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Given a Riemannian manifold \( (M,g)\) with a free \(U(1)\)-action, we may use the Cartan differentiation operator from equivariant cohomology and the Hodge star operator from Riemannian geometry to define a modified Laplacian, which acts on \(U(1)\)-invariant differential forms on \(M\), and which depends on a complex constant \(m\). We would like to study whether the order, in terms of \(|m|^2\), of the first eigenvalue of this modified Laplacian (which naively seems like it should depend on the choice of \(g\)) is related to a constant which comes from the equivariant cohomology of \(M\) (and hence is not related to the choice of \(g\)).

In the particular case when
(i) \(M\) is a contact manifold of dimension \(2n+1\) which admits a global contact form,
(ii) the \(U(1)\)-action is generated by the corresponding Reeb vector field,
(iii) \(g\) is compatible with the contact structure, and
(iv) we have some additional assumptions on how the Hodge star acts on wedge products of the contact form, then the order of the first eigenvalue of this modified Laplacian is \(O(|m|^{4n+2})\), and the largest (nonnegative) integer \(k\), so that there exists an \(\alpha\) in the equivariant cohomology of \(M\) for which \(z^k \cdot \alpha\) is also in the equivariant cohomology of \(M\), is \(k = n\).

In this seminar talk, I will first describe the setup for this problem. Then, I will discuss the proof of the above result. This project is based on a question posed by Tudor Dimofte.

There will be pizza.