Mirror symmetry is a mathematical and physical phenomenon discovered by
string theorists in 1989. In string theory, space-time is 10 dimensional
rather than 4 dimensional, with six of these dimensions being extremely
small. To picture this concretely, one might instead imagine the
universe in fact is shaped like a water hose. A small ant crawling on the
surface of the hose might perceive the universe as two-dimensional,
but a larger being might only see the length of the hose and not
the small circumference of the hose. Nevertheless, the geometry of
the circular cross-section of the hose impacts the physics of the
observed universe. In string theory, instead of the one-dimensional
circle, one has a very small six-dimensional geometric object,
known as a Calabi-Yau manifold. Mirror symmetry can then be expressed
as the observation that such Calabi-Yau manifolds come in pairs,
called mirror pairs, which give rise the same observed four-dimensional
physics.
While this phenomenon was originally very much part of theoretical
physics, mathematicians have become fascinated by it because it has
revealed profound insights into the geometry of Calabi-Yau manifolds
and beyond. Mirror symmetry identifies radically different calculations
carried out on the two members of a mirror pair, producing powerful tools
for carrying out difficult calculations. The subject has now become
central to modern mathematics, lying at the intersection of multiple
fields and driving innovation in all of these fields. These innovations
have lead to solutions to many diverse long-standing problems in mathematics.
One of the central approaches to understanding mirror symmetry from a
mathematical perspective has been developed by the PI and Bernd Siebert,
creating what is now known colloquially as the Gross-Siebert program.
The main goal of the grant is to both further develop the program
and to find applications of it to other areas of mathematics.
