Lagrangian Multiforms for Symmetries and Integrability: Classification, Geometry, and Applications

Whenever something moves or changes, it can be modelled mathematically using a differential equation. Solving a differential equation means determining the state of the system (the thing which is moving) at any time in the future from its current state. For many differential equations this is impossible: the systems they describe exhibit complicated behaviour (picture waves on a stormy ocean - their long-term movements are very hard to predict) and it is impossible to write a formula describing future states. Integrable systems are the exceptions: they are differential equations that can be solved and represent dynamics that look orderly (picture a wave on a canal produced by a boat that has suddenly stopped - it keeps traveling forward in a predictable way).

The orderly behaviour of an integrable system is caused by some hidden mathematical structure of the differential equation. This hidden structure can take many different forms. Some integrable systems possess several forms of hidden structures, but no form applies to all integrable systems. There is no universal theory of the mathematics of integrable systems.

This Fellowship explores a recent development in integrable systems, the central idea of which comes from physics. Almost every physical theory can be described by the fact that something is minimised. Such a description is called a variational principle. In optics, for example, a ray of light will always take the fastest possible path. In other cases, the quantity that is minimised may be less intuitive, but a variational principle always provides powerful mathematical tools.

The theory of "Lagrangian multiforms" uses variational principles to capture the hidden structures of integrable systems. It is a recent development, the advantages of which are only starting to come to light. One advantage is that Lagrangian multiforms apply to discrete systems in the same way as to continuous systems. This provides insight into relations between integrable systems of both types. Here, "discrete" means that space and time do not form a continuum, but work in fixed steps (like digital video consists of a finite number of pixels and a finite number of frames per second).
This Fellowship investigates the benefits of Lagrangian multiforms in three main areas:

1. Relations between integrable systems of different types and their classification. One example of such relations is found in the Lagrangian multiform theory of semi-discrete systems (which involve both discrete and continuous variables). Some semi-discrete Lagrangian multiforms exhibit surprising connections to fully continuous integrable systems. This is only one of several contexts in which Lagrangian multiforms provide relations between equations of different types. This Fellowship will deliver a broad investigation of this phenomenon, employ it to transfer insights between equations of different types, and classify equations of interest.

2. Geometry. In the theory of Lagrangian multiforms, parameters describing symmetries of the system are treated in the same way as the time-variable. Together they form "multi-time". This Fellowship will study geometric aspects of Lagrangian multiform theory. Of particular interest are geometric structures within multi-time, related to special solutions of the integrable system, as well as the geometry of multi-time itself. This will allow us to capture a larger class of differential equations and transfer the insights of Lagrangian multiforms beyond the realm of integrable systems.

3. Applications. This Fellowship will investigate applications of Lagrangian multiforms in computational science and in fundamental physics. Variational principles have many applications in both these areas, but not all are fully understood from a rigorous mathematical perspective. This research will employ Lagrangian multiforms as well as other recent developments to secure the mathematical foundation of these applications