Stephen Gelbart
The Nicki and J. Ira Harris Professorial Chair
The purpose of number theory is to
unravel the mysteries of the set of integers 1,2,3,..., and to explore the patterns
which arise from using integers to solve polynomial equations. For example,
it turns out that some integers can be written as the sum of two squares (e.g.
5), but others (like 7) cannot. Exactly which integers can be so expressed,
and what is their pattern? Questions like these are difficult to answer precisely
because there are infinitely many integers, and hence infinitely many equations
M = x2+y2 to solve.
In particular, checking things directly (say with a large computer) is useless
(since a computer can deal with only finitely many calculations in any given
time). Instead, answers to these questions come from esoteric mathematical theories
such as group representations and automorphic forms. These are my particular
areas of research. Though the motivation at present is theoretical, it is clear
that our work will eventually have applications to other branches of mathematics
and science, just as the number theory developed hundreds of years ago has already
had applications to such fields as cryptography.
Recent Publications
- [with J. Bernstein] Editors, An Introduction of the Langlands Program. Birkhauser, Boston, 2003, 281 pages.
- [with S. Miller] Riemann's zeta function and beyond. Bull. Amer. Math. Soc. 41, 1 (2004) 59-112.
- [with E. Lapid] Lower bounds for L-functions at the edge of the critical strip. Amer. J. of Math. 128, 3 (2006) 619-638.