Yongbin Ruan

A New Cohomology Theory of Orbifold

Based on the orbifold string theory model in physics, we construct a new cohomology ring for any almost complex orbifold. The key theorem is the associativity of this new ring. Some examples are computed.

Higher genus symplectic invariants and sigma models coupled with gravity

We define higher genus Gromov-Witten invariants and establish a mathematical theory of sigma model coupled with gravity over any semi-positive symplectic manifolds. As applications, we verify the stablizing conjecture of symplectic 4-manifolds for simply connected elliptic surfaces and construct smooth 6-manifolds admitting infinitely many deformation classes of symplectic structures.

Twisted Orbifold K-Theory

Abstract: We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition for K*orb(X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global quotient.