Prof. Paul Garrett
The cognitive and conceptual difficulties of abstract harmonic analysis are cousins of the difficulties in number theory. For example, the physicality of Fourier series is immediately appealing, while convergence properties are not trivial. Meanwhile, finitistic algebraic properties of integers are straightforward, while diophantine and asymptotic behaviors are mysterious. Riemann investigated the relation of complex analysis, a kind of harmonic analysis, to the zeta function and prime number theory. Dirichlet's theorem on primes in arithmetic progressions introduced harmonic analysis on finite abelian groups, and illustrated the point that more exotic zeta functions can have an impact on simpler objects, by using zeta functions attached to extension fields of the rational numbers. In the last 60 years, the work of many mathematicians has created a very broad, yet very detailed, harmonic analysis conjecturally enveloping very many number-theoretic problems. Sometimes this viewpoint goes by the convenient catchphrase "Langlands' program", although the ever-broadening use of this term is far-removed from the original specific issues. There is little immediate hope of proving grandiose conjectures such as the Riemann Hypothesis or Lindelof Hypothesis for the ordinary zeta, much less for more exotic zeta-functions and L-functions. Nevertheless, recent work has uncovered some tantalizing structures that illuminate these fundamental problems. In particular, we can start to see the impact that more exotic objects have on the simplest. I will try to explain these developments in accessible terms.
Go Back to Junior Coll. Web page
URL http://www.math.umn.edu/jrcoll/ The University of Minnesota is an equal opportunity educator and employer. © 2004, The Regents of the University of Minnesota |
|