**Title:** Fredholm theory for scattering on asymptotically conic spaces and applications

**Speaker:** Andras Vasy

**Speaker Info:** Stanford

**Brief Description:**

**Special Note**:

**Abstract:**

In this lecture I will discuss a new approach to scattering theory and some applications to the decay of waves on asymptotically flat, such as rotating Kerr black hole, spacetimes. Scattering theory in the spectral (time-independent) approach typically analyzes the near-spectrum behavior of the resolvent of a Laplacian-like operator on an asymptotically Euclidean space, including the existence of limiting resolvents in suitable function spaces. Here we set up a Fredholm theory on the spectrum, so the spectral family is inverted directly as an operator, rather than having to take limits of the resolvent.Indeed, it is natural to consider geometric generalizations of the asymptotically Euclidean setting to asymptotically conic spaces $(X,g)$ of dimension at least $3$ (with the asymptotic behavior at the `large endâ€™ of the cone). Then the setting is that of perturbations $P(\sigma)$ of the spectral family of the Laplacian $\Delta_g-\sigma^2$. The results include the inversion of $P(\sigma)$ for real $\sigma\neq 0$ on function spaces that precisely encode the behavior of outgoing spherical waves, as well as the uniform description of the inverse as $\sigma\to 0$. This then provides the analytic background for the linearized black hole stability result of the author with Haefner and Hintz.

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