## Project Details

### Description

The Atiyah-Singer index theorem for elliptic operators is one of the greatest results in mathematics, which builds an bridge between the analytic index of an elliptic opera- tor and the topological invariants of the underlying manifold. The Atiyah-Singer index formula includes many previous theorems, such as the Gauss-Bonnet-Chern theorem and the Riemann-Roch-Hizenbruch theorem, as special cases. Recently, van Erp and Baum generalized the Atiyah-Singer index theorem to subelliptic operators on contact mani- folds, their approach is to find a proper cohomology class based on the K-theory and the noncommunicative topology.

Besides Atiyah and Singer’s K-theory proof of their index theorem, Patodi developed an alternative approach based on the heat kernel asymptotic expansions. Patodi’s idea is to calculate the coefficients in the heat kernel expansions along the diagonal, and the index is involved in the coefficient of the (n/2)-th term, with n denoting the dimension of the manifold. Hence, the proof of Patodi and Atiyah-Bott-Patodi requires a huge calculations. Later on, Getzler introduced a rescaling on the variables and derivatives, which simplifies the heat kernel proof a lot.

This proposed project aims to prove the index theorem for the subelliptic operators on contact manifolds via the heat kernel approach. We shall use techniques in pseudo- differential operators to derive the heat kernel expansions. Based on the contact structure, we can incorporate Greiner’s approach for elliptic heat kernels and the Heisenberg calcu- lus. Getzler’s rescaling will also be used to simplify the derivation of the expression of the index. The index is given in terms of geometric quantities such as the curvature and its covariant derivatives, which are defined in the framework of subRiemannian geometries.

Besides Atiyah and Singer’s K-theory proof of their index theorem, Patodi developed an alternative approach based on the heat kernel asymptotic expansions. Patodi’s idea is to calculate the coefficients in the heat kernel expansions along the diagonal, and the index is involved in the coefficient of the (n/2)-th term, with n denoting the dimension of the manifold. Hence, the proof of Patodi and Atiyah-Bott-Patodi requires a huge calculations. Later on, Getzler introduced a rescaling on the variables and derivatives, which simplifies the heat kernel proof a lot.

This proposed project aims to prove the index theorem for the subelliptic operators on contact manifolds via the heat kernel approach. We shall use techniques in pseudo- differential operators to derive the heat kernel expansions. Based on the contact structure, we can incorporate Greiner’s approach for elliptic heat kernels and the Heisenberg calcu- lus. Getzler’s rescaling will also be used to simplify the derivation of the expression of the index. The index is given in terms of geometric quantities such as the curvature and its covariant derivatives, which are defined in the framework of subRiemannian geometries.

Status | Finished |
---|---|

Effective start/end date | 1/09/16 → 31/08/19 |

## Fingerprint

Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.