Program
The schedule is in local time (GMT-3).GAAG | Wednesday | Thursday | Friday | ||
10:30 — 12:00 | Araujo | Ortiz | Kinser | ||
12:00 — 14:00 | Lunch | ||||
14:00 — 15:00 | Kool | Neumann | Esposito | ||
15:00 — 15:30 | WebCoffee | ||||
15:30 — 16:30 | Gondim | Balibanu | Diogo |
Titles and Abstracts
Panoramic talks
- Carolina Araujo (IMPA). The Cremona group. Slides
When studying a projective variety $X$, one usually wants to understand its symmetries. The structure of the group of automorphisms of $X$ encodes relevant geometric properties of $X$. In birational geometry, however, the notion of automorphism is too rigid, and it is more natural to consider birational self-maps of $X$. Birational self-maps of the projective space $\mathbb{P}^n$ are called Cremona transformations. Describing the structure of the group of Cremona transformations of the plane ($n=2$) is a lovely problem that goes back to the 19th century. Nevertheless, only recently fundamental questions, such as (non-)simplicity, have been settled. In higher dimensions, new phenomena occur and the theory is undergoing rapid development. In this talk, I will discuss some remarkable results about the Cremona group, from the classical Noether-Castelnuvo Theorem (1870-1901) to exciting new developments.
- Cristian Ortiz (IME-USP). Symplectic Lie algebroids. Slides
Lie algebroids are geometric structures which have as particular instances Lie algebras, regular foliations, infinitesimal actions, Poisson structures, among others. The talk will give an introduction to symplectic Lie algebroids, which are Lie algebroids equipped with a non-degenerate 2-cocycle. Several examples will be presented and if time permits we will discuss the main ingredients of a reduction scheme for symplectic Lie algebroids.
- Ryan Kinser (University of Iowa). Quiver representations and multiple flag varieties. Slides
The goal of this panoramic talk is to survey my joint research program with Jenna Rajchgot. The focus of the program is to unify problems about the equivariant geometry of representation varieties of Dynkin quivers with the corresponding problems for Schubert varieties in multiple flag varieties. The specific problems we consider are:- characterizing singularities of orbit closures,
- combinatorial description of orbit closure containment,
- formulas for equivariant Grothendieck classes of orbit closures.
Afternoon talks
- Ana Balibanu (Harvard University). Steinberg slices in quasi-Poisson varieties. Slides
We consider a multiplicative analogue of the universal centralizer of a semisimple group $G$ — a family of centralizers parametrized by the regular conjugacy classes of the simply-connected cover of $G$. This multiplicative analogue has a natural symplectic structure and sits as a transversal in the quasi-Poisson double $D(G)$. We show that $D(G)$ extends to a smooth groupoid over the wonderful compactification of $G$, and we use this to construct a log-symplectic partial compactification of the multiplicative universal centralizer.
- Chiara Esposito (Salermo, Italia). [Formality, Redution]=0 ? First step. Slides
In this talk we propose a reduction scheme for multivector fields phrased in terms of L-infinity-morphisms. First, using geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to build up a suitable deformation retract of DGLA’s. As a second step, we construct a Poisson analogue of the Cartan model for equivariant de Rham cohomology. As a consequence we prove the existence of a curved L-infinity morphism between equivariant multivector fields and multivector fields on the reduced manifolds that coincides with the standard Marsden-Weinstein reduction.
- Frank Neumann (University of Leicester). Equivariant cohomology for smooth stacks and spectral sequences. Slides
We will study models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler.
We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group. If time permits I will indicate analogues of equivariant cohomology for actions of algebraic groups on algebraic stacks. Joint work with Luis Alejandro Barbosa-Torres (USP).
- Luis Diogo (UFF). Lagrangian submanifolds as modules. Slides
The problem of classifying Lagrangian submanifolds in symplectic manifolds is a central one in symplectic geometry. We will explain an algebraic approach to classification for the important class of monotone Lagrangians in cotangent bundles of spheres. The argument involves thinking of those Lagrangians as objects in a suitable category of modules over an algebra, and analyzing that category of modules. This is joint work with Mohammed Abouzaid.
- Martijn Kool (Universiteit Utrecht). Virtual Euler characteristics of moduli spaces of torsion free sheaves on general type surfaces. Slides
Moduli spaces of stable torsion free sheaves on general type surfaces are not necessarily smooth. Nonetheless, they are smooth in a "virtual" sense, which allows one to define their virtual Euler characteristics. I give a survey of recent conjectures for generating functions of such numbers, which have surprising connections with modular forms and physics. Joint work with L. Goettsche.
- Rodrigo Gondim (UFRPE). Waring problems and the Lefschetz properties. Slides
We study three variations of the Waring problem for homogeneous polynomials, concerning the Waring rank, the border rank and the cactus rank of a form. We show how the Lefschetz properties of the associated algebra affect them. The main tool is the theory of mixed Hessians and Macaulay-Matlis duality. We construct new families of wild forms, that is, forms whose cactus rank, of schematic nature, is bigger then the border rank, defined geometrically. Joint with T. Dias, UFRPE.