Frank Nijhoff (Leeds) – Lagrangian multiforms, integrability and applications

Lagrangian multiforms were introduced in 2009 together with a new variational principle that was suitable for describing the phenomenon of multidimensional consistency (MDC) within a variational framework. MDC is the key integrability aspect of families of simultaneous equations that can be imposed on one and the same dependent variable in a space of independent variables of in principle arbitrary dimension.
In Lagrangian multiform theory, not only the equations of motion are derived from one cohesive principle, but also the Lagrangian components themselves, of what has now become no longer a scalar object ("the Lagrangian of a theory") but a differential or difference form in a (multi-time) space of arbitrary dimension.

In the talk I will explain the principle, and present examples of Lagrangian 1-forms (compatible systems  of ODEs), 2-forms (hierarchies of PDEs or partial difference equations) and 3-forms, both  in the discrete as well as continuous realm. In the latter cases I will make a connection with topological field theory and an infinite-dimensional Chern-Simons theory.
(This comprises work with many collaborators).