Summary: The Atiyah-Singer index theorem is one of the monumental works in geometry and topology. My dream is to construct an infinite-dimensional version of it. Although this project is very hard, I constructed several core objects for an analytic index theory for infinite-dimensional manifolds. The problem is the following: For an infinite-dimensional “\(Spin^c\)”-manifold \({\mathcal M}\) on which a loop group of a circle acts, construct a \(C^*\)-algebra \(A\) which carries some information of \({\mathcal M}\), a Hilbert space which can be regarded as an “\(L^2\)-space consisting of sections of the Spinor bundle”, and an operator \({\mathcal D}\) which can be regarded as a Dirac operator on \({\mathcal M}\), and define a spectral triple coming from them for \(A\). The core idea for the construction comes from representation theory of loop groups and Higson-Kasparov-Trout’s algebra N. Higson et al. [Adv. Math. 135, No. 1, 1–40 (1998; Zbl 0911.46040)].
For the entire collection see [Zbl 1412.81008].