Monthly Archives: March 2015

Post Doc Poland, Research work

Mechanics on graded bundles

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My joint paper with K. Grabowska and J. Grabowski entitled “Higher order mechanics on graded bundles” has now been accepted for publication in Journal of Physics A: Mathematical and Theoretical. The arXiv version is arXiv:1412.2719 [math-ph].

I am very happy about this as it is my first joint paper to be published. The paper presents some novel and interesting ideas on how to geometrically formulate higher order mechanics, hopefully our expected applications will be realised.

One interesting possible application, as pointed out by one of the referees, is computational anatomy; this is the quantitative analysis of variability of biological shape. There has been some applications of higher derivative mechanics via optimal control theory to this discipline [1].

We were not thinking of such applications in the biomedical sciences when writing this paper. For me, the main motivation for higher order mechanics is as a toy model for higher order field theories and these arise as effective field theories in various contexts. It is amazing that these ideas may find some use in ‘more down to Earth’ applications. However, we will have to wait and see just how the applications pan out.

You can read more about the preprint in an earlier blog entry.

References
[1] F. Gay-Balmaz, D. Holm, D.M. Meier, T.S. Ratiu & F. Vialard, Invariant higher-order variational problems, Comm. Math. Phys. 309(2), (2012), 413-458.

General Mathematics

du Sautoy asks "can anyone be a maths genius?"

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Prof. du Sautoy asks this very question.

How many times have you heard someone say ”I can’t do maths”? Chances are you’ve said it yourself.

du Sautoy talking to the BBC

In all honesty I find myself thinking the above at least twice a day.

Genes or hard work
I am not an expert in how genes play a role in our intelligence, but for sure they do. That said, no-one is born an expert in mathematics and it takes a lot of hard work. Like everything in life, becoming proficient in mathematics to the level you set yourself is about perseverance and the willingness to struggle with things until you have mastered them.

Link
Can anyone be a maths genius? BBC iWonder

Physics

The original review of general relativity

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It has now been 99 years, to the day (20/03/2015) since Einstein published his original summary of general relativity [1].

Before that he had published some incomplete works that have the wrong field equation, but the key ideas were in place by 1914. The core idea is that space-time is dynamical and interacts with the matter and energy.

It is hard to believe that this theory of gravity has stood the test of time so well. We know for various reasons that general relativity cannot be the complete picture, but nature just refuses to give us hints on what could be the more complete theory.

Reference
[1] A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 354 (7), 1916, 769-822.

General Mathematics, Post Doc Poland

Two quotes on the philosophy of mathematics

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I gave a talk the other day based on our recent work on graded bundles in the category of Lie groupoids. Anyway, as part of the motivation I drew the audiences attention to two quotes…

“Mathematics is written for mathematicians.” Copernicus

“For the things of this world cannot be made known without a knowledge of mathematics.” Roger Bacon

They show the two different sides of mathematics; mathematics motivated by mathematics and mathematics motivated by applications. I think one should to some extent sit in the middle here, but ultimately it is nice when mathematics has something to to with the real world, even if that connection is somewhat loose.

My real motivation for these two specific quotes was that Copernicus was Polish and Bacon English!