example, there is mounting evidence that the same geometric flow techniques used in resolving the Poincaré conjecture can be applied to complex algebraic manifolds to understand the existence and nonexistence of canonical metrics—for example, the Kähler-Einstein metric—on these manifolds. Recently, ideas from algebraic topology dating back to the 1960s, A-algebras and modules, have been used in the study of invariants of symplectic manifolds and low-dimensional topological manifolds. These lead to the introduction of ever more sophisticated and powerful algebraic structures in the study of these topological and geometric problems.

In a different direction, deep connections have recently been discovered between random matrix theory, combinatorics, and number theory. The zeros of the Riemann zeta function seem to follow—with astounding precision—a distribution connected with the eigenvalues of large random matrices. This distribution was originally studied as a way of understanding the spectral lines of heavy atoms. The same distribution occurs in many other areas—for example with respect to the standard tableaux of combinatorics—and in the study of quantum chaos.

There are also connections recently discovered between commutative algebra and statistics. Namely, to design a random walk on a lattice is equivalent to constructing a set of generators for the ideal of a variety given implicitly; that is, solving a problem in classical elimination theory. This is of importance in the statistics of medium-sized data sets—for example, contingency tables—where classical methods give wrong answers. The classical methods of elimination theory are hard; the modern technique of Groebner bases is now often used.

Example 2: Interactions Between Mathematics and Theoretical High-Energy Physics

Certainly one of the most important and surprising recent developments in mathematics has been its interaction with theoretical high-energy physics. Large swathes of geometry, representation theory, and topology have been heavily influenced by the interaction of ideas from quantum field theory and string theory, and, in turn, these areas of physics have been informed by advances in the mathematical areas. Examples include the relationship of the Jones polynomial for knots with quantum field theory, and Donaldson invariants for 4-manifolds and the related Seiberg-Witten invariants. Then there is mirror symmetry, discovered by physicists, which, in each original formulation, led to the solution of one of the classical enumerative problems in algebraic geometry, the number of rational curves of a given degree in a quintic hypersurface in projective 4-space. This has been expanded conjecturally to a vast theory relating complex manifolds and symplectic manifolds. A more recent example of the cross-fertilization



The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement