- Magdalini Flari (Sheffield). Wednesday November 15th, 2pm. Venue and title/abstract to follow.
- Honglei Lang (MPIM Bonn)
Wednesday July 5th, 3.00pm, J 11.

*Double principal bundles*

(Note unusual day)We define double principal bundles (DPBs), for which the frame bundle of a double vector bundle, double Lie groups and double homogeneous spaces are basic examples.

It is shown that a double vector bundle can be realized as the associated bundle of its frame bundle. Also dual structures, gauge transformations and connections in DPBs are investigated.

The visit of Dr Honglei Lang is supported by the Sheffield MSRC.

- Matthew Peddie (Manchester)
Wednesday May 17th, 4pm, LT 5

*A super approach to Drinfeld doubles*

(Note unusual day, time and venue)Drinfeld's double construction for a Lie bialgebra produces a unique Lie bialgebra suitable for quantisation. With the introduction of Lie bialgebroids as linearisations of Poisson-Lie groupoids, followed the same question as to whether a double can be constructed. This proved to be not so straightforward, and indeed, can be considered to be only partially answered.

We will review these double constructions for Lie bialgebras and Lie bialgebroids using the language of supermathematics, and will discuss some of the problems encountered for the bialgebroid case. We will then define the Drinfeld double of a homotopy Lie bialgebra, or an $L_\infty$-bialgebra, and find a necessary condition for the existence.

- Sam Morgan (Sheffield)
Tuesday March 21st, 11am, J 11

*Double Lie groupoids and their double Lie algebroids, III*

(Note unusual day and time)In the third talk we will complete the construction of a double Lie algebroid of an LA-groupoid, and look at a specific example of an LA-groupoid arising naturally from a Poisson Lie group. We will finish by discussing the general notion of a double Lie algebroid of a double Lie groupoid.

- Sam Morgan (Sheffield)
Tuesday March 7th, 11am, J 11

*Double Lie groupoids and their double Lie algebroids, II*

(Note unusual day and time)In the second talk, we will briefly discuss some examples of Lie algebroids arising from Lie groupoids; this should tie in with the description of the Lie functor, given in the first seminar. We shall then continue the construction of a double Lie algebroid of an LA-groupoid.

We will complete the construction of a double Lie algebroid of an LA-groupoid, and look at a specific example of an LA-groupoid arising naturally from a Poisson Lie group. We will finish by discussing the general notion of a double Lie algebroid of a double Lie groupoid.

- Sam Morgan (Sheffield)
Tuesday February 28th, 11am, J 11 (Postponed from Tuesday 21st)

*Double Lie groupoids and their double Lie algebroids, I*

(Note unusual day and time)The series of two, possibly three, talks will consist of a precise formulation of the double Lie algebroid of a double Lie groupoid. We will also discuss some of the examples arising in Poisson geometry.

In the first talk we will consider the construction of the double Lie algebroid of an LA-groupoid. This will be a stepping stone in the general construction for a double Lie groupoid.

Knowledge of the standard formation of the Lie algebroid of a Lie groupoid will not be assumed, and the notions of a Lie groupoid and a Lie algebroid will be recalled.

- Iakovos Androulidakis (Athens)
Thursday January 12th, 2pm, LT 11,

*Almost regular Poisson structures and their holonomy groupoids*

(Note unusual day, time and venue)We introduce a big class of Poisson manifolds, the "almost regular" ones. Roughly, these are the Poisson manifolds whose symplectic foliation is regular in a dense open subset. All regular Poisson manifolds are included in this class, as well as all the log-symplectic manifolds and certain Heisenberg-Poisson manifolds. We are looking for desingularizations of such structures. A natural candidate is the holonomy groupoid of the symplectic foliation, which is always smooth in this category. We show that, moreover, this is a regular Poisson groupoid. In the case of log-symplectic manifolds it coincides with the symplectic groupoid constructed by Gualtieri and Li. And for the Heisenberg-Poisson manifolds it is Connes' tangent groupoid. All this hints that various blow-up constructions in Poisson geometry might be replaced by the systematic construction of the holonomy groupoid of a singular foliation.

The visit of Professor Androulidakis is supported by the Sheffield MSRC.

**When is a closed real-valued 2-form the curvature of a connection in a circle bundle (equivalently, a complex line bundle) ?**-
**In what sense are coadjoint orbits the universal examples of Hamiltonian symplectic manifolds ?**

**
The course will start on Monday 31 October and will last about five weeks,
with two lectures per week. The planned times are Mondays at 2pm and Thursdays at 10am.
**

This will be designed to be accessible to people who have not come to Ieke's lectures, but who are familiar with the basics of manifolds and Lie groups.

Roughly speaking the question is: when is a 2-form on a manifold, which takes values in a `bundle of Lie algebras', the curvature of a connection in a principal bundle? The answer is recognizably similar to the answer in the case of complex line bundles, but the techniques are substantially different.

I will cover principal bundles and their connections at the start.

(If you happen to be familiar with the classification of extensions of discrete groups, or Lie algebras, by cohomology, the differences are roughly the same as the differences between when the kernel is abelian and when it is not.)

Note unusual time and location

The proofs won't be complete in every detail, but should give an idea of what is involved.

I intend to be faster than last Friday but slower than Monday.

- All interested are welcome. The Hicks Building is 121 on the university map.