Geometry, Topology and Physics by M. Nakahara

nakaharabookGeometry, Topology and Physics Geometry and topology have become integral in the theoretical physicists tool kit. Ideas from geometry and topology are now fundamental in condensed matter physics, gravitational physics and particle theory.

Nakahara’s book Geometry, Topology and Physics provides an assessable introduction to these ideas.


The book is a massive expansion and revision of lectures given by Nakahara at the School of Mathematical and Physical Sciences, the University of Sussex back in 1986.

Chapters 1 and 2 provide a basic review of the physical and mathematical ideas need as a preliminary to the rest of the book. Most of this is familiar territory to a beginning graduate student in mathematical or theoretical physics.

Chapters 3 to 8 introduce some fundamental ideas in geometry and topology: homology groups, homotopy groups, manifolds, de Rham cohomology, Riemannian geometry and complex manifolds. Examples of the applications of these constructions in physics are given throughout these chapters.

Chapters 9 to 12 cover the topics of fibre bundles, connections & curvature, characteristic classes and the Atiyah-Singer index theorem. All these topics are fundamental in understanding gauge theories that form the backbone of the standard model.

The final chapters 13 and 14 describe anomalies in gauge theories and the bosonic string respectively. In a sense, these chapters bring all the ideas laid out earlier in the book together to describe some quite advanced mathematical physics.

Paperback: 596 pages
Publisher: Taylor & Francis; 2 edition (4 Jun 2003)
Language English
ISBN-10: 0750306068
ISBN-13: 978-0750306065

Higher contact-like structures and supersymmetry

In my latest preprint “Higher contact-like structures and supersymmetry” I provide a novel geometric view of N=1 supersymmety in terms of a polycontact structure on superspace. The preprint can be found at arXiv:1201.4289v1 [math-ph]

The conception of the idea to describe supersymmetry in terms of some contact-like structure came from understanding SUSY mechanics in terms of a contact structure. See my preprint “Contact structures and supersymmetric mechanics” arXiv:1108.5291v2 [math-ph] and an earlier blog entry here.

Young Researchers in Mathematics Conference 2012

Royal Fort House, University of BristolRoyal Fort House, University of Bristol. Picture courtesy of the YRM 2012 committee.

The Young Researchers in Mathematics Conference is an annual event that aims to involve post-graduate and post-doctoral students at every level. It is a chance to meet and discuss research and ideas with other students from across the country.


I will be attending the Young Researchers in Mathematics Conference 2012 to be held at Bristol University 2nd-4th April.  I have offered to give a talk and right now awaiting confirmation that my talk has been accepted. My talk would fit into the Geometry and Topology tract.


I will post more details in due course.








Mathematics the langauge of Physics

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

What is wrong with engineers?

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.”