I will be giving a talk at the “Seminarium Geometryczne” here in Warsaw on the first of October. The talk will be based my the recent work (arXiv:1409.0439 [math-ph]) with J. Grabowski and K. Grabowska.

Abstract
In this talk I will outline the theory of the recently discovered weighted Lie algebroids, which should be considered as a higher version of a Lie algebroid. We will then suggest how such structures can be employed in higher order Lagrangian mechanics. This is joint work with K. Grabowska and J. Grabowski.

Anyone who is interested is welcome to come along.

Link
Metody Geometryczne Fizyki

I have now placed a preprint on the arXiv entitled “Linear duals of graded bundles and higher analogues of (Lie) algebroids” (arXiv:1409.0439 [math-ph],) which is joint work with J. Grabowski and K. Grabowska.

In this preprint (which we will shortly submit for publication) we develop some technology based on n-tuple graded bundles as first studied by Grabowski & Rotkiewicz, to define the notion of a linear dual of a graded bundle. As graded bundles are not simply vector bundles, they are polynomial bundles, the notion of a dual is not immediately obvious. We propose that the linear dual of a graded bundle be a particular reduction of the cotangent bundle of the said graded bundle. Related to the linear dual is the notion of the linearisation functor which takes a graded bundle and produces a double graded bundle for which the two side bundles are vector bundles. The linearisation can also be understood in terms of a particular reduction of the tangent bundle of the graded bundle.

From there we define the notion of a weighted skew/Lie algebroid, which is loosely a skew/Lie algebroid carrying some extra gradings. Interestingly, these objects are closely related to higher Lie algebroids as defined by Voronov in terms of a weight-one homological vector field on a non–negatively graded supermanifold and the \(\mathcal{VB}\)-algebroids as studied by Mackenzie, Gracia-Saz & Mehta and most recently by Brahic, Cabrera & Ortiz.

There are plenty of canonical examples of weighted algebroids including tangent bundles of graded bundles, the linearisation of higher order tangent bundles and in particular the reduction of higher order tangent bundles on Lie groupoids, again via linearisation.

For all the details and proper references consult the preprint.

This work is purely theoretical mathematics, though we are now looking towards applications in geometric mechanics. So watch this space…

At the end of this month I will be attending a summer school “Introduction to the geometry of jet spaces and nonlinear differential equations” in Wisła, Beskid Mountains, Poland. The school is being organised by the Baltic Institute of Mathematics.

The summer school will consist of lectures by Joseph Krasil’shchik and Alexander Verbovetsky​, both of whom are world leading experts in the theory and application of jets to differential equations.

The period of the summer school is the 29th of June till the 6th of July.

Link
Introduction to the geometry of jet spaces and nonlinear differential equations

My work on curves and higher tangent bundles on supermanifolds has now been published as “On curves and jets of curves on supermanifolds“, Archivum mathematicum, Volume 50 (2014), No. 2.

Abstract
In this paper we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.
——————————————————

The main idea was to try to follow the classical definitions of a curve, jets of curves at a point and the geometric or kinematic definition of higher tangent bundles. One of the complications in the superworld is that supermanifolds are not just set theoretical objects. To overcome this the more categorical set-up of the “functor of points” and “internal Homs” is needed.

I posted a little about this before here.

My work on Odd Jacobi structures on supermanifolds and how to construct non skewsymmetric Poisson-like brackets from them has now been published in the Journal of Mathematics.

Odd Jacobi Manifold and Loday-Poisson brackets, Journal of Mathematics Volume 2014 (2014), Article ID 630749, 10 pages

I will be giving a talk at the Geometry Seminar of the Institute of Mathematics at Gdansk University on the 19th March.

Title: On the higher order tangent bundles of a supermanifold.

Abstract: In this talk I will introduce the notion of a supermanifold as a “manifold” with both commuting and anticommuting coordinates. After that I will discuss the notion of a curve on a supermanifold and how to construct their jets, this will lead to a kinematic definition of the higher order tangent bundles of a supermanifold. However, to formulate this properly we are lead to more abstract ideas that have their roots in algebraic geometry and in particular notion of internal Homs objects. I will not assume the audience to be experts in the theory of supermanifolds nor jets, only that they have some basic knowledge of standard differential geometry.

Link
Geometry Seminar Gdansk

I will be giving a talk at the Geometry and Differential Equations Seminar at IMPAN (Warsaw) on Wednesday 26th February 2014. The title is “A first look at N-manifolds”.

Abstract
In this talk I will introduce the concept of an N-manifold as refinement of the notion of a supermanifold in which the structure sheaf carries an additional grading, called weight, that takes values in the natural numbers. I will provide several motivating examples which largely come for the theory of jets, before discussing some generalities.

Link
Geometry and Differential Equations Seminar

I placed a preprint on the arXiv entitled “On curves and jets of curves and on supermanifolds” [1]. The abstract can be found below

Abstract
In this note we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.

Quick overview
The k-th order tangent bundle on a classical manifold can be defined as equivalence classes of curves at the points of the manifold that agree up to velocity, acceleration, rate of change of acceleration on so on up to k-th order. The tangent bundle which is much more widely known, is the “1st order tangent bundle”. These higher order tangent bundles have found applications in higher derivative Lagrangian mechanics for example and appear in the work of Grabowski & Rotkiewicz [2].

The main question posed and answered in this preprint is “can we define the k-th order tangent bundles of a supermanifold is such a kinematic way?”

The problems with simply generalising the classical notions directly are two fold:

1. Supermanifolds do not consist solely of topological point, thus point-wise constructions need some careful handling.
2. The naive definition of a curve as a morphism of supermanifolds \(\mathbb{R} \rightarrow M \) totally misses the odd dimensions of the supermanifold \(M\).

To resolve these issues we look at a “superised” version of curves defined via the internal Homs, a notion from category theory. In sort we consider curves that are paramaterised by all supermanifolds, however all the constructions should be functorial in this parametrisation. This allows us to define what I call S-curves that can “feel” both the even and odd dimensions of a supermanifold. Moreover, this allows us to think in terms of S-points and parallel the classical constructions of jets of curves rather closely.

This allow us to define the k-th order tangent bundle of a supermanifold at first as a generalised supermanifold (a functor from the opposite category of supermanifolds to sets ) and then we show that this is representable, that is a genuine supermanifold. Moreover, locally everything looks the same as the classical higher order tangent bundles.

References
[1] Andrew James Bruce, On curves and jets of curves and on supermanifolds, arXiv:1401.5267 [math-ph].
[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. geom. Phys. 62 (2012), 21-36.

On the 9th of January 2014 I will be giving a talk at the Algebra and Geometry of Modern Physics seminar here in Warsaw.

Title “What the functor is a superfield? ”

Abstract Physicists are usually quite happy to formally manipulate the mathematical objects that they encounter without really understanding the structures they are dealing with. It is then the job of the mathematician to try to make sense of the physicists manipulations and give proper meaning to the structures. (Confusing physicists is not the job of mathematicians, however mathematicians are good at it!) In this talk we will uncover the structure of Grassmann odd fields as used in physics. For example such fields appear in quasi-classical theories of fermions and in the BV-BRST quantisation of gauge theories. To understand the structures here we need to jump into the theory of supermanifolds. However we find that supermanifolds are not quite enough! We need to deploy some tools from category theory and end up thinking in terms of functors! :-O

I will try to post my notes here at some point after the talk…