Weighted algebroids: theory and outlook for applications

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.

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.

Metody Geometryczne Fizyki

'Banana skins are slippery', say Japanese physicists

Banana Kiyoshi Mabuchi, Kensei Tanaka, Daichi Uchijima and Rina Sakai have won the 2014 Ig Nobel Prize in physics for their paper on measuring the amount of friction between a shoe and a banana skin, and between a banana skin and the floor, when a person steps on a banana skin that’s on the floor [1].

Bananas are my second favourite fruit, just behind Polish apples, so it is nice to see them feature in world-class physics research. More than that, slipping on a banana skip is the pinnacle of slapsick comedy and deserves more scientific attention.

[1] Kiyoshi Mabuchi, Kensei Tanaka, Daichi Uchijima and Rina Sakai, Frictional Coefficient under Banana Skin, Tribology Online 7, no. 3, 2012, pp. 147-151. (Available online here)

The 2014 Ig Nobel Prize Winners

Nuclear theory research in the UK to be exanded

Flag A new nuclear theory group is going to be set up at the University of York. The Science and Technology Facilities Council (STFC) will make a special funding award to set up the group and will provide funding to appoint a nuclear physics theory chair and PhD studentships. Furthermore, the university of York will fund a nuclear physics theory lectureship.

The need to expand the UK’s capability in theoretical nuclear physics was part of the Institute of Physics review in October 2012. For sure, although the UK has some good researchers in this field, the numbers of people in theoretical nuclear physics is small. One number that has been suggestion is that there are about seven permanent researchers in the UK working on theoretical nuclear physics.

The establishment of a new group must be welcome news for the UK nuclear physics community.

Gap in nuclear physics research identified by IOP is to be filled.

Thyroxine levels while pregnant and offspring's mathematics capability

Flag A study presented at the European Society for Paediatric Endocrinology annual meeting in Dublin suggests that the thyroid function of pregnant women is associated with poorer mathematics skills in their offspring at primary school level.

Low levels of thyroxine are known to effect the mental development of the baby. The result of Dutch researchers’ study are that children whose mothers had low levels of thyroxine during pregnancy were twice as likely to have below average mathematics scores. Interestingly, language skills were not effected in the same way and there was no difference at the age 5.

Maternal Hypothyroxinemia in Early Pregnancy is Associated with Poorer Arithmetic Performance in a School Test in Offspring at Age 5 Years, ESPE Abstracts (2014) P-D-1-2-253.

The future of Scottish research?

Flag I have not yet really thought about this, but Scottish independence must bring with it the question of the role of the UK research councils in Scotland. I imagine that Scotland would have to set up its own research councils independently of the UK councils.

This then brings in the question of the future of science in Scotland, especially in the short to mid term during a transition period. It would then be a question of if the Scottish government and the people see science as a worthwhile investment. Coupled with that must be the long term stability of science funding.

Nearly everybody that has spoken to me is very worried if there’s a yes vote.

Prof Sir Paul Nurse, President, Royal Society talking to the BBC

There are some fantastic people in Scotland working in mathematical physics and related subjects, for example there is the Edinburgh Mathematical Physics Group, which part of the Maxwell Institute for Mathematical Science. It would be a shame if the already tough job of getting funding becomes harder for mathematicians and physicists working in Scotland.

Worrying times for researchers based in Scotland I would say.

Scottish independence: Future of science contested, BBC News.

Linearisation & linear duals of graded bundles and weighted algebroids

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…