#### Alladi Ramakrishnan Hall

#### Analytic surgery exact sequences and $\ell^2$ spectral invariants

#### Indrava Roy

##### ISI New Delhi

*The Atiyah-Singer index theorem describes a deep relation between the*

analytic and geometric-topological properties of a compact smooth manifold.

Further development of the theory by Atiyah, Patodi and Singer established

the existence of certain spectral invariants which are again intricately

related with the underlying geometry and topology. The Baum-Connes

conjecture is a far reaching generalization of the index theorem, and deep

results due to Keswani and Piazza-Schick assert that the spectral

invariants defined by Atiyah-Patodi-Singer have various stability

properties when the Baum-Connes conjecture holds. Similar stability results

hold for $\ell^2$ rho-invariants, a generalization of classical

rho-invariants that was first introduced by Cheeger and Gromov. Building on

earlier work by Higson and Roe on the analytic surgery exact sequence, we

give a more conceptual framework for the relation of $\ell^2$-rho

invariants with the Baum-Connes conjecture and give new proofs of the above

classical results, introducing new analytic and geometric invariants in the

process. (joint work with M.-T. Benameur)

Done