WHAT'S NEW?

FEB 01 2023 I have finally completed my reference work, GROUP TABLES. This gives detailed information about all finite groups of order up to 100 (excluding orders 64 and 96.

SEP 29 2022: I have revised the notes on Axiomatic Set Theory

I have revised Geometry, and renamed it Geometry vol 2. I have written a completely fresh set of notes, called Geometry vol 1. This is at the Second Year level and covers Euclidean Geometry, going beyond what is done at school. It contains the coordinate geometry of the line, circle, parabola, ellipse and hyperbola. The ruler and compass constructions have been transferred from volume 2 to volume 1. This now includes a brief description of string and pins constructions. It is expected that volume 1 will be extended in later editions, and will include Exercises and solutions. It is also intended to have some chapters on spherical geometry in a future edition.

I have revised both Topology and Galois Theory, correcting a number of layout problems with the diagrams.

Earlier in the year I did the following:

(1) I have completely revised the three elementary sets of notes, Basic Maths, Concepts of Algebra and Concepts of Calculus. With the help of my good friend and colleague, Liz Richards, I have made numerous corrections, and improved my explanations in many places.

(2) I have extended the chapter on Geometry, in my notes on Set Theory. In these notes I aim to set up all of mathematics entirely within set theory, assuming only the ZF axioms. Having developed the real and complex numbers from these axioms I define the Euclidean plane as a 2 dimensional vector space over the reals. I naively thought that all I had to do was to prove Euclid's axioms and his exposition in Elements would do the rest. But I soon realised that it isn't so easy. Euclid, careful and rigorous as he was, glosses over certain concepts and relies on intuition in many places. He is very vague about 'angle', defining it as 'the inclination of one line with another'. I think I have fixed this up by identifying the 2 dimeansional real vector space with the field of complex numbers, using the argument of a complex number. (All the trigonometry that I need can be set up using power series, which can be developed within set theory.) But I ground to a halt with 'area'. Euclid doesn't define area, but yet he has (Proposition 38), 'triangles a on the same base and between the same parallels are equal to one another'. Here 'equal' doesn't just mean 'similar' but rather 'equal in area'. Now I don't expect to define areas in general. But I believe that defining area as an integral might involve some circular reasoning and I certainly don't want to have to invoke measure theory! All I want is a definition of the area of any polygon. Yes I can define the area of a right-angled triangle in the obvious way, and divide a polygon into right-angled triangle, defining the area of the polygon to be the sum of the areas of the right-angled triangles. I think I could define such a dissection within set theory. My difficulty is in proving that area defined in this way is well-defined with the area being independent of the traingulation. Can anyone help me?

(3) I have started work on a second volume of geometry. The previous volume (projective geometry, isometries, symmetry and ruler and compass constructibility) is now Geometry volume 2. Volume 1 will consist of plane Euclidean geometry (lines, triangles, parallelograms and circles), coordinate geometry of the parabola, ellipse and hyperbola, and spherical geometry. Of course I don't begin the Euclidean geometry in this volume by attempting to set it up within set theory, though I briefly mention Euclid's Elements. In this volume anything goes! In proving the theorems I use whatever is simplest: a combination of coordinate geometry, vector algebra and complex numbers, as well as the usual methods of Euclid.

(4) I have revised the chapter on the Todd-Coxeter Algorithm in Group Theory volume 1 at the suggestion of my colleague Ross Moore. Although I believe my version, building up and contracting chains, is easier to understand than that in the original paper, it is rather unwieldy in large examples. So, after describing the algorithm, and doing a few small examples, I introduce the 'compact' version', which is pretty much how Todd and Coxeter set it out in their paper. The two versions are essentially the same algorithm, with the compact version requiring less writing.

(5) I have added some material to the chapter on power automorphisms in Group Theory volume 2, as in my paper from the 1970s, where I obtain the characterisation of Hamiltonian groups using power automorphisms.

(7) I have begun work on a new set of notes, Complex Variables. I believe that there are two areas of advanced mathematics that are so aesthetically perfect that all mathematic students should encounter them, irrespective of whether they will ever use this material. They are the theory of characters of a finite group over the complex numbers and the theory of complex variables.

So now I have three sets of notes under construction: Graph Theory, Geometry vol 1 and Complex Variables. I hope to complete these by the end of 2022.