundation for 2002-2006. His research has been continuously supported by the National Science Foundation through individual grants since 1996.

Sándor Kovács

Gunther Uhlmann

Inside-Out: Inverse Problems

In Inverse Problems one attempts to determine the internal properties of a medium by making observations outside the medium. In order to do this we measure the response of the medium probed with different kinds of waves, including X-rays, sound waves, electromagnetic waves, etc. I have done fundamental research in the mathematics of several inverse problems arising in different applications including medical imaging, seismic exploration, quantum scattering, non-destructive evaluation, and many others. My research has also given new insight into the question that has fascinated people for millennia as to whether one can make objects invisible to light and, more generally, to electromagnetic waves.

Mathematics of Tomography

A familiar inverse problem arising in medical imaging is Computed Tomography. In this imaging method the attenuation in intensity of an X-ray beam is measured, and the information from many X-rays from different sources is assembled and analyzed on a computer. Mathematically it is a problem of recovering a function from the set of its line integrals (or the set of its plane integrals). The mathematician Radon found in the early part of the 20th century a formula to recover a function from this information. The application to diagnostic radiology did not happen until the late 60s with the aid of the increasing calculating power of the computer. In 1970 the first computer tomograph that could be used in clinical work was developed by G. N. Hounsfield. He and A. M. Cormack, who independently proposed some of the algorithms, were jointly awarded the 1979 Nobel prize in medicine. Other familiar medical imaging techniques using X-rays to probe the medium are Positron Emission Tomography (PET) and Single Positron Emission Computed Tomography (SPECT). Magnetic Resonance Imaging (MRI) makes an image of tissue by measuring the body's response to strong magnetic fields. Ultrasound uses sound waves to make images of the body.

Mammography is undoubtedly a useful tool for early detection of breast cancer. Another new imaging technique called Electrical Impedance Tomography (EIT) has been proposed as a complementary medical imaging technique to mammography to improve on the rate of detecting breast cancer at an early stage. EIT works by applying tiny electrical currents through electrodes placed on the skin, measuring the corresponding voltage response, and then deducing the distribution of electrical conductivity and permittivity inside the body. (Roughly speaking, conductivity measures how easily charge moves through a medium, while permittivity measures the capacity of a medium to store electrical energy.) Since different parts of the body have different electrical properties, the computed distributions provide an image of the body's tissues and fluids. Since the conductivity of a breast tumor is much higher than that of the surrounding normal tissue, this technique could prove to be useful for early breast cancer detection.

Some of my colleagues and I have shown that, under some circumstances, it is indeed possible to solve the EIT problem, that is to determine the electrical conductivity inside a medium by making voltage and current measurements at the boundary (see e.g. [LU, SU]). This theoretical work has lead to algorithms that are being tried now for early breast cancer detection. David Isaacson and his group at RPI are actively involved in this medical imaging application.

Mathematics of Invisibility

The subject of invisibility has fascinated people for thousands of years and is the subject of many books, films, and television shows, ranging from H.G. Wells' The Invisible Man to J.K. Rowling's Harry Potter. There has also recently been considerable interest in the scientific community in the possibility of making objects "invisible," seemingly realizing the science fiction dreams. In particular there were two recent articles in Science (Pendry, et al, [PSS] and Leonhardt [L]) which discussed theoretical "cloaking" devices. These would shield an enclosed object from detection by electromagnetic (EM) waves. In principle, such devices could be constructed using "metamaterials," a catchall phrase coined about six years ago, which refers to composites which have physical properties, especially those having to do with the propagation of EM radiation, very different from their constituent materials.

The prescriptions for cloaking devices made of such materials described by Pendry, et al, turn out to be special cases of mathematical constructions of anisotropic conductivities (these depend on direction as well as position) that were given earlier in 2003 [GLU 1, GLU 2] for dimensions three and higher, using very similar methods. The anisotropic conductivities in these counterexamples are quite pathological - they exhibit perfect insulation in some directions and (in some cases) perfect conduction in others. In particular they don't satisfy the requirements of [LU] or [SU].

The space endowed with the conductivities constructed in [GLU 1, GLU 2] appear as vacuum in all external measurements, i.e., the inside of the ball is hidden inside a "cloak of invisibility." We draw the resulting current distribution of this example:

Picture of the currents for the homogeneous case (vacuum) and for the new conductivities that was constructed in [GLU 1, GLU 2]. An observer located outside the region cannot distinguish between the two.

In the figure we see that the current "bends" around the object being cloaked and then leaves as if a homogeneous object were there, without noticing the cloaked object.

In the future I plan to work on transferring the enormous theoretical progress that has been made in the mathematics of Inverse Problems into the practical applications. This will require a joint interdisciplinary effort with the researchers doing the concrete applications.

Gunther Uhlmann