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<H1 align=center> Timothy Gowers</H1>
<HR>

<H3> Part IA Probability, Lent 2008 </H3>

<P> The first examples sheet is now ready, either as a 
<A HREF="probex.2008.1.dvi"> dvi file </A> or as a
<A HREF="probex.2008.1.pdf"> pdf </A>. </P>

<P> And here is <A HREF="probex.2008.2.pdf"> the second </A>. </P>

<P> And <A HREF="probex.2008.3.pdf"> the third </A>. </P>

<P> And <A HREF="probex.2008.4.pdf"> the fourth </A>. </P>

<HR>

<H3> Informal discussions of mathematical topics </H3>

<P> 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 
<A HREF="http://www.maths.cam.ac.uk/undergrad/"> Cambridge 
Mathematical Tripos </A>. Another was to try to indicate,
in the spirit of <A HREF="http://www.math.twsu.edu/history/Men/polya.html">
George Polya </A>, 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 <A HREF="mathsindex.html"> 
here </A>. </P>

<HR>

<H3> Mathematics: A Very Short Introduction </H3>

<P> This is the title of a book that came out in 2002, which can
be ordered from <A HREF=
"http://www.amazon.co.uk/exec/obidos/ASIN/0192853619/qid=1031041815/sr=2-5/ref=sr_2_3_5/026-4472227-0791659"> www.amazon.co.uk </A> or 
<A HREF="http://www.oup.co.uk/isbn/0-19-285361-9"> directly from
the publisher </A>. It is available 
<A HREF="http://www.oup-usa.org/isbn/0192853619.html"> in the
USA </A> as well, and is now in stock at 
<A HREF="http://www.amazon.com/exec/obidos/ASIN/0192853619/qid=1033730307/sr=2-1/ref=sr_2_1/103-8197972-7070263"> Amazon </A> or
<A HREF="http://search.barnesandnoble.com/booksearch/isbnInquiry.asp?userid=2PBTV6NGBW&isbn=0192853619"> Barnes and Noble </A>. If you have read the 
book and would like to read similar material on this site, then I have some
<A HREF="vsipage.html"> suggestions </A>. </P>

<HR>

<H3> The Princeton Companion to Mathematics </H3>

<P> 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 <em> summarize </em>, 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.</P>

<P> There is a website where one can find out much more about the
book. A link to that can be found in a <A
HREF="http://gowers.wordpress.com"> blog </A> that I've just started
up, where you can, if you want, give your feedback, both before and
after publication. </P>

<HR>

<H3> <A HREF="naspage.html"> Part IA Numbers and Sets </A> </H3>

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

<HR>

<H3> Part II Topics in Analysis </H3>

<P> 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. </P>
<table BORDER WIDTH="0%" >
<tr>
<td>Sheet</td>

<td>Download</td>
</tr>

<tr> 
<td> 1 </td>
<td><A HREF="analtopics1.dvi">dvi</A>,<A HREF="analtopics1.pdf">pdf</A>
</td>
</tr>

<tr> 
<td> 2 </td>
<td><A HREF="analtopics2.dvi">dvi</A>,<A HREF="analtopics2.pdf">pdf</A>
</td>
</tr>

<tr> 
<td> 3 </td>
<td><A HREF="analtopics3.dvi">dvi</A>,<A HREF="analtopics3.pdf">pdf</A>
</td>
</tr>

<tr> 
<td> 4 </td>
<td><A HREF="analtopics4.dvi">dvi</A>,<A HREF="analtopics4.pdf">pdf</A>
</td>
</tr>

<tr> 
<td> Revision </td>
<td><A HREF="analtopics.extra.dvi">dvi</A>,
<A HREF="analtopics.extra.pdf">pdf</A>
</td>
</tr>

</table>

<HR>

<H3> Part III Reading Course: Asymptotic structure and Quasirandomness. </H3>

<P> 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
Szemer&eacute;di's regularity lemma. The first three topics can be
found in chapters 2, 5 and 6, respectively, of <A
HREF="addnoth.notes.dvi"> these notes on additive number theory </A>
(or you can try a <A HREF="addnoth.notes.ps"> ps version </A>) and the
last three can be found in sections 1-4 and 7 of <A
HREF="belapaper.dvi"> this paper </A> (which also comes in a <A
HREF="belapaper.pdf"> pdf version </A>). 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. </P>

<HR>

<H3> Part III Quasirandomness. </H3>

<P> I have an examples sheet for this course, which I gave in 2005, in
either a <A HREF="quasirandomex1.pdf"> pdf version </A> or a <A
HREF="quasirandomex1.dvi"> dvi version </A>. </P>

<HR>

<H3> Part III Additive and Combinatorial Number Theory </H3>

<P> 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. </P>

<P> <table BORDER WIDTH="0%" >
<tr>
<td>Sheet</td>

<td>Download</td>
</tr>

<tr> 
<td> 1 </td>
<td><A HREF="addnothex031.dvi">dvi</A>,<A HREF="addnothex031.pdf">pdf</A>
</td>
</tr>

<tr> 
<td> 2 </td>
<td><A HREF="addnothex032.dvi">dvi</A>,<A HREF="addnothex032.pdf">pdf</A>
</td>
</tr>

<tr> 
<td> 3 </td>
<td><A HREF="addnothex033.dvi">dvi</A>,<A HREF="addnothex033.pdf">pdf</A>
</td>
</tr>

</table>

<HR>

<H3> Links </H3>

<P> You can click <A HREF="links.html"> here</A> for some links, 
mostly to interesting home pages of mathematicians. </P>

<P> This sentence is here to provide a convenient route to
<A HREF="http://www.dpmms.cam.ac.uk/"> DPMMS </A> and the
<A HREF="http://www.cam.ac.uk/"> University of Cambridge </A>. </P>

<P> If you want to earn a million dollars, then as a preliminary
step you could try visiting 
<A HREF="http://www.claymath.org/millennium/"> this 
site </A>. </P>

<P> I recommend <A HREF="http://www.mathpages.com/home/"> this 
collection </A> 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. <A HREF="http://math.ucr.edu/home/baez/"> John
Baez </A> has also written several online expository articles in 
mathematics and physics, including a very clear discussion of
octonions. </P>

<P> This <A HREF="http://mathforum.org/library/"> Internet
Mathematics Library </A> has a huge collection of links to 
mathematics-related websites. The links are arranged by topic
and the sites are briefly described. </P> 

<P> Here is a useful and well-organized site on the 
<A HREF="http://www-history.mcs.st-andrews.ac.uk/history/">
history of mathematics </A>, which includes biographies of
a large number of mathematicians. </P>

<P> If you want to find out a British telephone number, then
these <A HREF="http://www.bt.com/directory-enquiries/dq_home.jsp">
directory enquiries </A> are free. </P>

<HR>

<H3> Preprints and Papers </H3>

<P> I have a few <A HREF="papers.html"> papers </A> available online,
and will add to them in due course. They include a preprint on the
general case of Szemer&eacute;di'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 <A HREF="papers.html#surveys"> survey
articles </A> and a videoed lecture, in which I explain my general
attitude to mathematics. </P> <HR>

<H3> Course Notes </H3>

<P> A few years ago I gave a <A HREF="http://www.maths.cam.ac.uk/CASM/"> 
Part III </A> course which included a section on the K-theory of
Banach algebras, for which I produced a set of <A HREF="Ktheory.dvi">
printed notes </A>. 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. </P>

<P> Similarly, I produced printed notes on <A HREF="3primes.dvi">
Vinogradov's three-primes theorem </A> 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. </P>
<HR>

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