Category Archives: General

Geometry and physics: Though lovers be lost love shall not

The title of this post comes from Dylan Thomas, And Death Shall Have No Dominion (1933). Here I give a non-technical essay on the interplay between geometry and physics, which I hope with give some of the readers a better idea of why I do what I do. Please enjoy and leave feedback if you like.

Geometry and physics: Though lovers be lost love shall not

To paraphrase a certain Polish mathematician: “the most important ideas in mathematics come from physics”. While there is no reason why mathematics — as mathematics — should come from physics, there is some deep connection between mathematics and our understanding of the Universe. Wigner in 1960 in his famous “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” article, noticed how the mathematical structure of physical theories can lead to new physical insight. And of course, physical insight can lead to new mathematics. By physics, I will mean the construction of mathematical models of natural phenomena and the comparing of the predictions of these models against nature.

From the very nature of physics it is clear that there is at least some superficial relation with mathematics. After all, physics uses mathematics. However, physics is not mathematics in the sense that mathematical constructions in physics should have (maybe not directly) some meaning. There must be some relation of the mathematics to a physical law. In mathematics, there is no such constraint that any of it have any meaning beyond what it mathematically means. It is a complete mystery as to why nature seems generally amenable to being understood in terms of abstract mathematics.

The deep interconnection between mathematics and physics seems especially true when focusing in on geometry: literally geometry means `Earth measurement’. At the most basic level, geometry is the study of spaces, which are understood as collections of points, together with a notion of points being `close to each other or not’, and usually with some further mathematical structures on them, such as a notion of the distance between to near by points. But this is definition in terms of points is not enough to cover the modern usage of `geometry’. So, what is geometry and where does it come from? Moreover, what has the study of spaces got to do with physics?

The first work on synthetic geometry is the book Elements written Euclid of Alexandria (c.325–265 BC). In this book an axiomatic approach to plane geometry, so parallel lines on flat surfaces etc., is established. For example, the internal angles of a triangle on the plane always add up to 180 degrees. However, curves, circles and spheres had been known about since antiquity. Solid geometry — the study of three dimensional objects — was needed as soon as humans started to imagine buildings such as domes and pyramids. In addition to this, the heavenly sky can be imagined as the inner surface of a dome speckled with stars — at least as we see it, and ancient astronomers saw it!

Methods of calculating the volume of simple regular three dimensional objects were developed. For example, the ancient Egyptians knew how to calculate the volume of pyramids and chambers therein: they were the mummy of all modern mathematicians! Archimedes (287–212 BC) in his `eureka’ moment realised that one could deduce the volume of three dimensional irregular objects based on the amount of water they displaced. However, Archimedes was unable to actually calculate volumes in any generality.

In another direction, Apollonius of Perga (c.262–190 BC) showed that the regular curves — circles; ellipses; parabola; and hyperbola — can be formed by cutting the cone, hence conic sections. Amazingly, in Newtonian gravity (circa 1686) the orbits of the two massive bodies are described by conic sections. This is part of the unifying power of mathematics: the mathematics involved in cutting cones is exactly the mathematics needed to describe orbits, for example the path of the Moon around the Earth! These mathematical coincidences are abound.

The most important mathematical works on conic sections — as far as our story goes — are that of Descartes (1596–1650) and Fermat (1601–1665), who in the 17th century brought algebra in to the game. Conic sections can be described by algebraic equations via coordinates — analytic and algebraic geometry were born! \par

The use coordinates (eg. x and x on the plane) opens up the use of calculus in geometry. Newton’s differential and integral calculus allows for methods of calculating gradients of curves, areas under curves, the volumes of objects etc. — calculus today is a common method of torturing undergraduate students! Differential geometry was born … or at least the seeds of the theory were planted by Newton (1642–1727) and Leibniz (1646–1716). One should not forget that much of Newton’s inspiration in developing calculus comes from his work on classical mechanics: so the mathematical description of the motion of massive bodies.

Curved surfaces – such as the sphere – represent non-Euclidean geometries. Lines drawn on them violate the axioms of Euclid’s plane geometry: this was seen as a real problem by mathematicians. It was Eugenio Beltrami (1835–1899) who showed that hyperbolic geometry is consistent: this is the geometry of surfaces of constant curvature for which the internal angles of a triangle add up to less that 180 degrees. Similar results were obtained for spherical geometries, so geometries of constant curvature for which the internal angles of triangles add up to more that 180 degrees.

Bernhard Riemann (1826–1866) in his PhD thesis extended the work of Beltrami to surfaces that have non-uniform curvature, and to higher dimensions. The work of Riemann allowed algebra and calculus to be applied to spaces known as smooth manifolds, i.e., spaces such that every `small piece’ of them looks like a `small piece’ of the n-dimensional plane for some integer n. One should keep in mind the relation between a globe and a map: any small piece of the globe can be represented on a sheet of paper as a map, and points on the globe are then represented by two numbers, the coordinates with respect to the given map. The notion of a smooth manifold underpins Einstein’s special and general relativity, as well as Maxwell’s theory of electromagnetism, Yang–Mills theories and classical mechanics: even thermodynamics has a geometric formulation!

It is worth saying a little more about Einstein’s general relativity (1916). This theory is a theory of gravity, and to date it is the most accurate theory of gravity we have. Moreover four dimensional smooth manifolds are central to the theory. Einstein took the earlier idea that space and time should be unified into space-time seriously, we have one time coordinate and three space coordinates. Einstein then told us that gravity is not your typical force, but rather it really is due to the local shape of space-time! The mathematical theory of curved smooth manifolds is vital to our understanding of gravity and the Universe as a whole, and vice versa, physics has been the impetus for many mathematical works on curved smooth manifolds.

There is a duality between a space and the algebra of functions on that space ( i.e., maps from that space to the real or complex numbers). Loosely, if you know the algebra of functions on a space, then you know the space. The algebra of functions on a classical space is commutative: the order of pointwise multiplication does not matter. We can imagine a more general notion of a `space’ by considering any algebra — not necessarily commutative — as the algebra of functions on some `space’. The phase space of quantum mechanics, that is the `space’ of positions and momenta of a quantum particle, is a noncommutative geometry.

A quantum theory of gravity could be some kind of noncommutative geometry: both string theory and loop quantum gravity suggest noncommutativity of space-time at some level — both the loopers and p-braners agree on this! Trying to make sense of physics at the smallest scales pushes what we mean by geometry well beyond our original understanding. Noncommutative geometries are in general not set theoretical objects, i.e., they do not consist of a collection of points — it is all rather pointless!

There is a kind of `halfway house’ between classical and quantum geometry: here I refer to supermanifolds as defined by Berezin and Leites in 1976. Without details and being very loose, a supermanifold is a `manifold-like object’ which comes with some coordinates that commute with all the coordinates, and some coordinates that anticommute amongst themselves. By anticommute we mean that they pick up a minus sign when we exchange the order they appear in expressions. In particular we have some coordinates that square to zero!

Supermanifolds play the role of manifolds when, for example, fermions such as the electron are present in the theory. If we want to develop a `classical’ theory of fermions then we must employ objects that anticommute: one can justify this using the Pauli exclusion principle — no more than one fermion can be in a given quantum state, while for bosons there is no such restriction. Heuristically, one can say that bosons like to be together, while fermions are rather more like hermits.

Supermanifolds offer a conceptual and geometric way to treat bosons and fermions on equal footing: supermanifolds are the geometry of supersymmetry. In short, supersymmetry is an operation that allows us to `rotate’ a boson into a fermion and vice versa. It turns out that this is not just a neat way of unifying bosons and fermions, but theories that posses supersymmetry can have remarkable mathematical and phenomenological properties — we await CERN’s confirmation that nature uses supersymmetry!

Another amazing link between geometry and physics can be found in mirror symmetry which relates pairs of particular manifolds called Calabi-Yau manifolds. Superstring theory is 10 dimensional, yet our physical world appears 4 dimensional — one time and three space dimensions. To overcome this discrepancy one can postulate that 6 of these dimensions is `scrunched up tightly’, and all we see is four dimensions on all but the very smallest scales. These compactifications as they are known, are Calabi-Yau manifolds, and different compactifications in general lead to different physics. However, it was noticed in the late 1980s by Dixon, Lerche, Vafa, and Warner that two different versions of superstring theory (type IIA and IIB) can be compactified on two different Calabi-Yau manifolds, yet lead to the same physics. In this case the two Calabi-Yau manifolds are said to be mirror duals, and the symmetry between the physics is known as mirror symmetry. This pairing of Calabi-Yau manifolds is now an active area of mathematical research with much effort devoted to carefully understanding the intuitive physics based picture.

In conclusion, not only has geometry been essential in developing physical theories, but these theories then push our understanding of geometry and lead to new mathematics. I have only touched upon a tiny part of this interrelation. There are a great number of other things I could have described and new links are being uncovered all the time. What will future mathematicians understand by the term `geometry’ is anyone’s guess. However, I am sure it will be closely related to our understanding of the physical Universe.

Trouble at the maths department at Leicester

This email made its way to me.

——————————————-

Twenty four members of the Department of Mathematics at the University of Leicester – the great majority of the members of the department – have been informed that their post is at risk of redundancy, and will have to reapply for their positions by the end of September. Only eighteen of those applying will be re-appointed (and some of those have been changed to purely teaching positions). This is supposedly because of a financial crisis at the University, though the union disputes this claim. It should be noted that there is no formal tenure in the UK, but such mass redundancies are highly unusual.

You can add your name to the online petition against this unusual
attempt at:

http://www2.le.ac.uk/institution/unions/ucu/news/no-redundancies-no-confidence

In this case it would be helpful to mention in the comments section that your signature is in support of the Mathematics Department (the petition is for the whole University, but apparently only the Math Dept has been formally notified of the redundancies at this stage).

You can also write directly to:

Professor Paul Boyle
President and Vice-Chancellor
University of Leicester
University Road,
Leicester, LE1 7RH,
United Kingdom

All mathematics is already known?

An old friend of mine, who went into teaching chemistry at high school, was surprised that I am involved in mathematics research, or more correctly that anyone is.

Paraphrasing what he said:

Surely all mathematics was worked out and finalised years ago?

I think he was willing to accept that there are still some classical open problems, but essentially he thought that mathematics was now ‘done and dusted’.

Of course this cannot be the case, as evidence I offer all the preprints that appear on the arXiv everyday. Mathematics departments are not full of people who just teach linear algebra and calculus to engineering students! I also submit that my boss Prof. Grabowski would be wondering what I am doing day in day out!

But why would he think mathematics research is over?

High School Mathematics
I think this belief stems from mathematics teaching in schools. Let me explain…

Let us start with physics and science in general. Students and the public at large know that scientists are working on open problems and discovering new things. For example we hear about new materials (eg. graphene); we know that the likes of Hawking are wrestling with the theory of black holes; we see images of all kinds of things in observational cosmology; we hear about medical scientists working on cancer cures; biologist discovering new species can make the news; CERN discovered the higgs boson…

High school students are aware that science is far from over and the syllabus for A-level physics is periodically updated to reflect some of these new discoveries.

But what about mathematics?

Linear algebra first emerged in 1693 with the work of Leibniz. By about 1900 all the main ingredients were know, so vectors have a modern treatment by 1900. This is all quite dated, but some open questions remain (for example in relation to quantum information theory).

Quadratic functions were solved by Euclid (circa 300 BC) and ‘the formula’ was known to Brahmagupta by 628 AD.

Calculus the foundations are from the 17th century in the works of Newton and Leibniz.

Plane geometry goes back to 300 BC and Euclid. Coordinate geometry is due to Descartes in the 1600’s.

Probability theory has it origins in Cardano’s work in the 16th century. Fermat and Pascal in the 17th century also made fundamental advances here.

Logarithms and exponentials in their modern form is due to Euler in the 18th century.

Trigonometry has roots going back to the Greek mathematicians from the 3rd century BC. Islamic mathematicians by the 10th century were using all six trigonometric functions.

So in sort, much of the typical pure mathematics syllabus at advanced level in high school is quite old. This I think, together with the ‘unchanging’ nature of mathematics (once proven a statement is always true) leads to the idea that it is all done already and nothing new can be discovered.

It also take from my friends question that he understood that the applications of mathematics are important and that plenty of work in applied mathematics is going on, for example in computational approaches to chemical dynamics. However, the ideas that mathematics as mathematics is finished remained.

For me personally, these applications of mathematics can lead to new structures in mathematics and this is worth studying. Indeed much of my professional work is in studying geometries inspired by applications in physics, particularly mechanics and field theory.

What can be done?
The ‘unchanging’ nature of mathematics is hard to get around. In science some new evidence could come to light and change our views. Indeed the scientific method is an integral part of teaching physics at advanced level in the UK. This ‘flexibility’ of science to adapt is important in student understanding of the philosophy of science.

So, we could try to promote new discoveries in mathematics to the general public, including high school students. The problem is that the background needed to understand the questions, let alone have any idea about the solutions prevents wide public engagement. Astronomers are lucky, we have all seen stars in the sky and can admire nice pictures!

Trying to start at a much higher level of mathematics would be futile, given the prerequisites that are needed. Moreover, most students will not become researchers in mathematics and will only need to be comfortable applying basic mathematics to their later field of study and work.

In short I have no idea how to promote the idea that mathematics research is not over, but please take my word it is not over!

An interview with Prof. Christopher Lintott

Prof. Christopher Lintott of Oxford University, winner of this years Kelvin Medal and Prize from the Institute of Physics and regular on the BBC’s Sky at Night agreed to answer a few questions.

Science and Popularisation

1. What first got you involved in science, and in particular astronomy?

I was a small kid who loved looking through telescopes – first of all a neighbour’s small reflector, then a larger telescope at school. I loved the idea that we could understand what’s happening in space despite being stuck on the surface of a small insignificant planet – and that there was lots left for us still to find out.

2. What was your first telescope?

The same one I have now, a 6” reflector. It’s nothing fancy – it doesn’t even have a motor – but it allows me to explore the sky. I’m a great fan of astrophotography, but I spend too much time looking at my computer as it is. When I’m observing I want the photons to be hitting my eyeballs!

3. How did you get involved in the BBC’s Sky at Night?

I’d been doing some science writing and got invited to be a guest on the show. From there, I was lucky enough to be part of the team and I gradually did more and more. I think Sky at Night’s a wonderful show, with the chance to explore so many fascinating aspects of our relationship with the Universe.

4. What is ‘Citizen Science’?

It’s a modern term for an old idea, which is that anyone can participate in the scientific process. These days, we use the term to cover the kind of projects we build on Zooniverse.org – projects which allows professional astronomers and volunteers together to comb through the vast stores of data which modern surveys produce.

5. Which medium do you think is the most effective at popularising science?

It depends what you’re trying to do, but one of the things that I think we need to remember is that we can’t rely on people choosing to seek out scientific content. A large proportion of the public have been put off science through experiences at school, or through a lack of confidence, and we need to find ways to reach them. In the old days, that meant big budget TV shows, but now that audience is fragmenting we need to find new ways for people to stumble across science. As an example, the Adler Planetarium in Chicago runs a Telescopes in the City program in which they take scopes (and astronomers) to random locations, surprising people with the sky. I think that kind of experience can be life-changing.

6. What, in your opinion, should be the ultimate goal of science popularisation?

I’m not sure it’s an ultimate goal, but there are lots of people who I believe would enjoy following science as it happens, and maybe even participating. I want that crowd to feel like they’re part of the journey, rather than just consumers of pre-packaged scientific results. We just reported on the New Horizons encounter with Pluto, which threw up all sorts of wonderful surprises. Someone said to me that they hadn’t realised that scientists smile when they say they don’t know something – I’d like more people to participate in the joy of not knowing.

Research

1. Can you say a few words about your research?

These days I’m interested in galaxies – in particular, we’re trying to find out why some galaxies form stars and why some appear to have shut down. Most of this work is done with the wonderful data provided by Galaxy Zoo volunteers.

2. Which one of your papers are you most proud of, and why?

The discovery paper for the Voorwerp – I had to learn a lot to write it, and we had the most tremendous battles with the referee, but it came out well in the end. Plus it’s about a wonderful object, and consisted of what I thought astronomers did when I was a kid. Find an interesting object, and point telescopes at it until you know what it is…

Links
Chris Lintott’s homepage at Oxford
BBC Sky at Night
Zooniverse.org

One nation science

The UK Science Minister Jo Johnson wants to reshape the funding of science and make sure that the funds are distributed wider.

Currently the ‘Golden Triangle’, Oxford, Cambridge and London’s UCL, Imperial and King’s receive 35% of the total £2.7bn annual funding.

I think that spreading the money could be a good idea, though it has to be done carefully and objectively.

Link
Science minister signals shift towards ‘one nation science’ BBC News

Publishing negative results

Recently the journal New Negatives in Plant Science, was launched with the aim of publishing negative, unexpected or controversial results in the field plant biology this.

This journal is aimed at plant science, but I have always thought that some kind of journal in mathematics that presents results that are ‘close but no cigar’ could be useful; for example one could present results of things that at first look should work, but do not. (Everybody’s note book is full of such things!) However, no-one would want to publish results that are not correct. The only way I can see to turn this around is to develop ‘no-go theorems’.

By ‘no-go theorems’ I mean clear mathematical reason why something the community expected to work does not. Such theorems are usually to be found in theoretical physics, but they can appear in pure mathematics also.

Such concrete statements are of course published in standard journals. Examples that spring to my mind are the Weinberg–Witten theorem, Coleman–Mandula theorem and the no-cloning theorem. Plenty of other examples exist.

Link
Why Science Needs to Publish Negative Results

John Nash killed in car crash

US mathematician John Nash was killed along side his wife in a taxi crash in New Jersey. Nash is best known for his work in game theory which lead him to be awarded the 1994 Nobel Prize for Economics. He is also more popularly known for the man who inspired the film ‘A Beautiful Mind’.

Nash battled with schizophrenia along side mathematics.

I know Nash’s work in differential geometry; his famous theorem states ‘every Riemannian manifold can be isometrically embedded into some Euclidean space’.

Our thoughts are with his friends and family.

Link
‘Beautiful Mind’ mathematician John Nash killed in crash

Plaque for Sir William Grove

Grove A plaque will be placed in Swansea’s Grove Place to commemorate the 19th Century scientist Sir William Grove.

Sir William, was the founder of the Swansea Literary and Philosophical Society, and managed to combine a legal career with several important scientific achievements. In particular he anticipated the conservation of energy and was a pioneer of fuel cell technology. He was the first to produce electrical energy by combining hydrogen and oxygen in 1842, a technology that went on to supply water and electricity for space missions.

Just another example of a Welsh person having done great things for the world.

Links
William Robert Grove, Wikipedia.

Blue plaque for Swansea scientist Sir William Grove, BBC News.

My Birthday

cake I did not realise this until today, but I share my birthday with John Wallis (23 November 1616 – 28 October 1703). Wallis made contributions to infinitesimal calculus, analytic geometry, algebra and the theory of colliding bodies. He is best known for the infinity symbol \(\infty\).

Wallis was also a code breaker and used his mathematical skills during the Civil war to decode Royalist messages for the Parliamentarians.

Link
John Wallis Wikipedia.

Engineers should embrace their artistic side.

? More warnings about a shortage of STEM graduates. Where are these messages coming from? There is no shortage, many graduates in engineering for example are forced to work in jobs that are nothing to do with their degrees, or even worse they are in non-graduate jobs.

Engineering needs to emphasise its creative side to encourage more young people to take it up as a career.

Sir John O’Reilly

This is fine. The fields of science and engineering do require some creative thinking. There is an element are art to this.

But I am worried about the continual message of a lack of STEM graduates where in reality many are unemployed. I really don’t understand the B.S. here.

Link
‘Embrace engineering’s creative side’ to fix skills crisis