[Alta-Logic] special seminar next week

[Kristine Bauer]

Dear colleagues,

Next week there will be a special meeting of the Peripatetic Seminar on Wednesday, July 12 at 1pm in MS 431.  Dr. Rory Lucyshyn-Wright from Mount Allison University is visiting for the week and will talk about his work related to the structure of differential geometry.  The talk may be of broad interest.  The title and abstract are below.

Many thanks,
Kristine

Title:  Simplicial calculus of higher-order differentials

Introduced by J. E. White in 1982, sector forms are certain real-valued functions on the n-fold iterated tangent bundle of a manifold, and they can be seen as generalized differential forms.  In joint work with Geoffrey Cruttwell, we showed that sector forms on a given manifold constitute a symmetric cosimplicial abelian group, and in fact we proved a similar result in an arbitrary tangent category.  As a corollary, sector forms constitute a cochain complex, with a subcomplex of alternating elements.  The latter subcomplex is isomorphic to the usual de Rham complex, so that the familiar exterior derivative is canonically induced by the cosimplicial structure on sector forms.

This talk will be an introduction to sector forms on manifolds, including a local description of sector forms on a given chart, as well as a comparison with the familiar description of differential forms via exterior algebra.  We shall see that sector n-forms constitute a locally free sheaf whose rank is related to the combinatorics of partitions, and we shall exhibit a specific basis on each chart.  The resulting local description makes it clear how sector forms provide a generalization of differential n-forms in which one finds not only first-order differentials and their n-th degree skew-symmetric products, but also higher-order differentials.  In particular, the cosimplicial structure on sector forms includes operations that capture the higher-order iterated partial derivatives of smooth functions, in coordinate-free form.

References:

G. S. H. Cruttwell and R. B. B. Lucyshyn-Wright, A simplicial foundation for de Rham cohomology in tangent categories, arXiv:1606.09080 (2016).

J. E. White, The method of iterated tangents with applications in local Riemannian geometry, Pitman, 1982.