Mathematics -- The Science of Patterns and Algorithms

Geometrization of Topology and Physics

The interaction between geometry, topology, high energy physics, and cosmology continues to enrich all these disciplines. Global structures are emerging that seem to connect these diverse fields of inquiry. One exciting development of this kind is "mirror symmetry," a subject which illuminates certain six-dimensional manifolds that arise in string-theoretic physics. This theory pairs each of these spaces with a twin that is geometrically different but carries an equivalent physical theory [TA]. Efforts to determine the origin of this symmetry, its intrinsic structure, and its implications are generating entirely new mathematics and breaking open old, hard problems on the number of solutions to polynomial and differential equations.

Differential-geometric methods are producing quite surprising information on four-dimensional manifolds, while at the same time increasing the mystery of spaces in this critical dimension. Unlike the situation in dimension three, where we have simply-stated conjectures describing the list of all possible three-dimensional manifolds and established programs under way to verify these conjectures, in dimension four every reasonable looking hypothesis to date has been shown to be wildly naive. A basic theme in these investigations is that the discovery of new differential equations, many originating in physics, leads to new understanding of the geometries. Symplectic structures, the geometries associated to Hamiltonian mechanics, have been especially fruitful and have inspired cunning new constructions of large numbers of new four-dimensional examples and new invariants derived from algebraic geometry and physical gauge theories [ST].