## Algebraic Topology, Geometry and Physics

### Past events

#### Spring 2013

11 Jul 2013
Mathieu Stiénon
Atiyah classes and homotopy algebras
The Atiyah class of a holomorphic vector bundle $E$ is the obstruction to the existence of a holomorphic connection on $E$. A theorem of Kapranov states that, for a complex manifold $X$, the Atiyah class of $T_X$ makes $T_X[-1]$ into a Lie algebra object in the derived category $D(X)$. Furthermore, Kapranov proved that, for Kaehler manifolds, this Lie algebra structure stems from an $L_\infty$ algebra structure on $\Omega^{0,*}[-1](T_X)$. In this talk, we will show how Kapranov's theorems can be extended to the more general setting of Lie pairs of algebroids so as to produce new homotopy algebras.
28 May 2013
Alessandro Zampini
Differential calculi on quantum groups
15 May 2013
Shuffle operads and their not yet discovered generalisations (II)
14 May 2013
Shuffle operads and their not yet discovered generalisations (I)
Shuffle operads can be used to approach questions about symmetric operads in a fairly efficient algorithmic way. The key idea behind the notion of a shuffle operad is to no longer allow the symmetric groups to act directly on operations, but only keep some of the indirect actions on results of compositions. I shall remind how that works, and then discuss two generalisations of symmetric operads for which it would be desirable to find "shuffle shadows", namely cyclic operads and wheeled operads. In the first case, I shall outline a solution, in second one, explain potential approaches to the problem.
07 May 2013
Alessandro Zampini
What is a quantum group? (II)
30 Apr 2013
Alessandro Zampini
What is a quantum group? (I)
I shall introduce the notion of quantum group and describe particularly the quantum deformation of the classical Lie group $SU(2)$.
From 12 Mar to 16 Apr Professor Pierre Schapira gave the minicourse “Introduction to D-modules on Ran spaces”. It consisted of 5 lectures. The schedule and the list of references can be downloaded here.
16 Apr 2013
Pierre Schapira
Ran spaces, 2-limits of categories, $\mathcal{D}$-modules on Ran spaces, a glance to the chiral product and to non-linear equations (II)
9 Apr 2013
Pierre Schapira
Ran spaces, 2-limits of categories, $\mathcal{D}$-modules on Ran spaces, a glance to the chiral product and to non-linear equations (I)
26 Mar 2013
Pierre Schapira
Internal operations $\mathcal{Hom}$ and $\otimes$ and external operations (inverse and direct images)
19 Mar 2013
Pierre Schapira
Characteristic variety, holonomic modules
12 Mar 2013
Pierre Schapira
Construction and main properties of the sheaf of rings $\mathcal{D}$ of differential operators, De Rham and Spencer complexes
01 Mar 2013
Jian Qiu
Lie theory for nilpotent $L_\infty$-algebras (IV)
26 Feb 2013
Jian Qiu
Lie theory for nilpotent $L_\infty$-algebras (III)
The simplicial set of MC elements serves as integration object. The main goal of the talk is to show that it is a Kan simplicial set, i.e. if the MC elements are defined on a horn of a simplex, one can always find an extension to the entire simplex. Furthermore, one can impose a gauge condition and show that to a certain extent the extension is unique, so that one actually gets a (weak) l-group(oid). The extension formula also provides a generalization of the CBH formula.
22 Feb 2013
Benoît Jubin
Lie theory for nilpotent $L_\infty$-algebras (II)
19 Feb 2013
Benoît Jubin
Lie theory for nilpotent $L_\infty$-algebras (I)
This is a presentation in two parts of Getzler's Lie theory for nilpotent $L_\infty$-algebras. I will give some motivation with the example of DGLAs and say in what sense we want to integrate them. Then I will review the notions of nerve of a category, of Kan complex and of higher groupoid. I will present Dupont's explicit proof of the simplicial De Rham theorem used in the sequel. I will finally introduce nilpotent $L_\infty$-algebras and their Maurer–Cartan simplicial sets, and show that they are Kan complexes. A specific subset of the Maurer–Cartan simplicial set can be considered as the integration of the given $L_\infty$-algebra, as will be explained next week.
12 Feb 2013
Jian Qiu
Derived symplectic geometry
The talk is a report on a mini course given during the Winter School 2013 in Les Diablerets. I start with some examples of intersection problems to motivate the idea of 'derivation' — see http://streamer.cit.utexas.edu/math-grasp/lurie.html. Then I sketch the notions of forms, vector fields, and Lagrangian structures in derived geometry, and provide examples.

#### Fall 2012

30 Nov 2012
Simon Covez
The local integration of Leibniz algebras
The goal of this talk is to present on an example how a Leibniz algebra integrates into a local Lie rack.
Nov 2012
Jumpei Nogami
Series of six lectures on Higher Categories, Higher Algebra, and Derived Algebraic Geometry.
30 Oct 2012
Benoît Jubin
Integrability of Lie algebroids (after Crainic and Fernandes), III
Integrability: smooth theory. Application to the integrability of Poisson manifolds.
26 Oct 2012
Benoît Jubin
Integrability of Lie algebroids (after Crainic and Fernandes), II
A few more examples: Atiyah sequence, Poisson manifolds. Connections on Lie algebroids. Representations and the general Lie algebroid. Integration of Lie algebroids: topological theory. The s-simply-connected Lie groupoid associated to a Lie groupoid. Uniqueness for the integrability problem. The “path space” (Duistermaat–Kolk) proof that any Lie algebra is the Lie algebra of a simply-connected Lie group. Notions of G-path, A-path, and A-homotopy. Construction of the Weinstein groupoid of a Lie algebroid. Monodromy groups of a Lie algebroid. Exponential map.
16 Oct 2012
Benoît Jubin
Integrability of Lie algebroids (after Crainic and Fernandes), I
Lie groupoids: definition and basic properties (s-fibers, isotropy groups, orbits, subgroupoids, morphisms). Examples (Lie group, pair groupoid, gauge groupoid, general linear groupoid, action groupoid, fundamental groupoid of a manifold and of a foliation). Representation of a Lie groupoid. Lie algebroids: definition and basic properties (isotropy Lie algebras, orbits, morphisms). Construction of the Lie algebroid of a Lie groupoid. Examples of this construction: Lie algebra, tangent bundle, action Lie algebroid.
09 Oct 2012
Jian Qiu
Model Categories II
This is a continuation of my previous seminar. I will explain how to formulate homotopy theory in a model category. This involves path and cylinder objects, as well as (co)fibrant replacements, and leads us naturally to derived functors. As an example, I work out, using the language of derived categories, the derived zero locus of a section in the purely algebraic setting.
02 Oct 2012
Tiffany Covolo
Higher trace and Berezinian of matrices over Clifford algebras
I will report on a recently published joint work with N. Poncin and V. Ovsienko. I will first introduce some notions of graded algebra (a generalization of superalgebra), then I will focus on the $(\mathbb{Z}_2)^n$ - graded version of the trace and of the Berezinian.
25 Sep 2012
Jian Qiu
Model categories
This talk is meant to be an introduction to model category theory. We start from the axioms of model categories, and illustrate them with two of the most prominent examples, topological spaces and chain complexes. Then we take a look at the formulation of homotopy in the model category and talk briefly about Quillen functors and other model categories.
18 Sep 2012
Benoît Jubin
Courant algebroids
I will show how Dirac structures and the Courant bracket on $TM\oplus T^*M$ appear as a common generalization of presymplectic and Poisson structures (after Courant, Weinstein). Then I will define general Courant algebroids and show how they arise as the double objects of Lie bialgebroids, in analogy to the Drinfeld double of a Lie bialgebra (after Liu, Weinstein, Xu). I will also discuss Severa's classification of exact Courant algebroids and the supergeometric interpretation of Courant algebroids (after Severa and Roytenberg).