Anthony D. Blaom (AUT)
Cartan connections on Lie algebroids
A Lie groupoid equipped with a multiplicatively closed, horizontal connection
may be viewed as a symmetry `deformed by curvature'. These connections ---
which we call {\em Cartan} connections --- are related to the connections
on principal bundles of the same name. However, unlike their classical namesakes,
no homogeneous model is fixed beforehand and intransitivity is allowed.
Most significantly, Cartan connections can be infinitesimalized to obtain
certain linear connections on the corresponding Lie algebroid. This introductory
talk discusses these latter connections, explaining the sense in which they
are infinitesimal symmetries deformed by curvature. We sketch the construction
of the canonical Cartan connection for Riemannian geometry, and mention
some applications to uniformization problems.
Robert Bryant (Duke University)
Henrique Bursztyn (IMPA)
Multiplicative geometric structures on Lie groupoids
In their many applications, Lie groupoids often come equipped with additional
geometric data (e.g. symplectic forms, Poisson structures, complex structures,
distributions etc) which are compatible with the groupoid structure, known
as "multiplicative". This talk will discuss a general result describing
multiplicative geometric structures at the infinitesimal level (i.e., in
terms of Lie algebroids),
with focus on examples.
Marius Crainic (Utrecht University)
Marco Gualtieri (Toronto)
Irina Kogan (North Carolina State University and IMA)
Invariant Variational Calculus
Many interesting variational problems arising in geometry and physics admit
a group of symmetries. As first recognized by S. Lie, these problems can
be rewritten in terms of group-invariant objects: differential invariants,
invariant differential forms, and invariant differential operators. It is
desirable from both computational and theoretical points of view to perform
variational calculus in terms of invariant objects. In this talk, we will
discuss invariant variational bicomplex, invariant Euler-Lagrange operators
and invariant Noether correspondence.
Boris Kruglikov (Tromso)
Kirill Mackenzie (Sheffield)
Multiple differentiation processes in differential geometry
Taking the Lie algebroid of a Lie groupoid encompasses many of the basic
differentation processes in differential geometry.
In this talk we give an overview of the second--order version of this process,
in which double Lie groupoids give rise, by two differentiation processes,
to double Lie algebroids. These encompass iterated tangent bundles and their
duals, and the classical canonical diffeomorphisms between them extend to
double Lie algebroids and their duals.
The cotangent bundle of a Poisson Lie group yields by differentiation the
Drinfel'd double of its tangent Lie bialgebra, and may be integrated to
a (symplectic) double Lie groupoid. Thus Poisson Lie groups are the intermediate
stage in the process of taking the double Lie algebroids of symplectic double
Lie groupoids. We describe this, and its extension to Poisson groupoids
and Lie bialgebroids.
In conclusion we outline the supergeometric formulation of double Lie algebroids
due to Th. Voronov, which extends naturally to $n$-fold Lie algebroids.
References:
1) K. C. H. Mackenzie.Ehresmann doubles and Drinfel'd doubles for Lie algebroids
and Lie bialgebroids.
J. Reine Angew. Math., 658:193--245, 2011.
2) K.C.H. Mackenzie. Double Lie algebroids and second-order geometry. II.
Adv. Math., 154(1):46--75, 2000.
3) Th.Th. Voronov. Q-Manifolds and Mackenzie Theory. Comm. Math.
Phys, 315(2):279--310, 2012.
Eckhard Meinrenken (University of Toronto)
On the Van Est homomorphism for Lie groupoids
Let G be a Lie groupoid, and A the corresponding Lie algebroid. In their
1991 article, Weinstein-Xu described a Van Est map from the complex of smooth
groupoid cochains on G to the Chevalley-Eilenberg Lie algebroid complex
of A. It is an algebra morphism on the normalized cochains. In this talk,
we will give an explanation of this Van Est map in terms of the fundamental
lemma of homological pertubation theory. (Joint work with David Li-Bland.)
Eva Miranda (Universitat Politècnica de Catalunya)
Symmetries of b-manifolds and their generalizations
The aim of this talk is to show some examples of simple Poisson manifolds
which have some common features with symplectic manifolds (including the
study of group actions). I will start presenting a Delzant theorem for toric
b-symplectic manifolds (joint work with Victor Guillemin, Ana Rita Pires
and Geoffrey Scott).
Time permitting, I will report on an ongoing project with Geoffrey Scott
on generalizing the notion of b-symplectic manifold. This notion includes
other Poisson manifolds which share good properties with b-symplectic manifolds
and seem to have less topological constraints.
Ercument Ortacgil (Bogazici University)
Higher order characteristic classes
Using the theory of prehomogeneous geometries that we defined earlier,
we outline the construction of the higher order Pontryagin classes as obstructions
to local homogeneity. The problem whether these invariants can be nontrivial
remains to be studied.
Juha Pohjanpelto (Oregon State and Aalto Universities)
Group Actions and Cohomology in the Calculus of Variations
I will discuss various new techniques based in part on the moving frames
method and Lie algebra cohomology for analyzing the cohomology of a variational
bicomplex and the associated edge, or Euler-Lagrange, complex invariant
under a continuous pseudo-group action. In particular, these methods in
many situations reduce the computation of the local cohomology of the two
complexes to an algebraic problem.
I will illustrate the new techniques with examples pertaining to $G$-structures
and integrable systems.
Colleen Robles, IAS/TAMU
Schubert varieties as variations of Hodge structure.
Variations of Hodge structure (VHS) are constrained by a system of differential
equations known as the infinitesimal period relation (IPR), or Griffiths
transversality. The IPR is a homogeneous system defined on a flag manifold
X = G/P. I will characterize the Schubert varieties that arise as variations
of Hodge structure (VHS). I will also discuss the central role that these
Schubert VHS play in our study of arbitrary VHS: infinitesimally their orbits
under the isotropy action `span' the space of all VHS. This yields a complete
description of the infinitesimal VHS. As a corollary we obtain sharp bounds
on the maximal dimension of a VHS, answering a longstanding question.
Maria Amelia Salazar,CRM-Barcelona
Abraham Smith (Fordham)
Ivan Struchiner (University of São Paulo)
Francis Valiquette (Dalhousie University)
Ori Yudilevich (Utrecht University)
Nguyen Tien Zung, University of Toulouse
Deformation coholomogy of singular foliations
The aim of this talk is to introduce the deformation cohomology for singular
foliations, to show how it's related to stability and deformation problems
of foliations, and how to compute it in various cases.
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