We will start to prove the Orbit theorem, Schur's theorem, Horn's theorem and finally the Convexity theorem for symplectomorphism groups of toric manifolds. The main technique will be analogues of the diagonalization and spectral theorems.

Cancelled.

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

In a recent paper, McDuff proves the connectedness of the symplectomorphism group of a resolution of a weighted projective space. Her argument is likely to adapt to some other orbifolds. As a stepping stone, we will explain the proof of the connectedness of Symp($CP^2$), a result originally proved by Gromov.

We begin with the sub-Laplacian on the Heisenberg group. The twisted Laplacian is then introduced by taking the inverse Fourier transform of the sub-Laplacian with respect to the center of the Heisenberg group. After a recapitulation of the spectral theory of the twisted Laplacian in terms of the Wigner transform, the spectral theory and number theory of the twisted bi-Laplacian obtained by Gramchev, Pilipović, Rodino and me are reported. We end the talk with a glimpse into a connection of the twisted bi-Laplacian with the Riemann zeta-function.

Accurate computer recognition of handwritten mathematics offers to provide a natural interface for mathematical computing, document creation and collaboration. Mathematical handwriting, however, provides a number of challenges beyond what is required for the recognition of handwritten natural languages. For example, it is usual to use symbols from a range of different alphabets and there are many similar-looking symbols. We present a geometric theory that we have found useful for recognizing mathematical symbols. Characters are represented as parametric curves approximated by certain truncated orthogonal series. This maps symbols to the low-dimensional vector space of series coefficients. The beauty of this theory is that a single, coherent view provides several related geometric techniques that give a high recognition rate and do not rely on peculiarities of the symbol set.