The symmetries to be studied in the this Summer Program naturally arise in several different ways. Firstly, there are the symmetries of a differential geometric structure. By definition, these are the vector fields that preserve the structure in question—the Killing fields of Riemannian differential geometry, for example. Secondly, the symmetries can be those of another differential operator. For example, the Riemannian Killing equation itself is projectively invariant whilst the ordinary Euclidean Laplacian gives rise to conformal symmetries. In addition, there are higher symmetries defined by higher order operators. Physics provides other natural sources of symmetries, especially through string theory and twistor theory.
These symmetries are usually highly constrained—viewed as differential operators, they themselves are overdetermined or have symbols that are subject to overdetermined differential equations. As a typical example, the symbol of a symmetry of the Laplacian must be a conformal Killing field (or a conformal Killing tensor for a higher order symmetry). The Summer Program will consider the consequences of overdeterminacy and partial differential equations of finite type.
The question of what it means to be able to solve explicitly a classical or quantum mechanical system, or to solve it in multiple ways, is the subject matter of the integrability theory and superintegrability theory of Hamiltonian systems. Closely related is the theory of exactly solvable and quasi-exactly solvable systems. All of these approaches are associated with the structure of the spaces of higher dimensional symmetries of these systems.
Symmetries of classical equations are intimately connected with special coordinate systems, separation of variables, conservation laws and integrability. But only the simplest equations are currently understood from these points of view. The Summer Program will provide an opportunity for comparison and consolidation, especially in relation to the Dirac equation and massless fields of higher helicity.
Parabolic differential geometry provides a synthesis and generalization of various classical geometries including conformal, projective, and CR. It also provides a very rich geometrical source of overdetermined partial differential operators. Even in the flat model G/P, for G a semisimple Lie group with P a parabolic subgroup, there is much to be gleaned from the representation theory of G. In particular, the Bernstein-Gelfand-Gelfand (BGG) complex is a series of G-invariant differential operators, the first of which is overdetermined. Conformal Killing tensors, for example, may be viewed in this way. Exterior differential systems provide the classical approach to such overdetermined operators. But there are also tools from representation theory and especially the cohomology of Lie algebras that can be used. The Summer Program will explore these various approaches.
There are many areas of application. In particular, there are direct links with physics and especially conformal field theory. The AdS/CFT correspondence in physics (or Fefferman-Graham ambient metric construction in mathematics) provides an especially natural route to conformal symmetry operators. There are direct links with string theory and twistor theory. Also, there are numerical schemes based on finite element methods via the BGG complex, moving frames, and other symmetry based methods. The BGG complex arises in many areas of mathematics, both pure and applied. When it is recognized as such, there are immediate consequences.
There are close connections and even overlapping work being done in several areas of current research related to the topics above. The main idea of this Summer Program is to bring together relevant research groups for the purpose of intense discussion, interaction, and fruitful collaboration.
In summary, topics to be considered in the Summer Program include:
- Symmetries of geometric structures and differential operators.
- Overdetermined systems of partial differential equations.
- Separation of variables and conserved quantities.
- Integrability, superintegrability and solvable systems.
- Parabolic geometry and the Bernstein-Gelfand-Gelfand complex.
- Interaction with representation theory.
- Exterior differential systems.
- Finite element schemes, discrete symmetries, moving frames, and numerical analysis.
- Interaction with string theory and twistor theory.