The first examples sheet is now ready, either as a dvi file or as a pdf .

And here is the second .

And the third .

And the fourth .

A few years ago I wrote several of these, for a variety of related reasons. One was to provide a more thorough discussion of definitions and basic results than I could normally hope to give in a lecture course from the Cambridge Mathematical Tripos . Another was to try to indicate, in the spirit of George Polya , how certain well-known proofs and definitions might have been discovered by anybody with just a few basic mathematical instincts. A general index to these discussions can be found here .

This is the title of a book that came out in 2002, which can be ordered from www.amazon.co.uk or directly from the publisher . It is available in the USA as well, and is now in stock at Amazon or Barnes and Noble . If you have read the book and would like to read similar material on this site, then I have some suggestions .

This is a book I am editing with the help of June Barrow-Green
and Imre Leader, which could be thought of as "Mathematics: A Very
Long Introduction". It has the aim of being a genuinely useful reference
work in mathematics. This is a difficult aim, since mathematics is
hard enough to explain at the best of times, and even more so if
one has limited space. Is there any point in trying to summarize
algebraic geometry in ten pages, for example? Probably not, but
the articles in the PCM don't try to * summarize *, so
much as to provide an initial overview and perspective. I like
to think of them as "prequels" to textbooks -- things you would
read to get an idea of why you were bothering to learn some concept
that your lecturer seems to take for granted is interesting.
Strenuous efforts have gone into making the book as accessible
as possible, which I hope will have been worth it when it comes
out, all going well, some time in the first half of 2008.

There is a website where one can find out much more about the book. A link to that can be found in a blog that I've just started up, where you can, if you want, give your feedback, both before and after publication.

This page is intended to provide modest back-up for Numbers and Sets, which I lectured from 2002-2004.

Just in case they are of any use, here are some examples sheets for this course, as I gave it in 2004 and 2005. They include a revision sheet for Easter term supervisions, which may still be useful as there are not yet that many Tripos questions on this course.

Sheet | Download |

1 | dvi,pdf |

2 | dvi,pdf |

3 | dvi,pdf |

4 | dvi,pdf |

Revision | dvi, pdf |

I have proposed this course for the academic year 2006-7. The syllabus is Roth's theorem, the geometry of numbers, Freiman's theorem, quasirandomness of graphs and 3-uniform hypergraphs, and Szemerédi's regularity lemma. The first three topics can be found in chapters 2, 5 and 6, respectively, of these notes on additive number theory (or you can try a ps version ) and the last three can be found in sections 1-4 and 7 of this paper (which also comes in a pdf version ). The course will be examined as a 24-lecture course, so it will have a three-hour exam. Links to examples sheets with many relevant questions can be found in the next couple of sections of this page.

I have an examples sheet for this course, which I gave in 2005, in either a pdf version or a dvi version .

Here are some question sheets from a course I have given a couple of times. I leave them here because some people have found them helpful for their understanding of this subject. They come with a health warning: sometimes when I invent a question it turns out to be trivial, false, or hard enough to count as a research problem. (In fact, one of them did end up as a published theorem of Green and Konyagin.) Also, some questions refer to particular comments or proofs from the lectures. I have not carefully gone through these sheets to weed out such questions.

Sheet | Download |

1 | dvi,pdf |

2 | dvi,pdf |

3 | dvi,pdf |

You can click here for some links, mostly to interesting home pages of mathematicians.

This sentence is here to provide a convenient route to DPMMS and the University of Cambridge .

If you want to earn a million dollars, then as a preliminary step you could try visiting this site .

I recommend this collection of reflections on miscellaneous mathematical topics by Kevin Brown. These are somewhat similar in spirit to my `informal discussions', but they are far more numerous, and on average shorter. John Baez has also written several online expository articles in mathematics and physics, including a very clear discussion of octonions.

This Internet Mathematics Library has a huge collection of links to mathematics-related websites. The links are arranged by topic and the sites are briefly described.

Here is a useful and well-organized site on the history of mathematics , which includes biographies of a large number of mathematicians.

If you want to find out a British telephone number, then these directory enquiries are free.

I have a few papers available online, and will add to them in due course. They include a preprint on the general case of Szemerédi's theorem, which recently appeared in GAFA, and another on a Banach-space dichotomy of mine, which has been accepted for publication and should appear reasonably soon. Any comments would be most welcome - I do not regard a paper as necessarily having reached its final form once it appears in a journal. There are also some survey articles and a videoed lecture, in which I explain my general attitude to mathematics.

A few years ago I gave a Part III course which included a section on the K-theory of Banach algebras, for which I produced a set of printed notes . I found Blackadar's book too compressed and Wegge-Olsen's not compressed enough, and was aiming at something like the geometric mean. What I produced has its faults, but may be useful when read in conjunction with those books. I should mention also that I gained greatly from reading some (I think still unpublished) notes of Bernard Maurey on this and related topics.

Similarly, I produced printed notes on Vinogradov's three-primes theorem for a Part III course last year. Again, I was aiming at something between two existing treatments: this time I found Vaughan (The Hardy-Littlewood Method - CUP) too compressed and Nathanson (Additive Number Theory Vol. I - Springer) not compressed enough, though this did not stop me finding both books very useful. As will be clear to anybody who reads them, my notes were aimed at those who had attended the lectures. I hope one day to rewrite them, but even in their current state they are tidy enough to make publicly available.

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