Title: Higher holonomy

Abstract: One of the most important aspects of Lie theory is the definition of holonomy, also known as parallel transport, or the path-ordered exponential. Lie’s definition of holonomy amounts to the statement that a (sufficiently small) connection on the unit interval 0≤t≤1, that is, system of linear first-order non-autonomous ordinary differential equations, is…

Title: The Gauss-Manin connection in noncommutative geometry

Abstract:
The noncommutative Gauss-Manin connection is a flat connection on the periodic cyclic homology of a family of dg algebras (or more generally, A-infinity categories), introduced by the speaker in 1991.

The problem now arises of lifting this connection to the complex of periodic cyclic chains. Such a…

Title: A Hopf index for isotropic sections of bundles with quadratic forms

Abstract: he Hopf index theorem counts solutions of equations via a winding number. I’ll review this and give 8 equivalent formulations. Then I’ll describe 8 analogues for equations which form “isotropic sections” of “bundles with quadratic form”. There are applications to “cosection localised virtual cycles”…

Title: Weight systems associated to Lie algebras

Abstract: V. A. Vassiliev's theory of finite type knot invariants allows one to associate
to such an invariant a function on chord diagrams, which are simple combinatorial
objects, consisting of an oriented circle and a tuple of chords with pairwise
distinct ends in it. Such functions are…