We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new class of elliptic operators associated to foliations.

We compute K-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of R. We discuss the relation between our results and the η-invariant.

We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P → M along with a G-invariant fiberwise Dirac-type operator D on P . The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory g...

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M...

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group action. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum o...

Observers for systems with Lie group symmetries are an active area of research that is seeing significant impact in a number practical domains, including aerospace, robotics, and mechatronics. This paper builds on the theory recently proposed Equivariant Filter (EqF), which general observer design homogeneous spaces takes advantage to yield performance advantages. It shown EqF error dynamics in...

For discrete groups G, we introduce equivariant Nielsen invariants. They are equivariant analogs of the Nielsen number and give lower bounds for the number of fixed point orbits in the G-homotopy class of an equivariant endomorphism f : X → X. Under mild hypotheses, these lower bounds are sharp. We use the equivariant Nielsen invariants to show that a G-equivariant endomorphism f is G-homotopic...

We present a version of twisted equivariant K-theory-K-twisted equivariant K-theory, and use Grothendieck differentials to compute the K -twisted equivariant K-theory of simple simply connected Lie groups. We did the calculation explicitly for SU(N) explicitly. The basic idea is to interpret an equivariant gerbe as an element of equivariant K-theory of degree 1.

We study the T−equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients.