Southern Regional Number Theory Conference

In my first year I attended the lectures of N. C. Ankeny on algebraic number theory, and fell in love with the subject. Jurgen Neukirch spent the following year as a visitor at MIT and gave a second-year course on algebraic number theory from the point of view of valuations. He invited me to return with him to Regensburg, Germany, but I couldn't accept at the time due to the draft. But he left me with a problem that I solved for my doctoral dissertation, formally under Ankeny. By 1972 the draft was by lottery and my number was high enough not to be called, so I was able to go to Regensburg as a post-doctoral assistant to Neukirch. At Neukirch's suggestion, I began to think about pairs K, L of number fields with identical Dedekind zeta functions. I was able to show that K and L shared many properties in common, and I called K and L "arithmetically equivalent". That name became very popular, and I received many invitations to talk on the subject. After many years of work, Bart de Smit and I were able to give an example showing that arithmetically equivalent fields did not have to have the same class numbers. In 1975 I went to Bonn University for a one-year postdoc at the SFB 40, a precursor to the present day Max Planck Institute. After one year, the director Fritz Hirzebruch invited me to be his assistant, but the German Labor Department would not approve it. For the next four years, without requesting it, I received a one-year extension from the SFB. I believe I was the longest short-term visitor in SFB history. In 1980 I accepted an offer to come to LSU as an associate professor. Here I met Pierre Conner, who asked me if I knew any interesting quadratic forms associated with algebraic number fields. This began a collaboration that bore lots of fruit, including our book on trace forms. The trace form of a field K is a quadratic form that naturally lives in a ring called the Witt ring of K. Later we became interested in the Witt rings themselves, and together with Rick Litherland and Kaz Szymiczek we proved a local-global principle for Witt rings of algebraic number fields. In 1985, T. Sunada showed that a construction analogous to the one I used to produce pairs of arithmetically equivalent number fields could be used to produce pairs of non-isometric isospectral Riemannian manifolds. As a special case, this construction produces pairs of graphs with identical Ihara zeta functions. Along with some REU students, I wrote a few papers related to this latter subject. At LSU, I directed the dissertations of seventeen students, including eight women and three African-Americans. I served as a program officer in the division of mathematical sciences at NSF from 1999 to 2000, and as departmental chair from 2010 to 2016. I enjoyed every moment of my mathematical life so far, and I look forward to continued involvement during my retirement.