# Planck backs the ideas of an inflationary epoch

The Planck space telescope has produced the most detailed picture yet of the cosmic microwave background radiation (CMBR) [1].

NASA/JPL-Caltech/ESA

Detailed analysis supports the idea that $$10^{-32}s$$, or there about, the Universe went into a phase of rapid expansion known as inflation. This rapid growth of the Universe explains why the Universe is so big and nearly flat, as well as providing an explanation as to why the CMBR temperature is uniform. More than this, the small anisotropies in the temperature are well explained by tiny quantum fluctuations in the early Universe that get blown-up by the inflationary phase. These small differences seeded the large scale structure of the Universe we see today.

ESA/PLANCK COLLABORATION

Hubble constant
From Planck we also know that the Universe is expanding today at a slightly lower rate than previous estimates have given. The Hubble constant is now revised to 67.3 kilometers per second per megaparsec, which makes the Universe about 80 million years older than WMAP data suggests.

Make up of the Universe
The new data has meant a revision in the proportions of “stuff” in the Universe:

Dark Energy – 68.3%
Dark Matter – 26.8%
Normal matter < 5%

Oddities in the CMBR
WMAP found and Planck has now confirmed, that there is an asymmetry between opposite hemispheres of the sky in the anisotropies of the CMBR. This suggests the rather unnatural possibility that there is a preferred direction in the cosmos. This does rule out some specific models of inflation, but the generic idea is still sound.

The cold spot

The CMBR cold spot is another strange feature that Planck has confirmed. This colder region of the CMBR, 70 µK colder that the average 2.7K was first discovered by WMAP. It is thought a possibility that the cold spot and the asymmetry maybe connected.

Planck telescope peers into the primordial Universe, Nature, 21 March 2013

References
[1] Planck Collaboration, Planck 2013 results. I. Overview of products and results, submitted to Astronomy & Astrophysics, 2013.

# The Royal Institution received a £4.4 million donation

On 19th March 2013, Sir Richard Sykes, Chairman of the Royal Institution (Ri) announced that the Ri has received a donation of £4.4 million. The announcement was made at a special general meeting for the Institution’s members. The donation was made by a foundation which will remain anonymous at this stage.

The Royal Institution of Great Britain, by Thomas Hosmer Shepherd, circa 1838

In January, the Ri said it was considering selling 21 Albemarle Street London home in order to ease it’s financial troubles. This £4.4 million gift will help give the RI some time in order to sort out their finances.

This donation is very timely and will clear the Ri’s bank debt, as well as giving us the breathing room to explore other options more fully. However, our financial issues are far from being resolved.

Sir Richard Sykes

The Royal Institution (Ri) is an independent charity dedicated to connecting people with the world of science. They are most famous for the Christmas Lectures which were started by Michael Faraday in 1825. The Christmas Lectures have been broadcast on television since 1966 and in 2011 the combined broadcast reached over 4 million viewers.

Lithograph of Michael Faraday delivering a Christmas lecture at the Royal Institution, by Alexander Blaikley circa 1856

# IOP Lab in a Lorry is comming to Wales

The Lab in a Lorry will be touring around Wales in April and June 2013. Follow the link below for more details

Lab in a Lorry is an interactive mobile science laboratory staffed by practising scientists and engineers.

The aim of Lab in a Lorry is to give young people aged 11-14 the opportunity to do experimental science in the way it actually happens; exploratory, accidental, informed by curiosity and intuition, but also bounded and guided by the experience and insight of practicing scientists.

Looking for volunteers
James Bamford, Senior Operations Coordinator – Lab in a Lorry, has made a call for volunteers to help run the events. A poster for the call can be found here (opens pdf).

Lab in a Lorry

# Jacobi algebroids and quasi Q-manifolds revisited

 I have already spoken about Jacobi algebroids and quasi Q-manifolds in earliear posts here and here. Details can be found in [1].

In the paper [1] I show that a Jacobi algebroid, which is nothing more than a linear odd Jacobi bracket on a vector bundle, is equivalent to a weight one quasi Q-manifold structure.

A little more specifically, consider the supermanifold $$\Pi E$$ build from a vector bundle $$E \rightarrow M$$. The supermanifold $$\Pi E$$ is equipped with natural coordinates $$(x^{A}, \xi^{\alpha})$$. Recall that $$\Pi$$ is the parity reversion functor and that it shits the parity of the fibre coordinates. So, is we have fibre coordinate $$(y^{\alpha})$$ on $$E$$ of parity $$\widetilde{y^{\alpha}} =\widetilde{\alpha}$$, then $$\widetilde{\xi^{\alpha}}= \widetilde{\alpha}+1$$. The weight is assigned naturally as zero to the base coordinates and one to the fibre coordinates. The parity reversion functor does not act on the weights.

A Jacobi algebroid is then in one-to-one equivalence with an odd vector field on $$\Pi E$$

$$D = \xi^{\alpha}Q_{\alpha}^{A}(x) \frac{\partial}{\partial x^{A}} + \frac{1}{2} \xi^{\alpha}\xi^{\beta}Q_{\beta \alpha}^{\gamma}(x) \frac{\partial}{\partial \xi^{\gamma}}$$,

and an odd function also on $$\Pi E$$

$$q = \xi^{\alpha}Q_{\alpha}(x)$$,

both of weight one and satisfy

$$\left[D,D\right] = 2 q D$$ and $$D(q)=0$$.

A supermanifold with such a structure I call a quasi Q-manifold.

Back to Lie algebroids
There is a well established one-to-one correspondence between Jacobi algebroids and Lie algebroids in the presence of a one cocycle [2,3]. A Lie algerbroid in the presence of a one cocycle is understood as a $$(\Pi E, Q, \phi)$$, where $$Q$$ is a homological vector field of weight one and $$\phi$$ is a weight one (linear) function on $$\Pi E$$. Now as we are in the category of supermanifold, we need to insist that the weight one function is odd. The structures here satisfy

$$Q^{2}=0$$ and $$Q(\phi) =0$$.

Now, given the initial data of a weight one quasi Q-manifold that encodes the Jacobi algebroid we can pass directly to a Lie algebroid in the presence of an odd one cocyle viz

$$Q = D {-} q \Delta$$
and set
$$\phi = q$$,

where $$\Delta$$ is the Euler vector field, which in local coordinates looks like

$$\Delta = \xi^{\alpha} \frac{\partial}{\partial \xi^{\alpha}}$$.

So, now what about the bracket on sections of this Lie algebroid and the anchor?

By thinking of the sections of our vector bundle $$E\rightarrow M$$ as weight minus one vector fields on $$\Pi E$$, we can use the derived bracket formalism. In particular

$$u = u^{\alpha}(x)s_{\alpha} \longrightarrow i_{u} = (-1)^{\widetilde{u}}u^{\alpha}(x)\frac{\partial}{\partial \xi^{\alpha}}$$

provides us with the appropriate identification. Then

$$a(u)(f) := \left[\left[Q, i_{u}\right],f \right] = [[D, i_{u}],f]$$

and

$$i_{[u,v]} := (-1)^{\widetilde{u}}[[Q, i_{u}], i_{v}] =(-1)^{\widetilde{u}} [[D,i_{u}], i_{v}] + i_{u}(q) i_{v} – (-1)^{\widetilde{u} \widetilde{v}}i_{v}(q)i_{u}$$,

where $$u,v \in \Gamma(E)$$ and $$f \in C^{\infty}(M)$$.

The interested reader can now work out all the local expressions if they want, it is not hard to do so.

The final remark must be that similar formula appear in the existing literature on Jacobi algebroids for the Lie bracket. This I may try to unravel at some point.

References
[1]Andrew James Bruce, Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids, Extracta Math. 27(1) (2012), 91-123.

[2]J. Grabowski and G. Marmo, Jacobi structures revisited, J. Phys. A: Math. Gen., 34:10975–10990, 2001.

[3]D. Iglesias and J.C. Marrero. Generalized Lie bialgebroids and Jacobi structures, J. Geom. and Phys., 40, 176–199, 2001.

# Prof. Martin Rees interviewed by IOP

Prof. Martin Rees, winner of the 2012 Isaac Newton medal for his “outstanding contributions to relativistic astrophysics and cosmology”, has given an interview for the Institute of Physics. Follow the link below to watch the video.

# Thoughts about Research – a list of interesting quotes

 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

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!

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

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