## Geometry seminar at Gdańsk

March 5th, 2014 by ajb
 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.

## Welsh parents are urged to be positive about maths

March 3rd, 2014 by ajb
 A campaign is going to be launched urging parents to have a more positive attitude towards mathematics and learning mathematics. This comes after a poll found three in 10 parents in Wales questioned admitted being negative about maths in front of their children.

Maths results are the worst of any core subject in Wales behind English, Welsh and science.

It’s fair to say that maths is suffering from an image problem, and as today’s poll demonstrates, there is still work to do to change the view in some quarters which is that maths isn’t really important and that it doesn’t really matter what we say to children about it.

We understand the value of strong numeracy skills, for life and for employment.

Education Minister Huw Lewis

The “What you say counts” campaign will be launched by the minister at a supermarket in Cardiff Bay on Monday.

## Improper use of school computers

March 2nd, 2014 by ajb

I am shocked. After the course, the teacher immediately came to me and told me he had watched porn on a school computer.

René Portenier, director KV Zurich Business School.

But that is not the end of it…he forgot to unplug the cable to the projector!

## Classification of low dimensional Lie superalgebras

March 1st, 2014 by ajb
 Just for fun I was thinking about how to classify Lie superalgebras of dimension 1|1, that is consisting of one even element and one odd element. The situation is similar, but different to the classification of 2 dimensional Lie algebras. Just to note, everything here will be over the real numbers. Also, I am sure this is well-know.

Lie superalgebras
I will define a Lie superalgebra to be a Lie algebra with a $\mathbb{Z}_{2}$-grading. That is I have vector space for which I can assign some elements to be even and some elements to be odd. The Lie bracket is an even binary operation on the vector space $V = V_{0}\oplus V_{1}$

$[\bullet, \bullet]: V \rightarrow V$

that satisfies

Skewsymmetry: $[x,y] = -(-1)^{\tilde{x} \tilde{y}}[y,x]$

Parity: $\widetilde{[x,y]} = \widetilde{x} + \widetilde{y}$

Jacobi Identity : $[x,[y,z]] = [[x,y] ,z] +(-1)^{\tilde{x}\tilde{y}} [y,[x,z]]$

where $\tilde{x} = 0,1$ depending on if $x \in V_{0}$ or $V_{1}$. Things with parity zero are called even and things with parity one are called odd.

The 1|0 and 0|1 case
As a “warm-up” let us consider the case of just one even or one odd element.

Just from the closure of the Lie superalgebra we know that for one even element $y$ the bracket has to be

$[y,y]= ay$,

for some real number a. However, the skewsymmetry forces $[y,y]= -[y,y]$ and so a=0 is the only possibility. In short we have a trivial abelian Lie algebra with one even generator.

What about the case with one odd element $x$ say? Well the skewsymmetry does not help us here as $[x,x] = [x,x]$. But the closure and the parity of the Lie bracket forces

$[x,x]=0$.

To summarise, if we have one even or one odd element then the corresponding Lie superalgebras must be trivial, but for slightly different reasons.

2 dimensional Lie algebras
Just to recall the situation for 2 dimensional lie algebras we have two and only two such algebras up to isomorphism. They are denoted $L_{1}$ which consists of two (even) elements that mutually commute $[x,y] =0$, and $L_{2}$ for which we have the non-trivial bracket $[x,y]=y$.

I will leave it as an exercise for those interested to prove this is the case. As a hint, start from $[x,y] = ax +by$ and you can show that under a change of basis you recover the two Lie algebras above.

The 1|1 case
Now to the main topic. Let $x$ be odd and $y$ be even. Then the only possibility for the non-trivial Lie brackets must be

$[x,y] = ax$ and $[x,x] = bx$,

from the parity of the Lie bracket. Now we need to look for consistency with the Jacobi identity. So,

$[x,[x,y]] = [[x,x],y] – [x,[x,y]]$.

This then means that we have

$2 a[x,x] = 2 ab y =0$.

Note that I have not assumed that a is not zero at this stage and so all I can conclude is the ab =0.

Thus we have three generic cases;
i) $a=0$ and $b =0$
ii) $a \neq 0$ and $b =0$
iii) $a = 0$ and $b \neq 0$

Then up to isomorphism (just a rescale of the basis) we have three Lie superalgebras whose non-trivial Lie brackets are

i) $[x,x]=0$ and $[x,y]=0$
ii) $[x,x] =0$ and $[x,y] = x$
iii) $[x,x] =y$ and $[x,y] =0$

All three of these have nice geometric realisations and physicists reading this might recognise iii) in connection with supersymmetry.

The geometric realisations
By a geometric realisation I mean in terms of vector fields on some supermanifold. For these 1|1 dimensional examples it is quite illustrative.

Translations on $\mathbb{R}^{1|1}$ are of the from

$t’= t + a$
$\tau’ = \tau + \epsilon$,

where where we have picked global coordinates $(t, \tau)$ consisting of one even and one odd function. These translations are generated by the vector fields

$X = \frac{\partial}{\partial \tau }$ and $Y = \frac{\partial}{\partial t}$.

Thus we see that we have the trivial Lie superalgebra i).

Note that we have both even and odd parameters for these translations and so we are not really talking about the categorical morphisms as linear supermanifolds here. They are slightly more general than this.

With this in mind let us look at the Lie superalgebra of the diffeomorphism supergroup of $\mathbb{R}^{0|1}$. The transformations here are of the form

$\tau’ = a \tau + \epsilon$,

which are generated by the vector fields

$X = \frac{\partial}{\partial \tau}$ and $Y = \tau \frac{\partial}{\partial \tau}$.

Thus we have the Lie superalgebra ii).

The last case is more interesting for physics. Here we have the N=1 d=1 SUSY-translation algebra

$t’ = t + \epsilon t\tau + a$
$\tau’ = \tau + \epsilon$,

which are generated by the two vector fields

$X = \frac{\partial}{\partial \tau} + \tau \frac{\partial}{\partial t}$ and $Y = \frac{\partial}{\partial t}$,

which up to a factor of 2 (we can rescale this away easily) we get the Lie superalgebra iii).

Conclusion
For dim 1|0 and dim 0|1 we only have the trivial abelian Lie superalgebras.

For dim 2|0 we have (up to isomorphism) two Lie (super)algebras.

For dim 1|1 we have (up to isomorphism) three Lie superalgebras.

## Nicolaus Copernicus’ birthday

February 19th, 2014 by ajb
 Today, the 19th February is the birthday of Mikołaj Kopernik, maybe better known in the west as Nicolaus Copernicus. Kopernik was born on the 19th February 1473 in the city of Toruń, in the province of Royal Prussia, in the Crown of the Kingdom of Poland.

The heliocentric hypothesis
De revolutionibus orbium coelestium (1543) is the book in which which Kopernik offered an alternative model of the Solar system to Ptolemy’s geocentric system. Kopernik’s new model places the Sun and not the Earth at the center of the Solar system and represented a new shift in thinking. Importantly, the heliocentric model fits the astronomical observations much more naturally than the geocentric model which required strange phenomena like epicycles.

Nicolaus Copernicus Monument in Warsaw
In Warsaw there is Bertel Thorvaldsen’s monument which was completed in 1830. The monument comes with the words “Nicolo Copernico Grata Patria” (Latin: “To Nicolaus Copernicus from a Grateful Nation”) and “Mikołajowi Kopernikowi Rodacy” (Polish: “To Mikołaj Kopernik from his compatriots”).

Early in the Nazi German occupation of Warsaw in 1939, the Germans replaced the Latin and Polish inscriptions on the monument with a plaque in German: “To Nicolaus Copernicus from the German Nation”.

On 11 February 1942 Maciej Aleksy Dawidowski removed the German plaque!

During the 1944 Warsaw uprising the momentum was damaged and shortly after the Germans decided to melt it down for scrap metal. The Germans sent the monument to Nysa (southwestern Poland), but they had to retreat before they could melt it down. The Polish people returned the monument to Warsaw on 22 July 1945. The monument was renovated and unveiled again on 22 July 1949.

In 2007 a bronze representation of Kopernik’s solar system, modeled on an image in his De revolutionibus orbium coelestium, was placed on the square in front of the monument.

You can see some pictures of me next to the monument here.

## Education Minister to visit China

February 18th, 2014 by ajb
 The UK Education Minister Elizabeth Truss is going to lead a fact-finding mission to Shanghai in order to find out how children there have become the best in the world at mathematics.

(They) have a can-do attitude to maths, which contrasts with the long-term anti-maths culture that exists here.

Ms Truss

In my opinion, there seems to be an acceptable level of mathematical ignorance in the UK and that needs to be addressed as a cultural issue as much as an educational one.

Let us hope that Ms Truss returns with some good ideas on how to revitalise mathematics education.

Shanghai visit for minister to learn maths lessons BBC News website.

## Peter Higgs talks to Jim Al-Khalili

February 18th, 2014 by ajb
 Prof. Peter Higgs is interviewed by Prof. Jim Al-Khalili on BBC Radio 4’s The Life Scientific this Tuesday, 18 February.

Higgs reveals that he did not see the full significance of his initial paper on symmetry breaking and how he got left behind with the further developments for a while before returning to the field in the 1970s.

Peter Higgs talks to Jim Al-Khalili on Radio 4 IOP website.

## A brief history of mathematics

February 17th, 2014 by ajb
 Prof. Marcus du Sautoy on BBC radio 4 a few of years ago gave a series of short accounts of some of the personalities that shaped modern mathematics. You can find these radio programs on the BBC i player by following the link below.

p.s. I must thank my brother for pointing this out.

## Galileo’s birthday

February 15th, 2014 by ajb
 Today, the 15th February is Galileo Galilei’s birthday. He is often referred to as the farther of modern physics. He is of course also know for his discoveries using his telescope including the Galilean Moons of Jupiter, the rings of Saturn, the phases of Venus many geographical features of the Moon. Galileo as born on the 15th February 1564 in Pisa, Italy.

His legacy for theoretical physics
Galileo’s legacy for physics was his blend of mathematics with experimentation. Most of the contemporary science at the time was rather qualitative and Galileo was one of the first to believe that the laws of nature can take a mathematical form.