Activities This Week

Symplectic function theory studies the space of smooth (compactly supported) functions on a symplectic manifold. This space is equipped with two structures: the Poisson bracket and the $C^0$ (or uniform) norm of the functions. Interestingly enough, while the Poisson bracket of two functions depends on their first derivatives, there are non-trivial restrictions on its behavior with respect to the $C^0$-norm. We will discuss various results of this kind and their applications to Hamiltonian dynamics.

Planar algebras are a type of diagrammatic graded algebra, introduced to axiomatize the standard invariant of subfactors. The fundamental example is the Temperley-Lieb algebra which can be constructed as End(X^n), where X is a quantum sl_2 module. Recently, there has been interest in a finite dimensional version of quantum sl_2, known as restricted quantum sl_2, and it has been conjectured that its representation theory is equivalent to a logarithmic conformal field theory. We aim to generalize the Temperley-Lieb construction to the restricted case, giving generators and relations of the planar algebra, and describing morphisms between indecomposable modules diagrammatically.