L3

We study Onsager’s model of isotropic–nematic phase transition with orientation parameter on a circle and sphere. We show the axial symmetry and derive explicit formulae for all critical points. Using the information about critical points we investigate a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the free energy is a functional of spatially-dependent orientation distribution, however, the spatial variation effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. As a particular example we consider a simplified model with orientation parameter on a circle and illustrate its relation to complex Ginzburg-Landau theory.

Let G be a connected reductive algebraic group over an

algebraically closed field of characteristic 0. A normal

irreducible G-variety X is called spherical if a Borel

subgroup of G has an open orbit on X. It was conjectured by F.

Knop that two smooth affine spherical G-varieties are

equivariantly isomorphic provided their algebras of regular

functions are isomorphic as G-modules. Knop proved that this

conjecture implies a uniqueness property for multiplicity free

Hamiltonian actions of compact groups on compact real manifolds

(the Delzant conjecture). In the talk I am going to outline my

recent proof of Knop's conjecture (arXiv:math/AG.0612561).