Background
Hejhal, Dennis Arnold was born on December 10, 1948 in Chicago, Illinois, United States. Son of Charles Jerry and Alice Ann Hejhal.
(During the last 10 years or so, mathematicians have becom...)
During the last 10 years or so, mathematicians have become increasingly fascinated with the Selberg trace formula. These notes were written to help remedy this situation. Their main purpose is to provide a comprehensive development of the trace formula for PSL(2,R). Volume one deals exclusively with the case of compact quotient space. Although the trace formula can be developed much more generally, there are severe limitations on what is known for the higher-dimensional groups. Under these circumstances, it makes sense to try to understand the simplest cases first. The main chapter in volume one is chapter 2. In this chapter, we study the trace formula using the techniques of analytic number theory. By focusing on the Selberg zeta function, we can prove some deep results about the distribution of eigenvalues and pseudoprimes (lengths of closed geodesics) for a compact Riemann surface. This is the first time most of this material has appeared in print. Roughly speaking: 1/3 of it is fairly obvious, 1/3 is due to Selberg (unpublished from 1950-53), and 1/3 is new. Although I have made a reasonable attempt to push the estimates in chapter 2 as far as they can go, further improvements might still be possible. There are definitely several loose-ends which deserve further investigation. I am inclined to say that the techniques are just as important as the theorems. Another important feature of volume one is the derivation of a trace formula for modular correspondences, in which the analytic and non-analytic automorphic forms are treated simultaneously. The analytic component of this formula corresponds to the well-known Eichler-Selberg trace formula. These notes grew out of an attempt to understand the relationship between trace formulas and the Riemann zeta functions. To make a long story very short: the relationships considered in volume one proceed mainly from zeta to the trace formula, while those in the other direction first appear in volume two.
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Hejhal, Dennis Arnold was born on December 10, 1948 in Chicago, Illinois, United States. Son of Charles Jerry and Alice Ann Hejhal.
Hejhal graduated from the University of Chicago in 1970 with a Bachelor"s degree and from Stanford University in 1972 with a Doctor of Philosophy in mathematics under the direction of Menahem Max Schiffer.
In his mathematical research he frequently uses extensive computer calculation. He became an assistant professor at Harvard, then in 1974 an associate professor at Columbia University and starting in 1978 a professor at the University of Minnesota, where he now works. Additionally he has been since 1994 a professor at the University of Uppsala and since 1986 a fellow of the Minnesota Supercomputing Institute.
He was in 1993 a guest professor at Princeton University and several times (for the first time in 1983) at the Institute for Advanced Study.
Hejhal works on analytic number theory, automorphic forms, the Selberg trace formula and quantum chaos. In 1968 he was a Putnam Fellow, from 1972 to 1974 a Sloan Fellow.
In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley (Zeros of Epstein Zeta Functions and Supercomputers). In 2012 he became a fellow of the American Mathematical Society.
Among his successful former doctoral students is Persi Diaconis.
He also supervised undergraduate honors thesis research of James Z. Wang.
(During the last 10 years or so, mathematicians have becom...)
Member Mathematics Association of America, American Mathematics Society, Swedish Mathematics Society.