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General Mathematics

Mathematics the langauge of Physics

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It is a rather indisputable fact for physicists that mathematics really is the correct language   of physics.  Without mathematics one could not formulate physical theories and then make prediction to be tested against nature.  Indeed, the formulation of physical theories has required the development of new mathematics.  Theoretical physics is really the construction of mathematical models to describe nature.

Even the experimentalist cannot avoid mathematics.  One has a lot of analysis of results and statistics  to preform in order to make sense of the experiments.

It is rather clear then, that without mathematics one will not go very far in physics. Any understanding of nature is going to be rather superficial without some mathematics.

A little deeper than this I believe that mathematics is more than just a language for physics, or indeed all science. The structures, patterns and rules of mathematics can guide one in constructing/analysing theories. The notion of symmetry is so fundamental in modern theoretical physics and at its heart is group theory.  Understanding physics can be driven my mathematical beauty. Given a new theory the first question to ask is what are the symmetries?

One has to ask why mathematics is the language of the physical sciences? Can we understand why mathematics has been just so useful and powerful in structuring our understanding of the Universe?

Eugene Wigner in 1960 wrote an article The Unreasonable Effectiveness of Mathematics in the Natural Sciences which was published in Communications on Pure and Applied Mathematics.  Wigner argues that mathematics has guided many advances in the physical sciences and that this suggests some deep link between mathematics and physics far beyond mathematics simply being a language.

A very extreme version of this deep interconnection is Max  Tegmark’s mathematical universe hypothesis, which basically states that all mathematics is realised in nature.  What this hypothesise also suggests is that the Universe really is mathematical. We uncover this mathematical structure rather than impose it on nature. This would explain Wigner’s “unreasonable effectiveness”.

We are now close to having to think about the philosophy of mathematics and in particular Platonism. I am certainly no big thinker on philosophy and so will postpone discussion about the philosophy of mathematics.

I would not go as far as to say I believe in Tegmark’s hypothesis, but it is for sure an interesting and provocative idea.  It certainly makes one think about the relation between mathematics, physics  and the nature of our Universe.

 

Wikipedia Links

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Mathematical universe hypothesis

General

What is wrong with engineers?

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Here are a few comments on understanding engineers. I will tell you that an engineer sent them to me.

 Understanding Engineers: One


Two engineering students were walking across a university campus when one said, “Where did you get such a great bike?”
The second engineer replied, “Well, I was walking along yesterday, minding my own business, when a beautiful woman rode up on this bike, threw it to the ground, took off all her clothes and said, “Take what you want.”
The first engineer nodded approvingly and said, “Good choice; the clothes probably wouldn’t have fit you anyway.”


Understanding Engineers: Two


To the optimist, the glass is half-full.
To the pessimist, the glass is half-empty.
To the engineer, the glass is twice as big as it needs to be.


Understanding Engineers: Three


A priest, a doctor, and an engineer were waiting one morning for a particularly slow group of golfers.
The engineer fumed, “What’s with those guys? We must have been waiting for fifteen minutes!”
The doctor chimed in, “I don’t know, but I’ve never seen such inept golf!”
The priest said, “Here comes the green-keeper. Let’s have a word with him.”
He said, “Hello George, what’s wrong with that group ahead of us? They’re rather slow, aren’t they?”
The green-keeper replied, “Oh, yes. That’s a group of blind firemen. They lost their sight saving our clubhouse from a fire last year, so we always let them play for free anytime.”
The group fell silent for a moment.
The priest said, “That’s so sad. I think I will say a special prayer for them tonight.”
The doctor said, “Good idea. I’m going to contact my ophthalmologist colleague and see if there’s anything he can do for them.”
The engineer said, “Why can’t they play at night?”

Understanding Engineers: Four


What is the difference between mechanical engineers and civil engineers?
Mechanical engineers build weapons. Civil engineers build targets.


Understanding Engineers: Five


The graduate with an engineering degree asks, “How does it work?”
The graduate with a science degree asks, “Why does it work?”
The graduate with an accounting degree asks, “How much will it cost?”
The graduate with an arts degree asks, “Do you want fries with that?”


Understanding Engineers: Six


Three engineering students were gathered together discussing who must have designed the human body.
One said, “It was a mechanical engineer. Just look at all the joints.”
Another said, “No, it was an electrical engineer. The nervous system has many thousands of electrical connections.”
The last one said, “No, actually it had to have been a civil engineer. Who else would run a toxic waste pipeline through a recreational area?”


Understanding Engineers: Seven


Normal people believe that if it ain’t broke, don’t fix it.
Engineers believe that if it ain’t broke, it doesn’t have enough features yet.


Understanding Engineers: Eight


An engineer was crossing a road one day, when a frog called out to him and said, “If you kiss me, I’ll turn into a beautiful princess.”
He bent over, picked up the frog and put it in his pocket.
The frog then cried out, “If you kiss me and turn me back into a princess, I’ll stay with you for one week and do ANYTHING you want.”
Again, the engineer took the frog out, smiled at it and put it back into his pocket.
Finally, the frog asked, “What is the matter? I’ve told you I’m a beautiful princess and that I’ll stay with you for one week and do anything you want. Why won’t you kiss me?”
The engineer said, “Look, I’m an engineer. I don’t have time for a girlfriend, but a talking frog, now that’s cool.”

 

 

 

General

Response to "String Bashing" by M.J. Duff

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If you have read Smolin’s and/or Woit’s books arguing against string theory then please read Duff’s response  here.

Duff’s “String and M-theory: answering the critics” is quite accessible and does a good job explaining why people are interested in string theory, both from a physics and mathematics point of view. One major point he makes is the unfair coverage of “anti-string theorists” and  works that are wrong.

For example, Lisi’s theory of everything based on \(E_{8}\).  This theory is mathematically wrong, it does not describe  the correct matter content of the universe.  Also, the use of the exceptional groups  in theories of everything, including string theory pre-dates    Lisi’s work.  I would say that the media and the “blogosphere”  was too quick to hail Lisi’s work and too slow in pointing out the errors.

Duff also points out how quickly the attacks on string theory become personal attacks on string theorists.

Is it important that the general public has a reasonable understanding of string theory and supports such reserach?

I would have to say  yes.

Not that science or mathematics is a popularity contest that will be won via the general media, it will be won via peer-reviewed papers. However, the general public pays for almost all fundamental science research and thus it is vital to keep the public on board. There will always be speculation, disagreements and conflicting points of view in science at the frontiers of our knowledge, but this should not devolve into personal attacks. This only weakens the position of  science in wider society.

String does  have many attractive features and seems to be our best hope at understanding the Universe.  The best response to  the critics is come up with an alternative!

General

CERN to announce glimpse of the Higgs.

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Today at CERN scientists will present reports on the progress of the hunt for the Higgs boson.

I await the news…

For now read the BBC report here.

CERN’s public website can be found here.

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Update

Both the Atlas and CMS experiments at CERN independently suggest that the Higgs has been observed and has a mass of about 125 GeV.  However, the statistical uncertainly in the data means that the Higgs has not truly been discovered.

There will be further experiments and lots of data analysis before the claim of discovery will be made.

Optimistically,  some time next year we may have confirmation of the Higgs.

Find out about the press release  here.

Read the BBC News report here.

 

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Now we await news of supersymmetry.

 

General

Bridging mathematics and art

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We all know that art,  symmetry,  beauty and mathematics are well intertwined.  Since 1998 the Bridges Organization organised an annual conference bringing together mathematicians and artists. All very interesting stuff and shows that mathematics can be appreciated by those who are not traditional mathematicians.

The next conference is July 25-29, 2012 at Towson University, located in the Baltimore Metropolitan Area.

I guess I will now have to go away and create some interesting computer graphics or something! (Not that I think I will be attending this conference)

General

2015, The International Year of Light

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The European  Physical Society has initiated a drive to get the United Nations to proclaim 2015 as the International Year of Light.

 

Light plays a central role in human activities in science, technology and culture. Light itself underpins the existence of life, and light-based technologies will guide and drive the future development of human society. Light and optics have revolutionized medicine, have opened up international communication via the Internet, and continue to be central to linking cultural, economic and political aspects of the global society. Advances in lighting and solar energy are considered crucial for future sustainable development.

Follow this link to the full statement.

 

I think making 2015 the International Year of Light is a great idea. Of all the physics, understanding light and more generally electromagnetic radiation,  has had a huge impact on society as well as all branches of  science and engineering.

2005 was the International Year of Physics, 2009 was the International Year of Astronomy and 2011 is the International Year of Chemistry. It seems fitting that 2015 be the year of light, lets see why…

  • 1815  Fresnel published his first works on light as a wave.
  • 1865 Maxwell mathematically described electromagnetic phenomena via his now famous equations.
  • 1915 Einstein developed general relativity, which shows that light is fundamental in understanding space-time and gravity.
  • 1965  Penzias and Wilson discover the Cosmic Microwave Background Radiation (CMBR).

 

Let us hope the United Nations agree on the importance of  light in all our lives and declare 2015 The International Year of Light.

Research work

Jacobi algebroids and quasi Q-manifolds

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In “Jacobi algebroids and quasi Q-manifolds”  (arXiv:1111.4044v1 [math-ph]) I reformulate the notion of a Jacobi algebroid (aka generalised Lie algebroid or Lie algebroid in the presence of a 1-cocycle) in terms of an odd Jacobi structure of weight minus one  on the total space of the “anti-dual bundle” \(\Pi E^{*}\). This mimics the weight minus one Schouten structure associated with a Lie algebroid. The weight is assigned as zero to the base coordinates ans one to the (anti-)fibre coordinates.

Recall that a Lie algebroid can be understood as a weight one homological vector field  on the “anti-bundle” \(\Pi E\). What is the corresponding situation for Jacobi algebroids?

Well, this leads to a new notion, what I call a quasi Q-manifold…

A quasi Q-manifold is a supermanifold equipped with an odd vector field \(D\) and an odd function \(q\) that satisfy the following

\(D^{2}= \frac{1}{2}[D,D] = q \: D\)

and

\(D[q]=0\).

The extreme examples here are

  1. Q-manifolds, that is set \(q=0\). Then \(D^{2}=0\).
  2. Supermanifolds with a distinguished (non-zero) odd function, that is set \(D=0\).  (This includes the cotangent bundle of  Schouten and higher Schouten  manifolds)
  3. The entire category of supermanifolds if we set \(D=0\) and \(q =0\).

 

The theorem here is that a Jacobi algebroid,  understood as a weight minus one Jacobi structure on \(\Pi E^{*}\) is equivalent to  \(\Pi E\) being a weight one  quasi Q-manifold.  I direct the interested reader to the preprint for details.

A nice example is \(M:= \Pi T^{*}N \otimes \mathbb{R}^{0|1}\), where \(N\) is a pure even classical manifold.  The supermanifold \(M\) is in fact an odd contact manifold or equivalently an odd Jacobi manifold of weight minus one, see arXiv:1101.1844v3 [math-ph]. Then  it turns out that \(M^{*} := \Pi TN\otimes \mathbb{R}^{0|1}\)  is a weight one quasi Q-manifold. It is worth recalling that \(\Pi T^{*}N\) has a canonical Schouten structure (in fact odd symplectic) and that \(\Pi TN\) is a Q-manifold where the homological vector field is identified with the de Rham differential on \(N\).  Including the “extra odd direction” deforms these structures.

As far as I can tell quasi Q-manifolds are a new class of supermanifold that generalises Q-manifolds and Schouten manifolds.  It is not know if other examples of such structures outside the theory of Lie and Jacobi algebroids are interesting. Only time will tell…

General

Faster Than the Speed of Light?

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Professor Marcus du Sautoy offered a rather sobering view of the results of OPERA on the BBC last night. The BBC iPlayer version is available to view  until the 31st October, follow this link.

Until now I have resisted posting anything about the superluminal speed of neutrinos as measured by OPERA. There are plenty of blogs about this.  All I really want to say is that Marcus du Sautoy does a great job in approaching the topic from a science/maths point of view and does not “fan the flames” of the media hype.  His programme, in my opinion combats some of the hysteria and plain rubbish out there.

Please take the time to watch it.

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du Sautoy’s Oxford homepage

Wikpedia article about du Sautoy

 

General Mathematics

Number theory and numerology

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In a similar way to the historical link between  astronomy and astrology the subjects of number theory and numerology are also  linked.  The very early impetus for number theory was numerology.  The Pythagorean  school (500BC) were interested in the philosophical and mystical properties of numbers. Plato was influenced by this and mentions numerology in his works, notably The Republic (380BC).  Judaism ,  Christianity and Islam all have elements of numerology.

Number theory itself is probably older than this and goes back  almost to counting and simple arithmetic in  prehistory.

In the same way as astronomy developed into a science, so did number theory.

Definition Number theory is the branch of mathematics that deals with the study of numbers, usually the integers, rational numbers ,  prime numbers etc.

Definition Numerology is the study of the supposed relationship  between numbers, counting and everyday life.

There is another kind of numerology that is the study of numerical coincidences. This happens a lot in physics, where a series of apparent coincidences  can occur between various rations of physical constants or physical observables.

The famous example of this is Dirac’s large number hypothesis which enforces a ration between the cosmic scale and the scale of fundamental forces. Dirac’s hypothesis predicts that  Newton’s constant is varying in time. There has been some work in understanding the implications of physical constants changing in time.

Although Dirac’s hypothesis is the most famous, it was Eddington and Weyl who first noticed such numerical coincidences.

The trouble is that this cannot really be called science.  Physics is all about mathematical models that can be used to explain physical phenomena.  Noticing numerical coincidences by itself does not really add to our understanding of nature.  One would like to explain the  coincidences clearly and mathematically within some theory.  Generally these coincides are interesting, but it is not clear how they are fundamental. Of course, this is apart from those that are really just due to our choices in units etc.

Number theory also has some intersection with physics. Recently there has been some considerable crossover between   arithmetic and algebraic geometry  and string theory (via modular forms largely). I will have to postpone talking about this.