Fall 2011 Abstracts
Anton Alekseev, Université de Genève
Tropical avatar of the Gelfand-Zeitlin integrable system
We recall the definition and elementary properties of the Gelfand-Zeitlin (generalized eigenvalue map) on the space of $n$ by $n$ Hermitian matrices. By the classical result of Guillemin-Sternberg, it defines a completely integrable system, and its image is a polyhedral cone singled out by the interlacing inequalities between the eigenvalues.
We suggest a "tropical version" of the Gelfand-Zeitlin integrable system where the generalized eigenvalue map is replaced by a certain piecewise linear (tropical) map. It turns out that its image coincides with the Gelfand-Zeitlin cone. As an application, we obtain a new description of the Horn cone (formed by eigenvalues of triples of Hermitian matrices adding up to zero). The talk is based on a joint work with M. Podkopaeva and A. Szenes.
Matthias Franz, University of Western Ontario
Equivariant cohomology and syzygies
The GKM method is a powerful way to compute the equivariant (and ordinary) cohomology of many spaces with torus actions. So far it has been applied to so-called equivariantly formal $T$-spaces, which include compact Hamiltonian $T$-manifolds.
In this talk I will explain that the GKM method is valid for a much larger class of $T$-spaces. The explanation is based on a new interpretation of a sequence originally due to Atiyah and Bredon, and involves the notion of syzygies as used in commutative algebra. I will also exhibit a surprising relation between the equivariant Poincaré pairing and the GKM description.
This is joint work with Chris Allday and Volker Puppe.
Rajan Mehta, Pennsylvania State University
The fundamental group(oid) of a Lie groupoid
There is a notion, originating in the work of Haefliger, of generalized "paths" and "homotopies of paths" on a Lie groupoid, allowing one to define the "Haefliger fundamental group(oid)" of a Lie groupoid $G$. Moerdijk and Mrčun proved that the Haefliger fundamental group is "correct," in the sense that it agrees with the fundamental group of the classifying space of $G$. I will describe an alternate construction of the Haefliger fundamental groupoid, which both explains and simplifies Haefliger's generator-and-relation construction. The key tool is a general procedure taking double Lie groupoids to Lie 2-groupoids. This is joint work with Xiang Tang.
Eckhard Meinrenken, University of Toronto
Group-valued moment maps revisited
In her 1989 paper, Jiang-Hua Lu introduced the notion of moment maps for actions of Poisson Lie groups $G$, taking values in the dual Poisson Lie groups $H=G^*$. If $G$ carries the trivial Poisson structure, the Poisson Lie group $G^*$ is the dual of the Lie algebra of $G$, and one recovers the usual moment maps from symplectic geometry. A different kind of group valued moment maps appeared in 1998 in the AMM theory of quasi-Hamiltonian actions. Here, the given structure on $G$ is a bi-invariant pseudo-Riemannian metric, and the moment map takes values in $H=G$. In this talk, based on joint work with David Li-Bland, I will show that the two types of group-valued moment maps fit into a common framework, of moment maps with values in Dirac Lie groups $H$.
Ana Rita Pires, Cornell University
Symplectic Geometry on $b$-manifolds
In 2002, Radko gave a complete classification of the Poisson structures on a surface that vanish transversally on a union of curves. For higher dimensional manifolds, this notion generalizes to the top power of the Poisson bivector vanishing transversally on a union of hypersurfaces. In this talk, we will see that this is the analogue of being symplectic when one uses the $b$-tangent and $b$-cotangent bundles introduced by Melrose, Nest and Tsygan originally in the context of manifolds with boundary. Using this framework, we will see what classification result can be obtained for these higher dimensional "symplectic $b$-manifolds", and how it extends Radko's result for surfaces. This is joint work with Victor Guillemin and Eva Miranda.