Abstract: Given a $G$-invariant almost CR structure on a manifold $M$ one can construct a first-order differential operator whose restriction to the fibres of the CR structure resembles the Dolbeault-Dirac operator on an almost Hermitian manifold. This operator defines a virtual $G$-representation that is infinite-dimensional; however, given an additional assumption on the group action we can compute the character of this representation as a generalized function on $G$. To justify the use of the word "quantization" I'll sketch some parallels with geometric quantization that occur when some additional structure is imposed on the almost CR structure. When the almost CR structure is integrable, we will see the appearance of the tangential Cauchy-Riemann complex in this approach.

Abstract: The theory of p-compact groups deals with the homotopy analogues of compact Lie groups, and has been traditionally studied from the homotopy theory point of view. In this talk I will present some connections between this theory and the Schubert calculus, in a more algebro-combinatorial style. In particular I will focus on the different descriptions of the torus-equivariant cohomology of p-compact flag varieties, generalizing the theory of Bruhat graphs and results of Goresky-Kottwitz-MacPherson.

Abstract: We'll see a relationship between some elementary geometry and the discrete Fourier transform, which offers a starting point for excursions into polynomials, complex analysis, interpolation and circulant matrices. It has turned up in practical statistics, fluid mechanics, sculpture, and elsewhere, as well as providing intriguing pictures for the motions of the $N$-body problem. It's behind what has been popularized by Kalman as the most marvelous theorem in mathematics.

Abstract: We consider weighted projective spaces and homotopy properties of their symplectomorphism groups. In the case of one singularity, the symplectomorphism group is weakly homotopy equivalent to the Kahler isometry group of a certain Hirzebruch surface that corresponds to the resolution of the singularity. In the case of multiple singularities, the symplectomorphism groups are weakly equivalent to tori. These computations allow us to investigate some properties of related embedding spaces.