Friday, 12 October 2012

What Next after Topology

The question was asked in the Topology forums what happens next. What is the point of say Cauchy series and completeness and connectedness. Here is my personal take on where I want to go next.

The next step is to finally tackle functional analysis this is a combination of metric space theory and linear algebra. The concept of a vector space of linear algebra is extended to spaces of functions, orthogonality being defined by an integral over the functions. The simplest example being the trig functions which satisfy

$$\int^{\pi}_{-\pi}sin(nx)sin(mx) dx = 0$$ .

if m is not equal to  n or $$\pi$$ if m = n

The Integral can be seen as a type of scalar product or Norm for the trig functions.
Thus after suitable normalisation the set of functions sin(mx) m = 0,1.... can be said to form a set of orthonormal functions. Also sin(mx) is an eigenfunction of the differential operator

$$\frac{d^2}{dx^2}$$ with eigenvalue $$-m^2$$ so that the solution space mapped out by the differential equation

$$\frac{d^2y}{dx^2}+ m^{2}y = 0$$ is precisely the set of orthonormal trig functions.

The generalisation of this to all sorts of differential operators and the abstract study of such relations forms a large part of functional analysis.

The OU MSc course does a course on this and the recommended text book is by Maddox which is out of print. I managed to get a copy from Amazon but it is really dry, more importantly from my own perspective, namely trying to understand the applications of Pure Maths to physics there is nothing on which most physicists would call the essential point namely the spectral theorem for both finite and unbounded operators. This is important for understanding quantum mechanics.

A more accessible book on Functional Analysis is Kreyszig

This has an introductory chapter on metric spaces, open sets compactness and convergence but also(eventually !)  proves the Spectral theorem for unbounded operators which is important in understanding quantum mechanics.

The definitive book on the applications to quantum mechanics being Von Neumann's book

But some understanding of Lebesgue Integration may be necessary the Old OU course used to be based on Weir

Unform convergence and Cauchy sequences as I discussed with Neil and the other Edinburgh M338 er's in the Diggers after the exam  play an important part of showing rigourously why Fourier Series converge. The MSc course in Approximation theory covers some of this. It's covered in Kreyszig as well.

Topology is of course used in General relativity especially the proof of the singularity theorems the definitive book on the subject being Hawking and Ellis.

However the book is extremely concise some familiarity with differential geometry in it's modern form is needed and an ideal book would seem to be the second edition of O'Neill's book on differential geometry

If that is considered to simple then either read chapter 2 of Hawking and Ellis or

A slightly more accessible account of the singularity theorems seems to be given by Wald

There appears to be an extension of the notion of compactness to differerntiable manifolds called para compactness
And this would appear to be an essential condition for defining a Riemannian Manifold.

Another area of study is the relationship between Lie Groups and Topology a book on which is given by Gilmore

And there are books on the application of Topology to physics the most accessible of which is probably by Nash and Sen

So plenty to build on from the topology course  is a stepping stone to understanding the pre-requisites to understand the pre-requisites for books like Hawking and Ellis or Von Neumann

I intend to tackle Kreyszig and O'Neill and possibly Weir next

In the mean time I shall have my pudding and get stuck into to some past papers for MST324 a weekend solving differential equations both partial and ordinary. Doing Fourier transforms and solving problems in Lagrangian Mechanics absolute Bliss. I hope I can get up to speed.

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