Annual Newsletter of the University of Wisconsin Math Department

Eighth Wolfgang Wasow Memorial Lecture

The year 2003 marked the tenth anniversary of the death of Wolfgang Wasow. The 8th Memorial Lecture, funded by a generous endowment established by Wolfgang's children, was given on November 12, 2003 by Nancy Kopell on Rhythms in the nervous system: From cells to behavior via dynamics.

Nancy Kopell is the William Goodwin Aurelio Professor of Mathematics and Science at Boston University and is one of the leading biomathematicians in the world. She is co-director of the Center for BioDynamics (CBD) at Boston University, a multidisciplinary center which aims to train undergraduates, graduates, and postdoctoral fellows in leading techniques from dynamical systems theory and its applications to biology and engineering. A former MacArthur Foundation Fellow, Professor Kopell was elected to the National Academy of Sciences in 1996.

The abstract of her lecture was:

The nervous system produces rhythmic electrical activity in many frequency ranges, and the rhythms displayed during waking are tightly tied to cognitive state. This talk describes ongoing work whose ultimate aim is to understand the uses of these rhythms in sensory processing, cognition and motor control. The method used is to address the biophysical underpinnings of the different rhythms and transitions among them, to get clues to how specific important subsets of the cortex and hippocampus process and transform spatio-temporal input. We focus on the gamma rhythm (30-80 hz), which is associated with attention and awareness, and theta (4-12), associated with active exploration and learning of sequences. Via case studies, we show that different biophysics corresponds to different dynamical structure in the rhythms, with implications for function. The mathematical tools come from dynamical systems, and include the use of low-dimensional maps, probability and geometric singular perturbations.

Professor Kopell also gave a more technical lecture Ä neuron as a chain of oscillators: dynamics of a dopaminergic neuron" in the Applied Math/PDE seminar on November 13, 2003.

Eighth LAA Lecture

The 8th annual LAA Lecture was given by Peter Sarnak of Princeton University and New York University on October 17, 2003. The title of his lecture was Ramanujan Graphs and his abstract reads as follows:

Highly connected but sparse graphs (ëxpanders") have proven to be fundamental building blocks in many algorithms in computational complexity theory. We review the definition(in terms of eigenvalues of the adjacency matrix) and construction of Ramanujan Graphs, these being optimal in this connection. We then discuss recent developments related to the eigenvalues and random graphs and matrices as well as higher dimensional Ramanujan Buildings.

Peter Sarnak is Eugene Higgins Professor of Mathematics at Princeton University and Professor of Mathematics at the Corant Institute of Mathematical Sciences, NYU. Professor Sarnak has made many major contributions to number theory and to problems in analysis motivated by number theory. He is a 2003 winner of the AMS's Levi L. Conant Prize, a 2001 winner of the Ostrowski Prize, and a 1998 winner of SIAM's Polyá Prize. He was elected to the U.S. National Academy of Sciences and the U.K. Royal Society in 2002. He has supervised the theses of over 30 graduate students.