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There is an interesting article on the BBC website about crackpots at scientific conferences. The likes of Bayard Peakes are mentioned, as is the interesting case of Prof Schwartz who won a Nobel prize. |
I have fortunately, not seen many crackpots at conferences that I have attended, but I have seem some…
Follow the link below to read more.
Link
‘Crackpot’ science and hidden genius at physics meeting
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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 $latex\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)$, 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 $latex(x^{A})$ on \(M\) and \(\tau \) on \(\mathbb{R}^{0|1}\). Then
\(\gamma^{*}(x^{A}) = x^{A}(\tau) = x^{A} + \tau \; \; v^{A}\),
where $latex\widetilde{v^{A}} = \widetilde{A}+1$. 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.
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)
Links
Euler Biography (The MacTutor History of Mathematics archive)
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.
Link
Faces of Mathematics, London Mathematical Society
The International Space Station, image courtesy of NASA
Scheidl, Wille and Ursin [1] have proposed using the International Space Station to test the limits of spooky action at a distance. These experiments could help develop global quantum communication systems.
Part of their plans include a Bell test experiment which probes the theoretical contradiction between quantum mechanics and classical physics. A pair of entangled photons would be generated on the Earth. One of these would then be sent to a detector aboard the International Space Station, while the other photon would be measured locally on the ground for comparison.
According to quantum physics, entanglement is independent of distance. Our proposed Bell-type experiment will show that particles are entangled, over large distances — around 500 km — for the very first time in an experiment…
Professor Ursin
It is also not really known if gravity plays any role in quantum entanglement. These experiments would be the first to really probe the potential effects of gravity.
References
[1] T Scheidl, E Wille and R Ursin, Quantum optics experiments using the International Space Station: a proposal, 2013, New J. Phys. 15 043008 (online here)
Link
“Spooky action at a distance” aboard the ISS IOP News
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The Newton lecture 2012 is now available to watch on the IOP website. Professor Martin Rees, the winner of the 2012 Newton medal, gave the lecture entitled Form Mars to the multiverse. |
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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
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.
Links
Planck Science Team Home
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.
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
About the Ri
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
