Principal Investigator: Yongbin Ruan
During the past twenty years, there has been a great deal of interaction between mathematic and physics. Various correspondences or dualities arising from physics have had a great impact on mathematics. One of these correspondences is the Landau-Ginzburg/Calabi-Yau correspondence. This proposal offers a comprehensive program to establish this correspondence in mathematics with mathematical rigor. There is a wide range of applications such as global mirror symmetry, the computation of Gromov-Witten invariants of compact Calabi-Yau manifolds, and to modular forms.
Since the era of Newton, mathematics has been a key tool in helping us to comprehend the universe. An example is differential geometry via Einstein's theory of general relativity. During the last twenty years there has been a great deal of activity in building so-called string-theoretic models of the universe. The string-theoretic model of the universe is ten-dimensional, our usual 4-dimensional space-time togeher with a very small 6-dimensional space called a Calabi-Yau manifold. Such a small internal space will affect our space-time through certain mathematical quantities such as Gromov-Witten invariants. The computation of these invariants has been a central problem in geometry and physics. This is the area in which mathematical tools can be very useful. The current proposal envisages a theoretical framework as well as developing technical tools to compute Gromov-Witten invariants