A simply connected domain in C is analytically equivalent to the unit disc, by the Riemann mapping Theorem. The only exception is the whole plane. The situation in C2 is much more complicated. It was already observed in the 1920s and 30s that there are subdomains of C2 which are analytically equivalent to all of C2. These so-called Fatou Bieberbach domains are small enough that you can have two disjoint ones in C2. In fact, in the ground-breaking paper from 1988 by Rosay and Rudin they found infinitely many parallel ones. The space C2 is thus more mysterious than one might think at first. This has become an interesting issue in complex dynamics. Stable manifolds of complex dimension two ''ought to be'' C2s. But are they?
Wolfgang Wasow was on our faculty for 23 years retiring as the Rudolf E. Langer Professor of Mathematics in 1980. A generous endowment established by his children is responsible for this series of Wolfgang Wasow Memorial Lectures.
Professor Thomas Kailath of the Department of Electrical Engineering of Stanford University gave the Seventh Annual LAA Lecture on April 26, 2002. The title of his lecture was Displacement Structure: From Theory to Application. Professor Kailath has worked in a number of areas including information theory, communications, computation, control, signal processing, VLSI design, statistics, linear algebra and operator theory; his recent interests include applications to problems in semiconductor manufacturing. He has held Guggenheim and Churchill fellowships, among others, and received outstanding paper prizes from the IEEE Information Theory Society, the American Control Council, the European Signal Processing Society, the IEEE Signal Processing Society and the IEEE Transactions on Semiconductor Manufacturing. He holds honorary doctorates from Linkoping University, Sweden, Strathclyde University, Scotland, and the University of Carlos III, Madrid, Spain. He has had over 70 PhD students at Stanford University and more than 40 postdocs and research scholars. He is a member of the National Academies of Science and of Engineering, and the American Academy of Arts and Sciences.
In his talk, Professor Kailath related how the presence of special structure, and its exploitation, occupies engineers and applied mathematicians in fields where standard algorithms (like Gaussian elimination) are well known. The concept of displacement structure is a uniform and powerful way of capturing and exploiting both explicit (like Toeplitz structure) and implicit structure in matrix and operator theory.
During the week of April 22, 2002, George Papanicolaou, Robert Grimmett Professor of Mathematics at Stanford University, visited us. Professor Papanicolaou has had a great influence on applied mathematician through his research, PhD students, and postdoctoral fellows. He has developed and applied stochastic techniques to significantly advance our understanding of physical, biological, economic and other phenomena. His contributions to Science have been recognized by election to the National Academy of Sciences and the National Academy of Arts and Sciences. While in Madison, Professor Papanicolaou gave three lectures with the title: Time Reversal, Imaging and Communications in Random Media The first was to the Applied Math/PhD Seminar and had the subtitle Mathematical Analysis. The second was a general colloquium. The final talk was to the Graduate Seminar and it had the subtitle Communication Applications.
Another series of distinguished lectures was given by Armand Borel of the Institute for Advanced Study in Princeton. He visited the department during the week of May 6, 2002 and gave two lectures. The first was to the Lie Theory Seminar on Moduli Spaces for Flat Bundles over two and three dimensional Tori. The second was a fascinating talk on The scientific relations between Elie Cartan and Hermann Weyl, 1922-1930. Dr. Borel is the author of the book on the history of Lie groups is Essays in the history of Lie groups and algebraic groups. History of Mathematics, 21. American Mathematical Society, Providence, RI; London Mathematical Society, Cambridge, 2001. A recent article by Borel on E. Cartan and H. Weyl is ``Elie Cartan, Hermann Weyl et les connexions projectives'', in Essays on geometry and related topics, Vol. 1, 2, 43-58, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001.
Professor Borel was born in Switzerland, and did his undergraduate work at the Federal School of Technology (ETH) in Zürich. He obtained his doctorate degree at the University of Paris in 1952, He has been a professor at the Institute for Advanced Study since 1957.