## Archive for the ‘General Mathematics’ Category

### G-torsors

Tuesday, May 14th, 2013
 Let me quickly define a G-torsor.

Definition A G-torsor is a non-empty set X on which a group G acts freely and transitively.

A little more explicitly, X is a G-torsor if X is a nonempty set that is equipped with a map X × G → X such that
1) x·1 = x
2) x·(gh) = (x·g)·h

for all x in X, and h,g in G such that

(x,g) → (x, g·x)

is an isomorphism of sets.

Note that here we have picked the right action of G on X.

Remark One can modify these definitions to include categories other than sets, for example topological groups and spaces or even Lie groups and spaces.

Note that as we have an isomorphism, as sets, between X and G, they are equivalent objects. However, the subtlety is that there is no preferred identity point in X.

Ethos A G-torsor is a group that has lost its identity.

Once you have picked an identity element in X, you get an isomorphism as groups between X and G. This means that X and G are isomorphic as groups, but not canonically, a choice is needed.

What is the point of all this?
So it seems at first glance that torsors are very abstract objects far too complicated to be of much use to anyone. That is, until you realise that you have been using torsors without knowing it.

A good example of the use of a torsor is the potential difference in electromagnetism. When you measure a voltage, you in fact measure the difference of some voltage relative to some other fixed voltage. In practice one takes the ground to be zero, but this is a choice. Other values would work just as well. You can think of voltages as being elements of a torsor as there is no fixed identity voltage to measure against.

Energy in classical physics is very similar. The energy of a specified isolated system only really makes sense when one has set the “zero point energy”. One can only really measure energy differences relative to the “zero point energy”. This is why one can arbitrarily shift energies without effecting the physics. Actually, this is important when looking at the notion of energy in quantum field theory, but that is another story. Anyway, energies can be viewed as being elements of a torsor, you have no fixed “zero point energy” to measure all other energies against.

Physics is littered with similar examples.

A counter example would be temperature. We have a zero point temperature, that is absolute zero fixed for us.

Mathematics of course has lots of its own examples of torsors.

Consider a vector space V we can take G to be the general linear group GL(V) and X to be the set of all ordered bases of V. The group G acts transitively on X since any basis can be transformed via G to any other basis. In essence, one can take a specified basis and transform it into any other basis. Thus, one can consider all other bases as transformed versions of the initial basis. However, there is no natural choice of this “identity frame”. The set of bases do not form a group, but rather a torsor.

I will let the John Baez explain further here.

### Compatible homological vector fields

Tuesday, April 23rd, 2013
 In an earlier post, here, I showed that the homological condition on an odd vector field $Q \in Vect(M)$, on a supermanifold $M$, that is $2Q^{2}= [Q,Q]=0$, is precisely the condition that $\gamma^{*}x^{A} = x^{A}(\tau)$, where $\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)$, be an integral curve of $Q$.

A very natural question to answer is what is the geometric interpretation of a pair of mutually commuting homological vector fields?

Suppose we have two odd vector fields $Q_{1}$ and $Q_{2}$ on a supermanifold $M$. Then we insist that any linear combination of the two also be a homological vector field, say $Q = a Q_{1} + b Q_{2}$, where $a,b \in \mathbb{R}$. It is easy to verify that this forces the conditions

$[Q_{1}, Q_{1}]= 0$, $[Q_{2}, Q_{2}]= 0$ and $[Q_{1}, Q_{2}]= 0$.

That is, both our original odd vector fields must be homological and they mutually commute. Such a pair of homological vector fields are said to be compatible. So far this is all algebraic.

Applications of pairs, and indeed larger sets of compatible vector fields, include the description of n-fold Lie algebroids [1,3] and Q-algebroids [2].

The geometric interpretation
Based on the earlier discussion about integrability of odd flows, a pair of compatible homological vector fields should have something to do with an odd flow. We would like to interpret the compatibility of a pair of homological vector fields as the integrability of the flow of $\tau = \tau_{1} + \tau_{2}$. Indeed this is the case;

Consider $\gamma^{*}_{\tau_{1} + \tau_{2}}(x^{A}) = x^{A}(\tau_{1} + \tau_{2}) = x^{A}(\tau_{1}, \tau_{2})$, remembering that we define the flow via a Taylor expansion in the “odd time”. Expanding this out we get

$x^{A}(\tau_{1}, \tau_{2}) = x^{A} + \tau_{1}\psi_{1}^{A} + \tau_{2}\psi_{2}^{A} + \tau_{1} \tau_{2}X^{A}$.

Now we examine the flow equations with respect to each “odd time”. We do not assume any conditions on the odd vector fields $Q_{1}$ and $Q_{2}$ at this stage.

$\frac{\partial x^{A}}{\partial \tau_{1}} = \psi_{1}^{A} + \tau_{2}X^{A} = Q_{1}^{A}(x(\tau_{1}, \tau_{2}))$
$= Q_{1}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}}$,

and
$\frac{\partial x^{A}}{\partial \tau_{2}} = \psi_{2}^{A} {-} \tau_{1}X^{A} = Q_{2}^{A}(x(\tau_{1}, \tau_{2}))$
$= Q_{2}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}}$.

Then equating coefficients in order of $\tau_{1}$ and $\tau_{2}$ we arrive at three types of equations

i) $\psi_{1}^{A} = Q_{1}^{A}$, $\psi^{B}_{1} \frac{\partial Q_{1}^{A}}{\partial x^{B}}=0$ and $\psi_{2}^{A} = Q_{2}^{A}$, $\psi^{B}_{2} \frac{\partial Q_{2}^{A}}{\partial x^{B}}=0$.

ii) $X^{A} = \psi^{B}_{2} \frac{\partial Q_{1}^{A}}{\partial x^{B}}$ and $X^{A} = {-}\psi^{B}_{1} \frac{\partial Q_{2}^{A}}{\partial x^{B}}$.

iii) $X^{B} \frac{\partial Q_{1}^{A}}{\partial x^{B}} =0$ and $X^{B} \frac{\partial Q_{2}^{A}}{\partial x^{B}} =0$.

It is now easy to see that;

i) implies that $[Q_{1}, Q_{1}] =0$ and $[Q_{2}, Q_{2}] =0$ meaning we have a pair of homological vector fields.

ii) implies that $[Q_{1}, Q_{2}]=0$, that is they are mutually commuting, or in other words compatible.

iii) is rather redundant and follows from the first two conditions.

Thus our geometric interpretation was right.

References
[1] Janusz Grabowski and Mikolaj Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59(2009), 1285-1305.

[2]Rajan Amit Mehta, Q-algebroids and their cohomology, Journal of Symplectic Geometry 7 (2009), no. 3, 263-293.

[3] Theodore Th. Voronov, Q-Manifolds and Mackenzie Theory, Commun. Math. Phys. 2012; 315:279-310.

### Odd curves and homological vector fields

Wednesday, April 17th, 2013
 On a supermanifold one has not just even vector fields but also odd vector fields. Importantly, the Lie bracket of an odd vector field with itself does not automatically vanish.

This is in stark contrast to the even vector fields on a supermanifold and indeed all vector fields on a classical manifold. Odd vector fields that self-commute under Lie bracket are known as homological vector fields and a supermanifold equipped with such a vector field is known as a Q-manifold.

In the literature one is often interested in homological vector fields from an algebraic perspective. Indeed, the nomenclature “homological” refers to the fact that on a Q-manifold one has a cochain complex on the algebra of functions on the supermanifold. You should have in mind the de Rham differential and the differential forms on a manifold in mind here.

In fact, if we think of differential forms as functions on the supermanifold $\Pi TM$, then the pair $(\Pi TM, d)$ is a Q-manifold.

But can we understand the geometric meaning of a homological vector field?

Odd curves and maps between supermanifolds
Consider a map $\gamma : \mathbb{R}^{0|1} \longrightarrow M$, for any supermanifold $M$. We need to be a little careful here as we take $\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)$, that is we include odd maps here. Informally, we will use odd parameters at our free disposal. More formally, we need the inner homs, which requires the use of the functor of points, but we will skip all that.

Let us employ local coordinates $(x^{A})$ on $M$ and $\tau$ on $\mathbb{R}^{0|1}$. Then

$\gamma^{*}(x^{A}) = x^{A}(\tau) = x^{A} + \tau \; \; v^{A}$,

where $\widetilde{v^{A}} = \widetilde{A}+1$. This is why we need to include odd variables in our description. Note that as $\tau$ is odd, functions of this variable can be at most linear.

Aside One can now show that $\Pi TM = \underline{Map}(\mathbb{R}^{0|1}, M)$. Basically we have local coordinates $(x^{A}, v^{A})$ noting the shift in parity of the second factor. One can show we have the right transformation rules here directly.

Odd Flows
Now consider the flow on odd vector field, that is the differential equation

$\frac{d x^{A}(\tau)}{d \tau} = X^{A}(x(\tau))$,

where in local coordinates $X = X^{A}(x)\frac{\partial}{\partial x^{A}}$.

From our previous considerations the flow equation becomes

$v^{A} = X^{A}(x) + \tau v^{B} \frac{\partial X^{A}}{\partial x^{B}}$.

Thus equating the coefficients in order of $\tau$ shows that

$X^{A}(x) = v^{A}$ and $v^{B} \frac{\partial X^{A}}{\partial x^{B}}=0$.

Then we conclude that

$X^{B} \frac{\partial X^{A}}{\partial x^{B}}=0$, which implies that $[X,X]=0$ and thus we have a homological vector field.

Conclusion
The homological condition is the necessary and sufficient condition for the integrability of an odd vector field. Note that in the classical case there are no integrability conditions on vector fields.

### Leonhard Euler’s birthday

Monday, April 15th, 2013

Today, the 15th of April, is Euler’s birthday. Euler, a pioneer of modern mathematics, was born on April 15 1707, in Basel, Switzerland. His work introduced much of today’s modern notation. He worked on quite diverse areas such as mathematical analysis, geometry, number theory, graph theory and so on, as well as making massive impact in the world of physics in areas such as mechanics and optics.

Portrait by Johann Georg Brucker (1756)

Euler Biography (The MacTutor History of Mathematics archive)

### Faces Of Mathematics

Wednesday, April 10th, 2013

The London Mathematical Society has a webpage in which they “take a look at some of the people who’ve made major contributions in the world of mathematics”. Indeed, it is always nice to put a face to the names we read about. My only comment about the page is that I do not think all the information is up to date, but never mind.

Faces of Mathematics, London Mathematical Society

### 60 years of DNA

Saturday, April 6th, 2013
 This month marks the 60th anniversary of the discovery of deoxyribonucleic acid, or DNA to most of us. In the USA, there is DNA Day, which is a holiday celebrated on the 25th of April. The holiday commemorates the day in 1953 when James Watson and Francis Crick published their paper in Nature on the structure of DNA. It is also the 10th anniversary of the first sequencing of the human genome.

Knot theory
One area of mathematics that has been rather useful in the study of DNA, and in particular how it tangles is knot theory. DNA is tightly packed into genes and chromosomes. This packing can be thought of as two very long strands that have been intertwined many times and tied into knots. Before the DNA can replicate it needs to be arranged much neater than that and so needs to be unpacked. Thus knot theory is important in understanding this “unknotting” of DNA.

The way the knots were classified had nothing to do with biology, but now you can calculate the things important to you.

Nicholas Cozzarelli, in [1].

A knot is just a embedding of a circle in 3d.

The knot diagram of the Trefoil knot

The classification of knots has been a harder problem that one might expect. The general idea is to construct ways to see if two knots are equivalent, meaning they are the same knot. More mathematically two knots are equivalent if they can be transformed into each other via a special kind of transformation known as an ambient isotopy. Such transformations are really just “distortions” of the knot without any cutting.

A powerful way of deciding of knots are the same or not, is to calculate their Jones polynomial [2]. Interestingly, there is a relation between the Jones Polynomial and Chern-Simons gauge theory, which was first discovered by Witten [3].

References
[1] David Austin, That Knotty DNA, Feature Column of the AMS.

[2] Jones, V.F.R. (1985),A polynomial invariant for knots via von Neumann algebra, Bull. Amer. Math. Soc.(N.S.) 12: 103–111

[3]Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399

### Thoughts about Research – a list of interesting quotes

Thursday, March 14th, 2013

 Professor Piotr Pragacz, a mathematician working in the area of algebraic geometry here at IMPAN, has collected a few quotes on mathematics and science a little more generally.

Some of my favorites listed include

Nicolaus Copernicus: “Mathematics is written for mathematicians.”

Godfrey H. Hardy: “Young men should prove theorems, old men should write books.”

Albert Einstein: “The important thing is not to stop questioning; curiosity has its own reason for existing.”

David Hibert: “One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.”

Henri Poincaré: “The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful.”

And my personal favorite

Winston Churchill: “Success consists of going from failure
to failure without loss of enthusiasm. ”

### Yes, it is Pi Day

Thursday, March 14th, 2013

Today is Pi Day,  an earlier post can be found here and the official Pi Day website can be found here.

I have found a recipe for szarlotka, which is a traditional Polish apple pie. It looks light just the right thing to celebrate Pi Day.

### The most irrational day of the year!

Tuesday, March 12th, 2013

14th March has been officially designated Pi Day, a day for which we can celebrate the glorious number that starts with 3.14. Coincidentally, the 14th of March is also Albert Einstein’s birthday.

$\pi$ -the ratio of the circumference of a circle to its diameter- has been calculated to over one trillion decimal places. The record as far as I know belongs to Alexander J. Yee & Shigeru Kondo, who have calculated $\pi$ to 10 trillion digits [1]. As an irrational and transcendental number, $\pi$ will continue infinitely without any repetition or patterns emerging.

 The “pi man” Larry Shaw The first Pi Day was organized by Larry Shaw and held in San Francisco in 1988. In 2009, the US House of Representatives backed its official designation.

What to do for Pi Day?
Suggestions include bake a pie for Pi Day, or be artistic and write a piece of music, a poem or make a painting. You can find lots more suggestions by following this link.

The Welsh connection
The earliest known use of the symbol $\pi$ to represent the ratio of the circumference of a circle to its diameter is by Welsh mathematician William Jones FRS (1675 – 3 July 1749) in 1706 [2].

Portrait of William Jones by William Hogarth, 1740 (National Portrait Gallery)

Jones was a close friend of Sir Isaac Newton and Sir Edmund Halley. In November 1711 he became a Fellow of the Royal Society, and was later its Vice-President.

References
[1] Alexander J. Yee & Shigeru Kondo, Round 2… 10 Trillion Digits of Pi 2013.

[2] William Jones, Synopsis Palmariorum Matheseos, 1706.

Pi Day

### National Science and Engineering Week 2013

Thursday, March 7th, 2013

National Science and Engineering Week 2013 in the UK is running from the 15th to the 24th March. The events are coordinated by the British Science Association, though it is other organisations and community groups that actualy run the events and activities. The theme this year is invention and discovery.

For those of you in the UK, follow the link below and get involved in something near you.

National Science & Engineering Week shines the spotlight each March on how the sciences, technology, engineering and maths relate to our everyday lives, and helps to inspire the next generation of scientists and engineers with fun and participative events and activities.

Last year’s National Science and Engineering Week consisted of something like 500 events and activities from thousands of different organisers. More than 2 million people at schools, museums, universities, shopping centres, cafes etc. attended the various events.

Engineering Education Scheme Wales Awards & Presentation Day 2013
Wednesday, March 20, 2013 – 10:00 to 16:00
Celtic Manor Resort, Newport