After introducing the notion of a quantum dynamical system and ergodicity of such systems, I will explain the example of the quantum Kronecker map on noncommutative 2-torus. If time allows I will also explain the notion of semi-classical ergodicity (quantum ergodicity) that arises in quantization of chaotic Hamiltonian systems.

A comparison of the etale homotopy type of a representable geometrically pointed simplicial sheaf X and that of its standard and twisted fibred sites is demonstrated. This result is applied to yield an etale van Kampen theorem for representable geometrically pointed connected simplicial sheaves under suitable hypotheses on the ambient site.

I will continue last week's talk by listing the axioms for the topological index, and then sketching how these axioms can be used to prove the index theorem (with emphasis on the sketching).

I shall first recall the classical theory of Ray-Singer analytic torsion, and conformal anomaly, for families of elliptic operators. I will mostly focus on families of elliptic operators on Riemann surfaces. The methods used here are based on ideas of spectral geometry and hence stand a chance of extension to a noncommutative setting. The extensions, when possible, are however quite nontrivial and involve many new elements and difficult computations. I shall then look at some known examples of noncommutative Riemann surfaces, the noncommutative elliptic curves equipped with curved metrics, and sketch the progress made in the last few years in understanding their conformal and spectral geometry. Scalar curvature can be defined by study of special values of spectral zeta functions. In particular I shall explain a formula for scalar curvature obtained in my joint work with F. Fathizadeh (and independently by Connes and Moscovici). This formula plays an important role for further study of noncommutative spectral geometry of noncommutative tori.

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.

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.

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.

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.