Event № 879
An index theory for elliptic operators on a closed manifold was developed by M. F. Atiyah and I. M. Singer. For a family of such operators parametrized by points of a compact space X, the K^0(X)-valued analytical index was computed there in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by M. F. Atiyah, V. K. Patodi, and I. M. Singer; the analytical index of a family in this case takes values in the K^1 group of a base space.
If a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes more complicated. The integer-valued index of a single boundary value problem was computed by Boutet de Monvel, who developed a special pseudodifferential calculus on manifolds with boundary. This result was recently generalized to K^0-valued family index by S. Melo, E. Schrohe, and T. Schick. The case of self-adjoint operators, however, remained open; it seems that Boutet de Monvel's calculus is not adapted to it.
In a series of two talks, I present a first step towards a family index theorem for self-adjoint elliptic operators on manifolds with boundary. A simplest non-trivial case of such a manifold is a compact surface with boundary. As it happens, for an X-parametrized family of such operators over a surface, the K^1(X)-valued analytical index can be computed topologically, without using of pseudo-differential operators. The second result is universality of the index: I show that it is a universal additive invariant for such families, if the vanishing on families of invertible operators is required.
The talks are based on my preprint arXiv:1809.04353.
All necessary notions will be explained during the first talk.