The first Heidelberg Laureate Forum is going to place on September 22-27, 2013. The forum aims to bring together the best students and early stage researchers in mathematics & computer science with winners of the most prestigious awards in these disciplines – Abel, Fields, and Turing Laureates.
Applications to attend must be submitted by the 15th February 2013. Please see the website below for details.
Link
A student recently ask me about to explain what mathematicians mean by a corollary, so I thought I would quickly explain here.
Image by Free-StockPhotos.com
The four labels given by mathematicians to statements that can be shown to be true are Lemma, Theorem, Proposition and Corollary. They all basically mean the same thing: some mathematical statement that is true, given some starting axioms or previous true statements. Showing that these statements are true constitutes a proof. I will say something about proofs another time.
Everything I say here will be rather informal.
Lemma
A lemma is some statement that can be shown to be true, starting from some previously accepted statements, that is used to show that some other statement or statements are true.
In essence, a lemma is a “stepping stone” to some other statement or statements that are regarded more important in the current context.
However, lemmas can turn out to be very important statements in many cases. They often develop some line of attack that can be applied to a wider context than they were originally intended. Some of the most famous results in mathematics are lemmas.
Theorem
A theorem is some statement that can be shown to be true, starting from some previously accepted statements.
This is almost the same as a lemma, but a theorem is deemed to be of primary importance within the context it is established. Though note that one theorem can be used to prove another theorem, so the distinction between lemma and theorem is loose.
You can find a list of theorems here.
Proposition
A proposition is some statement that can be shown to be true, starting from some previously accepted statements.
Hang on a moment, is this not just the same as a theorem?
In short, yes a theorem and a proposition are the same thing. Though, and this is rather subjective, a theorem is deemed to be a more important result than a proposition.
So the distinction between a lemma, a theorem and a proposition is rather loose.
Corollary
A corollary is some statement that is true, that follows directly from some already established true statement or statements.
Typically, a corollary will be some statement that is easily derived from a theorem or a proposition. Often corollaries are “specialisations” of a theorem or a proposition. They key thing is they follow naturally from some established statement.
But again, this is not really any different to a theorem or proposition, or indeed a lemma if the corollary is used to establish some other statements.
In conclusion
So we see that the distinction between a lemma, a theorem, a proposition and a corollary is not very strict and depends largely on the context as well as the whims of the mathematicians writing papers.
One man’s lemma is another man’s theorem.
The University of Buckingham, which is the only private research and teaching university of the United Kingdom, is to stop accreditation of courses run at the Victoria University in Kampala, Uganda.
This is in direct response to proposed legislation against homosexuality in Uganda. Homosexuality is already against the law in Uganda, but the new bill would increase the punishment, including the possibility of a life sentence.
Under both UK and Ugandan law discrimination on a variety of grounds is prohibited; however there are fundamental differences between the two nations’ respective laws regarding equality and diversity, which cannot be reconciled. After seeking legal guidance from both UK and Ugandan lawyers, Victoria University and University of Buckingham have concluded that as the laws of Uganda and UK presently stand, Victoria University cannot comply with both sets of laws.
Victoria University Press Release 8 Jan 2013
Not quite terminator the musical, but it is a start!
[youtube:http://www.youtube.com/watch?v=3RBSkq-_St8]
They are a genuine heavy metal band!
Prof. Grabowski has placed a review of the various brackets found in geometry and physics [1]. He also covers some of the ideas of superalgebra and graded differential geometry as many of the brackets really have their roots there. The review is based on a mini-course held at XXI Fall Workshops on Geometry and Physics, Burgos (Spain), 2012.
I have posted here about the review here as it contains a lot of the background material needed to understand my own research. In particular I am interested in brackets found in supergeometry, including super versions of Poisson, Jacobi and Loday brackets.
Brackets?
Rather generally, a bracket is understood as a non-associative operation on a vector space or a module. The principle example here is a Lie bracket. The review focuses on Lie brackets, such as Poisson and Jacobi brackets as well as Loday brackets, which are a non-skewsymmetric generalisation of a Lie bracket.
Interestingly, various forms of brackets arise in a wide context in contemporary mathematics. For example, Poisson brackets are found in classical and quantum mechanics as well as the theory of cluster algebras and geometric representation theory.
Prof. Janusz Grabowski
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Prof Grabowski is Head of the Department of Mathematical Physics and Differential Geometry at the Institute of Mathematics within Polish Academy of Sciences. His personal homepage can be found here. |
Reference
[1] Janusz Grabowski, Brackets, arXiv:1301.0227 [math.DG], 2012.
Last year we lost three very important people in the field of space exploration and astronomy.
Sally Ride
May 26, 1951 – July 23, 2012 (aged 61)
Ride in 1984
She was the first American woman to enter into low Earth orbit in 1983 and remains the youngest American astronaut to be launched into space.
Niel Armstrong
August 5, 1930 – August 25, 2012 (aged 82)
Armstrong in 1969
He was the first man to step foot on the Moon way back in 1969. Armstrong’s first space flight, as command pilot of Gemini 8, in 1966, made him NASA’s first civilian astronaut to fly in space.
Patrick Moore
4 March 1923- 9 December 2012 (aged 89)
Patrick Moore
Known as the presenter of the world’s longest-running television series with the same original presenter, the BBC’s The Sky at Night. Moore was a former president of the British Astronomical Association. He was a co-founder and former president of the Society for Popular Astronomy (SPA). As an author he wrote over 70 books on astronomy. He was the reason many astronomers here in the UK became astronomers.
Prof. Higgs, whom the Higgs boson is named after, is now a member of the Order of Champions of Honour. The honour has no title, but consists of the Sovereign, plus no more than 65 Companions of Honour.
Prof. Higgs
Other Companions of Honour include Prof. Stephen Hawking and Sir David Attenboroug.
The Higgs Boson
Prof. Higgs developed the idea of electroweak symmetry breaking to explain the masses of the Z and W bosons [1]. The Higgs-Kibble mechanism gives rise to the mass of all the massive elementary particles and predicted the existence of a new particle given the title “the Higgs boson”.
Computer simulation of particle traces from an LHC collision in which a Higgs Boson is produced. © CERN. Image credit: Lucas Taylor
Only recently at the LHC has there been evidence that the Higgs boson is realised in nature. CERN announced on 4 July 2012 that they had experimentally established the existence of a Higgs-like boson and that further study is needed to established if this really is a standard model Higgs boson, [2,3].
The Nobel prize?
Is a Nobel prize in Physics the next big award for Prof. Higgs? We will have to wait and see.
References
[1] Peter Higgs, Broken Symmetries and the Masses of Gauge Bosons, Physical Review Letters 13 (16): (1964) 508–509.
[2] The ATLAS Collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys.Lett. B716 (2012) 1-29. (arXiv:1207.7214 [hep-ex])
[3] The CMS Collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys.Lett B716 (2012) 30–61. (arXiv:1207.7235 [hep-ex])
Links
The Order of Champions of Honour
Prof. Higgs Website at the University of Edinburgh
The Chancellor, George Osborne, has announced a £21.5m investment fund to boost the development of the new wonder material graphene.
The £21.5m investment fund would aim to take the technology from the lab to the factory floor.
George Osborne
Graphene
Graphene is a material made carbon atoms arranged in a regular hexagonal pattern, similar to graphite, but in a one-atom thick sheet. Graphene has many unusual electrical and mechanical properties that make it interesting from a fundamental science point or view, but also lend this wonder material to applications.
Image courtesy of AlexanderAlUS
Andre Geim and Konstantin Novoselov, both based at the University of Manchester won the 2010 Nobel Prize in Physics for demonstrating some of the properties of graphene.
Andre Geim
Funding
EPSRC identified the most promising graphene-related research projects in British universities that would benefit from state funding.
For example, Cambridge will receive more than £12m for research into graphene flexible electronics and opto-electronics. Other UK universities, including Manchester will also receive money.
We need to support our universities, they’re one of the jewels in the crown of the British economy
George Osborne
Link
BBC News
Physicists from Japan have grown crystals in near zero gravity by using a diving airplane [1]. The idea was to examine crystal growth under such conditions to uncover phenomena usually masked by gravity’s effects.
[youtube:https://www.youtube.com/watch?v=xkjzKjGo40A&feature=player_embedded]
For more details follow the link below or the original paper cited.
This man had nothing to do with it…
Link
IOP News Physics on a plane: crystals made under zero gravity
Reference
[1] Takuya Takahashi, Haruka Ohuchi, Ryuji Nomura and Yuichi Okuda, Ripening of splashed 4He crystals by acoustic waves with and without gravity, 2012 New J. Phys. 14 123023 (link)
Physics World (IOP’s magazine) have collated 12 images that demonstrate the visual beauty that can be found in physics.
My personal favorite is this the Weaire–Phelan foam, which I reproduce below.
The Weaire–Phelan foam is believed to be the lowest energy structure for a foam formed of equal-volume bubbles. The foam was created using a special template in a solution of water and Fairy Liquid, with bubbles being introduced by releasing nitrogen gas from a glass capillary. The resulting foam was photographed using a digital SLR camera.
Link
Our favourite pictures of 2012 (Physics World)