Van Daele's lectures were concerned with the generalization of the following classical result. For any locally compact abelian group $G\,$ the dual group $\hat G\,$ of characters is again a locally compact abelian group. The dual of $\hat G\,$ is canonically isomorphic to $G\,$. If $G\,$ is compact, then $\hat G\,$ is discrete. To generalize this to the setting of compact groups, one uses the notion of a Hopf algebra. In the first lecture it was shown that to obtain a duality theory for discrete groups and the (non-unital) algebra $A\,$ of finitely supported functions $G\rightarrow C\,$, it is profitable to introduce the notion of a multiplier Hopf algebra. In this theory the comultiplication is a homomorphism $A\rightarrow M(A\otimes A)\,$ to the multiplier algebra of $A\otimes A\,$. Assuming that $A\,$ has a non-zero left invariant functional one can establish a biduality theorem analogous to Pontyagin's.
In the second lecture the $*$-algebra case was considered in detail. When a positive invariant functional is present one can form a GNS representation of $A\,$ and create an analogue of the left regular representation. Both talks were evidently very carefully prepared and were smoothly delivered.
The next day's lectures dealt with a contrasting but equally fashionable topic, namely the differentiability of functions on Banach spaces. This topic originates in the calculus of variations where minimisation problems are used to solve non linear differential equations. Although this can be a rather technical topic, Deville gave an accessible and clear account of some recent results. He began by presenting an elementary example of a differential equation with no classical solution but a continuum of weak solutions. This motivated the introduction of viscosity solutions defined in terms of sub and super-differentials. It was shown that for a Banach space $X\,$ of type (H1) (admitting a $C^1$ smooth bounded Lipschitz bump function) any lower semicontinuous positive $u:X\rightarrow R\,$ may be perturbed by a smooth function $\varphi\,$ of arbitrarily small supremum norm so that $u-\varphi\,$ attains its minimum.
In the second lecture the speaker considered how to prove uniqueness of solutions of the Hamilton Jacobi equations. To achieve a smooth variational principle with constraints it appears necessary to suppose either that $X\,$ is finite dimensional or that the functions involved are uniformly continuous. This surprising fact nevertheless leads to a satisfactory answer to the original problem. The combined attendance for the lectures was over 26, with an unusually large number of graduate students. Dismal weather forced cancellation of some intended hill walks, but the mathematics should have been a good compensation.
Dr. G. Blower (Secretary),