AT&T Bell Laboratories
Murray Hill, NJ
The 3x+1 problem and its generalizations
A new view on the Hirsch Conjecture
The Hirsch conjecture states that any d-dimensional bounded polytope
with n facets has an edge-path between any two vertices of length at
most n-d. The d-step conjecture is the special case n=2d, and is known
to be equivalent to the general conjecture. It has long been suspected
to be false in high dimensions. In joint work with N. Prabhu and J.
Reeds, we discovered striking evidence that it is true in all
dimensions, in a strong form. This evidence was based on a connection
with Gaussian elimination of a set of $(d!)^2$ matrices constructed
from the d-polytope with 2d facets, and massive computational
I got my degrees from Massachusetts Institute of Technology
(S.B./S. M. 1972 Ph.D. 1974) all in mathematics. My thesis advisor
was Harold M. Stark, with a thesis in Algebraic Number Theory.
I am a pretty quiet guy who enjoys reading and
going for nice leisurely runs.
I'm kind of the traditional type; I am most attracted to the
permanence and stability of mathematics: I don't even own