Beckman Fellow 2002-03

Matthew A Ando



Topology is the study of features of shapes that remain unchanged by small distortions. Professor Ando works in a subset of this field, algebraic topology. Shapes are complicated, so one analyzes them with instruments that measure some features while filtering away others. The approach of algebraic topology is to use instruments whose readings take discrete values: under small distortions, the instrument’s reading will not change, because it cannot leap to another value.

Recently, Professor Ando has been studying a particular class of instruments known as elliptic cohomology theories. These mysterious cohomology theories were discovered in the mid-1980s. It is still not known precisely what topology these theories probe, but they exhibit a profound and previously unexpected relationship between topology, number theory, and the physics of string theory. During his Center appointment, he will pursue several projects in elliptic cohomology. One goal is to prove, with M.J. Hopkins, a conjecture that has motivated much of their collaboration. This conjecture expresses the relationship between physics, topology, and elliptic curves in particularly sharp form, and reveals previously hidden structure in the topology of manifolds. A second goal is to investigate the elliptic cohomology of shapes with symmetries. A significant portion of the work will take place at a special program on elliptic cohomology at the Isaac Newton Institute at Cambridge University.