 
Autumn 2017  Summer 2018
Since autumn 2017, the research group has been running weekly research seminars with the goal to discuss research that is being conducted within the group.
Past events
Spring 2017
Thursday, 18 May 2017 
14:00 to 16:00 
Room MNO 6A (06.15.440) 
Pablo Guzman

University of Luxembourg 

Thursday, 4 May 2017 
14:00 to 16:00 
Room MNO 6A (06.15.440) 
Thursday, 11 May 2017 
14:00 to 16:00 
Room MNO 6A (06.15.440) 
Vladimir Salnikov

University of Luxembourg 
Discussing the
Stasheff's secret paper
These two weeks we will be discussing the paper by Jim Stasheff 
"The (secret?) homological algebra of the BV approach"
(arXiv:hepth/9712157v1).
The first day: some introduction to BRST and BV.
The second day: actually the results of the paper, eventually tranlated to
"modern" language.
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Thursday, 16 Mar. 2017 
14:15 to 15:45 
Room MNO 6A (06.15.440) 
Thursday, 23 Mar. 2017 
14:15 to 15:45 
Room MNO 6A (06.15.440) 
Thursday, 30 Mar. 2017 
14:15 to 15:45 
Room MNO 6A (06.15.440) 
Thursday, 6 Apr. 2017 
14:15 to 15:45 
Room MNO 6A (06.15.440) 
Andrew
Bruce 
University of Luxembourg 
Mathematical aspects of supersymmetry
Lecture 1: A first look at supersymmetry I.
I will introduce some of the basic features of supersymmetry
concentrating on the geometry and how to construct quasiclassical actions. In
particular I will focus on toy mechanical models as these allow us to present
some of the generic features and ideas without the need to understand Clifford
algebras and spinors (which should be a subject for another seminar).
Lecture 2: A first look at supersymmetry II.
We continue in our adventure in superspace by looking at
supermanifolds, and in particular the use of the functor of points. This will
allow us to give a clear notion of a Lie super group. We will also look at using
generalised supermanifolds to make sense of superfields as they appear in the
physics literature. I will assume some knowledge of basic differential geometry
and an acquaintance with the notion of a sheaf.
Lecture 3: A first look at supersymmetry III.
Physical fermions, such as the electron, are spin 1/2 particles. This means that
they are describes by spinors. In this lecture I will present the bare minimum
to get an understanding of spinors. I will cover the Lorentz Lie algebra,
Clifford algebras, the Dirac matrices and finally spinors.
Lecture 4: A first look at supersymmetry IV.
In the last of this series, I will bring everything together and present a
real version of N=1 superMinkowski spacetime, which we can view as an
extension of Minkowski spacetime by appending extra anticommuting coordinates
that transform as Majorana spinors under Lorentz transformations. This version
of `superspace' is not used in particle theory, there Weyl spinors are
typically used. The use of Majorana spinors is more convenient for
supergravity, and as geometers we follow the lead of supergravity community.
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Thursday, ?? ??? 2017 
14:15 to 15:15 
Room B27 
Marius
Crainic

Utrecht University 

Fall 2016
Thursday, 6 Oct. 2016 
14:15 to 15:15 
Room B27 
Florian Schätz

University of Luxembourg 
The Eulerian idempotent revisited
The Eulerian idempotent is a canonical map from the free algebra on generators
x_1,...,x_n to the space of Lie words on x_1,..,x_n. Besides its importance in
Lie theory, it also plays a central role in the theory of linear ODEs, due to
its relation to the Magnus expansion (it is therefore also studied by numerical
analysts). I will report on joint work in progress with Ruggero Bandiera
(Sapienza  University of Rome), whose main point is to establish a (to the best
of our knowledge) new formula for the Eulerian idempotent. The derivation of
this formula relies on the notion of (and computations within) preLie algebras.
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Monday, 10 Oct. 2016 
14:15 to 15:15 
Room B24A 
Igor Khavkine

University of Rome 2 
Applications of compatibility complexes and their
cohomology in relativity and gauge theories
I will discuss the Killing operator ($K_{ab}[v] = \nabla_a v_b + \nabla_b v_a$)
as an overdetermined differential operator and its (formal) compatibility
complex. It has been recently observed that this compatibility complex and its
cohomology play an important role in General Relativity. In more general gauge
theories, an analogous role is played by the "gauge generator" operator and its
compatibility complex. An important open problem is to explicitly compute the
tensorial form of the compatibility complex on (pseudo)Riemannian spaces of
special interest. Surprisingly, despite its importance, the full compatibility
complex is known in only very few cases. I have recently reviewed one of these
cases, constant curvature spaces, where this complex is known as the Calabi
complex, in arXiv:1409.7212. I will also mention a connection with the problem
of intrinsic local characterization of isometry classes of (pseudo)Riemannian
geometries. The specific case of cosmological spacetime geometries was recently
attacked with G. Canepa (MSc, Pavia).
References:
Khavkine, I. Covariant phase space, constraints, gauge and the Peierls formula.
International Journal of Modern Physics A 29, 1430009 (2014)
http://dx.doi.org/10.1142/s0217751x14300099
http://arxiv.org/abs/1402.1282
Slides
Khavkine, I. The Calabi complex and killing sheaf cohomology. Journal of
Geometry and Physics (2016)
http://dx.doi.org/10.1016/j.geomphys.2016.06.009
http://arxiv.org/abs/1409.7212
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Thursday, 20 Oct. 2016 
14:15 to 15:15 
Room B27 
Andrew
Bruce 
University of Luxembourg 
Curves and Mechanics on Supermanifolds
`Supermechanics' understood as Grassmann algebra valued mechanics has been
around for decades. However, a careful geometric understanding of mechanics on a
supermanifold is generally lacking from the literature. In particular the notion
of phase dynamics and solutions thereof require some thought. We will show how
the geometric approach of Tulczyjew generalises to the case of supermanifolds.
In order to do this we first need to define curves on supermanifolds, a task
that is not a simple as one might first think and requires some tools from
category theory and algebraic geometry!
I will only assume rudimentary familiarity with the notion of a supermanifold
and Lagrangian mechanics throughout the talk.
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Thursday, 27 Oct. 2016 
14:15 to 15:15 
Room B27 
Vladimir Salnikov

University of Luxembourg 
What I (want to) understand about supersymmetrization
A year ago in my talk about graded geometry in gauge theories, I mentioned
some work in progress related to possible extensions of the formalism to the
supersymmetric setting  I would like to give some details about this work.
From the mathematical perspective, I will introduce the notions of Qbundles,
morphisms and homotopies for multigraded manifolds, as well as present
the "supersymmetric" generalization of the
AleksandrovKontsevichSchwarzZaboronsky procedure. As for physics, I will
explain what I mean
by supersymmetric, how the above mentioned contructions can be useful, and what
is complicated in the problem.
For those who were not here last year I will certainly recall the "standard"
formalism.
The talk is mostly based on the preprint:
arXiv:1608.07457
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Week 7  11 Nov. 2016 
Erasmus+ 
Tuesday, 8 Nov. 2016 
14:00 to 15:30 
Room B27 
Thursday, 10 Nov. 2016 
14:00 to 15:30 
Room B27 
Thursday, 10 Nov. 2016 
15:30 to 17:00 
Room B27 
Visit to RMATH of

Emil Akhmedov

HSE/ITEP Moscow

Joint session with the Algebra, Geometry and
Quantization Seminar

On geometry and dynamics of fields in de Sitter and anti
de Sitter spacetimes
The lectures will be mostly addressed to graduate students. I am with
physics background,
hence, the presentation will be without any pretends for the real
mathematical rigor. However, I will do my best to keep my presentation
as rigorous as my education allows.
I will start with the describtion of the geometry of de Sitter and
anti de Sitter
spacetimes. My goal will be to present the way we can quickly (with minimal
efforts) see most of the properties of these spacetimes. Then I will
continue with the derivation
of the free massive scalar modes and their properties in these
spacetimes. I will end up my lectures
with the derivation of the Green functions in these spacetimes in two
different ways.
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Tuesday, 6 Dec. 2016 
16:00 to 17:00 
Room B02 
Joint session with the General Maths Seminar
and
Geometry
and Topology seminar

Catherine Meusburger

ErlangenNürnberg 
Generalised shear coordinates for (2+1)spacetimes
The diffeomorphism invariant phase space of (2+1)gravity is a
moduli space of maximal globally hyperbolic constant curvature
Lorentzian (2+1)spacetimes with the curvature given by the cosmological
constant.
We consider spacetimes with cusped Cauchy surfaces S and parametrise
these moduli spaces in terms of shear coordinates and measured geodesic
laminations on S. This leads to a simple description of their
symplectic structure in terms of the cotangent bundle of Teichmueller
space and can be viewed as analytic continuation of shear coordinates.
We describe the mapping class group action on these moduli
spaces and show that it is by symplectomorphisms. This leads
to three different mapping class group actions on the cotangent bundle
of Teichmueller space, which involve the cosmological constant as a
parameter and are generated by Hamiltonians.
This is joint work with Carlos Scarinci, arXiv:1402.2575,
J. Differential Geometry 103 (2016) 425474
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Spring 2016
Thursday, 11 Feb. 2016 
Research meeting 
N. Poncin, J. Grabowski, S. Kwok, V. Salnikov, K. Grabowska.

Tuesday, 23 Feb. 2016 
14:15 to 15:45 
Room G002 
Thursday, 25 Feb. 2016 
14:15 to 15:45 
Room B27 
Tuesday, 1 March 2016 
11:30 to 13:00 
Room B23 
Tuesday, 22 March 2016 
15:30 to 17:00 
Room A16 
Thursday, 24 March 2016 
14:15 to 15:45 
Room B27 
Thursday, 14 April 2016 
14:00 to 17:00 
Room B27 
Tuesday, 26 April 2016 
11:30 to 13:00 
Room B23 
To be continued... 


Alessandro Zampini 
University of Luxembourg 
Minicourse  On the physical background of BRST/BV
During the first three talks I shall mainly focus on the definition of path
integral quantization in mechanics and field theory.
Talk 1. Classical mechanics: Lagrangian and Hamiltonian formalism.
Basics of quantum mechanics.
Talk 2. Examples of the free particle and of the harmonic
oscillator.
Idea of Feynman integral.
Talk 3. Feynman Green's function, quantization.
Talk 4.
Talk 5. Only Alessandro knows where this goes
Talk 6. Maybe... (It was about quantizing Maxwell)
Talk 7. Actually he does and he does not  that's all quantum now.
(And this one was about principal bundles)
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Wednesday, 9 March 2016 
14:30 to 15:30 
Room B14 
Thursday, 10 March 2016 
14:30 to 15:30 
Room B27 
Friday, 11 March 2016 
11:45 to 13:00 
Room B17 
Theodore Voronov 
Manchester 
Minicourse  Microformal geometry
In search of constructions that may give Linfinity morphisms for algebras of functions on homotopy Poisson (or odd Poisson) manifolds,
I discovered a generalization of familiar pullbacks with respect to smooth maps. Surprisingly, these new pullbacks are *nonlinear*,
actually formal, mappings of the vector spaces of functions given by formal nonlinear differential operators. As nonlinear,
they cannot be algebra homomorphisms in contrast with the ordinary pullbacks; however, their derivatives at each point are algebra homomorphisms.
Underlying these "nonlinear pullbacks", there is a formal category (actually, there are two parallel versions of such a category giving pullbacks of even and odd functions),
which is a formal neighborhood of the semidirect product of the usual category of (super)manifolds with algebras of smooth functions.
Morphisms in this formal category  "microformal" or "thick" morphisms of (super)manifolds, as I call them, 
are formal canonical correspondences between the cotangent bundles and are described by formal generating functions.
(A close nonformal category based on germs of symplectic manifolds was introduced for different purposes by CattaneoDherinWeinstein,
but their theory does not have pullbacks.) For homotopy Poisson supermanifolds, nonlinear pullbacks of functions induced by Poisson thick morphisms
(which are easy to define) are indeed Linfinity morphisms for the corresponding algebra structure. Another application is related with vector bundles.
By generalizing ordinary maps to thick morphisms, it is possible to have *adjoints for nonlinear operators* on vector bundles.
This has applications to Linfinity (bi)algebroids. The most recent development is related with a *quantum version* of this theory.
Namely, it is possible to define "quantum thick morphisms", which are particular oscillatory integral operators, so that "classical"
thick morphisms arise as their limits as Planck's h goes to zero. This is work in progress.
Plan of lectures (tentative):
1. Thick morphisms of (super)manifolds and nonlinear pullbacks of functions;
2. Applications to homotopy Poisson structures, vector bundles and Linfinity (bi)algebroids;
3. Quantum thick morphisms.
References:
arXiv:1409.6475,
arXiv:1411.6720,
arXiv:1506.02417,
arXiv:1512.04163.
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Thursday, 17 March 2016 
14:15 to 15:45 
Room B27 
Pavol Ševera 
Geneva 
Integration of differential graded manifolds
I will describe a procedure that integrates differential graded manifolds
(also known as higher Lie algebroids, or NQ manifolds) to higher Lie groupoids.
The main technical result is the fact that solutions of a generalized MaurerCartan
(MC) equation form an infinitedimensional manifold. When we consider those solutions
of the generalized MC equation on simplices which also satisfy a gauge condition
(following an idea of Ezra Getzler), we get a finitedimensional Kan simplicial manifold,
i.e. a Lie ngroupoid. I will also explain to which extent this procedure is functorial,
and how symplectic forms on dg manifolds get integrated to A_infinity functors.
The talk is based on a joint work with Michal Siran.
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Thursday, 7 April 2016 
14:15 to 15:45 
Room B27 
Olga Chekeres 
Geneva 
Wilson surface observables from equivariant cohomology
Our construction of the Wilson surface observable for gauge theories is based on path integral descriptions of Wilson lines.
The first presentation of a Wilson line (due to AlekseevFaddeevShatashvili) uses path integrals over coadjoint orbits.
The second one (due to DiakonovPetrov) replaces a 1dimensional path integral with a 2dimensional topological sigmamodel.
We show that this sigmamodel is defined by the equivariant extension of the Kirillov symplectic form on the coadjoint orbit.
This allows to define the corresponding observable on arbitrary 2dimensional surfaces, including closed surfaces.
We give a new path integral presentation of Wilson lines in terms of Poisson sigmamodels,
and use it to test our observable in the framework of the 2dimensional YangMills theory.
On a closed surface, the Wilson surface observable turns out to be nontrivial for G nonsimply connected
(and trivial for G simply connected), in particular we study in detail the cases G=U(1) and G=SO(3).
The talk is based on a joint work with A. Alekseev and P. Mnev,
arXiv:1507.06343,
Original source (JHEP).
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Thursday, 9 June 2016 
14:15 to 15:45 
Room B17 
Andrew
Bruce 
Warsaw 
Kirillov and Jacobi structures up to homotopy.
In this talk I will present the notion of a homotopy
Kirillov structure on the sections of an even line bundle over a
supermanifold. When the line bundle is trivial we have a homotopy
Jacobi structure. These structures are understood furnishing the
module of sections with an L_\inftyalgebra; which is a 'higher' or
'homotopy' version of a Lie algebra.
The listener is only expected to have some rudimentary familiarity
with Poisson and hopefully Jacobi manifolds. No prior knowledge of
L_\inftyalgebras will be assumed and our treatment of
supermanifolds will be elementary.
The talk will be based on the preprint 'Jacobi structures up to
homotopy' (arXiv:1507.00454 [math.DG]) which is joint work with
Alfonso Tortorella, Universita degli Studi di Firenze, Italy.
Slides
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Thursday, 28 April 2016 
14:15 to 15:45 PM 
Room B27 
Name 
Affiliation 

Fall 2015
Tuesday, 20 Oct. 2015 
2:00 to 3:00 PM 
Room B27 
Joint session with the Algebra, Geometry and Quantization Seminar

Vladimir Salnikov 
University of Luxembourg 
Graded geometry in gauge theories and beyond (part 1)
We study graded geometric constructions appearing naturally in the context of gauge
theories. We introduce the language of Qbundles convenient for description of symmetries
of sigma models. Inspired by a known relation of gauging with equivariant cohomology we
generalize the latter notion to the case of arbitrary Qmanifolds introducing thus the
concept of equivariant Qcohomology.
This notion turns out to be useful for analysis of such theories as the (twisted) Poisson
sigma model and the Dirac sigma model. We obtain these models by a gaugingtype
procedure of the action of a group related to Lie algebroids and nplectic manifolds.
We also show that the Dirac sigma model is universal in spacetime dimension 2.
On top of applications to gauge theories (time permitting) I will comment on a possible
definition of equivariant cohomology for Courant algebroids.
This is a joint work with Thomas Strobl, and in part with Alexei Kotov.
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Tuesday, 27 Oct. 2015 
2:15 to 3:15 PM 
Room B27 
Vladimir Salnikov 
University of Luxembourg 
Graded geometry in gauge theories and beyond (part 2: multiple gradings and SUSY)
In this second part of the talk I am going to address the question of generalizing the
results of the first part to "supersymmetric" theories. This is mostly work in progress
related to introduction of multiple gradings in the context of the Poisson sigma model
and the ChernSimons theory. The global goal is to approach a clean construction of
an appropriate multigraded analog of the AKSZ procedure. I will also provide some
(rather toy model) examples.
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Thursday, 5 Nov. 2015 
2:15 to 3:15 PM 
Room B27 
Florian Schätz 
University of Luxembourg 
Variations on the BRST construction
The classical BRST construction provides a homological framework for symplectic reduction.
After recalling how this works in the simplest setting, I will outline a generalization  known as the BFV construction 
to arbitrary coisotropic submanifolds and its use in deformation theory. I will finish with some thoughts on Poisson submanifolds. abstract
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Tuesday, 10 Nov. 2015 
Time 4:00 to 5:00 PM 
Room B02 
Joint session with the General Mathematics Seminar

Rita Fioresi 
University of Bologna, Italy 

Tuesday, 24 Nov. 2015 
4:00 to 5:00 PM 
Room B02 
Joint session with the General Mathematics Seminar

Vladimir Dotsenko 
Trinity College, Dublin 
Noncommutative analogues of moduli spaces via brick manifolds
I shall talk about a remarkable series of algebraic
varieties that resemble the DeligneMumford compactifications of
moduli spaces of curves of genus zero with marked points. They admit
three equivalent descriptions: as "brick manifolds" recently defined
by Escobar, as toric varieties of Loday's realisations of
associahedra, and as De ConciniProcesi wonderful models of certain
subspace arrangements. The talk will introduce, on a level accessible
to a general mathematics audience, these three setups from scratch,
and outline remarkable properties behind those spaces. It is based on
a joint work with Sergey Shadrin and Bruno Vallette
http://arxiv.org/abs/1510.03261.
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Thursday, 3 Dec. 2015 
2:15 to 3:45 PM 
Salle des conseils 
Alessandro Valentino 
Max Planck Institute for Mathematics, Bonn 
Central extensions of mapping class groups from characteristic classes
I will discuss a functorial construction of extensions of mapping class groups of smooth manifolds which are induced by extensions of (higher)
diffeomorphism groups via the group stack of automorphisms of manifolds equipped with higher degree topological structures.
The problem of constructing such extensions arises naturally in the study of topological quantum field theories, in particular in 3d ChernSimons theory.
Based on joint work with Domenico Fiorenza and Urs Schreiber.
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