Mathematical Physics Seminar 

As was already discussed in this seminar, a theorem of Delzant states that any symplectic manifold $(M,\omega )$ of dimension $2n$ equipped with an effective Hamiltonian action of the standard $n$-torus $T^n$, i.e. any symplectic toric manifold, is completely determined (as a Hamiltonian $T^n$-space) by the image of the moment map, a convex polytope $P$ in ${ℝ}^n$. In this talk I will explain, using symplectic (action-angle) coordinates, how all $\omega$-compatible toric complex structures on $M$ can be effectively parametrized by smooth functions on $P$. I will also sketch Donaldson’s formal general framework underlying this symplectic coordinates approach to Kähler toric geometry, and discuss some topics suited for application, namely: explicit construction of extremal Kähler metrics, spectral properties of toric manifolds and combinatorics of polytopes.

We describe the classification of equivalence classes of symplectic toric manifolds by their moment polytopes and prove the existence part in the classification theorem by constructing symplectic toric manifolds with symplectic reduction. This construction also yields an invariant Kähler structure on every toric manifold. We conclude by analysing the transformation of a symplectic toric manifold which corresponds to a cutting of the corresponding polytope.

To define and study toric manifolds from the algebro-geometric viewpoint, we review basic notions of algebraic geometry, define toric varieties and explain the correspondence with fans. Several examples will be given.

Native to algebraic geometry, the study of toric manifolds has many different entrances and has been scoring a wide spectrum of applications. For symplectic geometers, they provide examples of extremely symmetric (completely integrable) hamiltonian spaces, encoded in a neat combinatorial fashion.

To define and study toric manifolds from the symplectic viewpoint, in this informal talk we review basic symplectic notions and tools, such as moment maps and symplectic reduction. We conclude by stating the list of these manifolds and by illustrating it with some famous examples.

This will be the first talk of a series in a reading seminar devoted to Toric Manifolds.

General ellipsoids (with three distinct axes) are shown to be equilibrium configurarions of a self-gravitating fluid mass. Some comments on existence and stability of other equilibrium configurations (not necessarily convex) are made.

Second part of a two-part seminar on the paper Hyperelliptic Prym Varieties and Integrable Systems

Algebraic cycles are classical objects in algebraic geometry. In this talk we will briefly review their definition and discuss how they can be studied using homotopy theoretic techniques.

First of two sessions on the paper Hyperelliptic Prym Varieties and Integrable Systems by R.Fernandes and P. Vanhaecke.