Table of Contents
Foreword
If you're looking for my professional Web site, it is
elsewhere.
As the title indicates, this is my “old” math page. It is
a repertoire of various mathematical texts I wrote when I was student
(at the ENS essentially,
or before), which I keep for its “historical” interest (some people
find certain things on this page interesting for various reasons, so I
don't want to simply remove it from the Web). This is not my
current mathematical research, only old stuff: for my publication list
and whatnot, please see
my professional Web site
(which is also not up to date, unfortunately).
For the same reason, I am generally not interested in possible
errors in these papers: I know there are some, but this is old stuff,
and the style of TeX and the mode of compilation is such that it would
be very inconvenient for me to make any changes now (in a few cases, I
may even have lost the source).
There are more math
pages on this site, some being possibly more up to date than this
one (such as the math posts
on my blog).
Research
See my professional Web
site.
Dissertations
My doctoral dissertation, done
under Jean-Louis
Colliot-Thélène's supervision,
entitled Hypersurfaces
cubiques : équivalence rationnelle, R-équivalence et approximation
faible (pdf)
My magistère dissertation includes a
copy of both my master's dissertation and my DEA dissertation (below).
It also contains a simplified presentation of the latter as well as a
few notes on intuitionist logic as a kind of summary of the workshop I organised on the subject last
year. (ps (compressed); French and
English)
My DEA dissertation, done under Michel
Raynaud's (michel![[dot]](../images/full_stop.png)
raynaud![[at sign]](../images/commercial_at.png)
mathu-psudfr)
supervision, in which I prove two results of Raynaud's on theta
divisors, and ordinarity of curves and coverings. Actually, this
paper is not strictly my DEA dissertation, but the text of a talk I
gave a month later at the CIRM in Luminy on the same subject. (LaTeX, dvi or ps (compressed); English)
My master's dissertation, done jointly with
Jean Marot (marotquatramaranensfr),
under Yves Laszlo's (laszlomathpolytechniquefr)
supervision on Belyj's theorem (concerning
étale coverings of the projective line minus three points).
(ps; French)
- The text of a talk I gave during Professor Illusie's algebraic
geometry course on associated prime ideals and primary
decomposition. (dvi or ps (compressed); French)
- The text of a talk I gave during Vincent Lafforgue's introduction
to Lie groups, on tensor products and representations of sl2(C)
(dvi or ps
(compressed); French), and the initial (much more complete)
preversion of the same (dvi or ps (compressed); French).
I have organized at the ENS a workshop on
logic, set theory and forcing (both classical and
intuitionist) during the 1998–1999 term. Here is the list of
talks that took place:
- Prolegomena (by myself). Available (dvi or ps
(compressed); French).
- The Constructible Universe (by Itaï Ben-Yaacov).
Not available.
- Large Cardinals (by Benoît Collins).
Available (dvi or ps (compressed); French).
- Introduction to categorical logic (by myself). Available (dvi or ps
(compressed); French).
- Classical Forcing àla Cohen (by Jean Marot).
Please refer to Jech's book.
- Topos semantics and intuitionist forcing (by
Frédéric Déglise).
No notes were written but my magistère dissertation (see above) might play that role; otherwise refer
to MacLane's book.
Thoughts
Starting at the end of 2001, I have been keeping a mathematical
diary with various (cross-referenced) thoughts on various questions.
You can now download it (dvi, pdf or ps (compressed);
English)
I have written a rather long sequence of thoughts on the
hypothetical theory of noncommutative algebraic geometry. Now I don't
agree with all these thoughts any more (and in particular I think that
my emphasis on a non-symmetry-breaking way of doing things is stupid
— though I still agree with what I said about the category of
rings having unsatisfactory colimits). Anyway, here are most of these
thoughts (in order of decreasing interest):
Also related to noncommutative algebraic geometry is discussion thread which
occurred on the ENS local newsgroup, and in one message of
which I presented several of my ideas on the subject.
A newer programme for
algebraic geometry was posted more recently to that same
newsgroup.
I have amused part of Orsay's algebraic geometry department
with my thoughts on placid schemes (dvi or ps
(compressed); French).
Trying to make some sense of categorical logic (dvi or ps (compressed);
French).
Here are now some thoughts on the philosophy of
mathematics and the notion of Truth: (dvi or ps
(compressed); English).
And, last but not least, I have discovered the meaning of
Life, the Universe, and Everything (dvi or ps
(compressed); English).
Some posts I made to the students' forum (local newsgroup) of the ENS,
included in my Best Of
are of mathematical interest (note that all are in
French):
- A rant about
algebraic geometry.
- Some comments on the
EPR paradox and “negative probabilities”.
- A rant about
cohomology.
- Constructing a
certain function.
- Srinivasa Ramanujan in memoriam:
concerning the number 1729.
- Remarks on
transitive group actions.
- A programme for
algebraic geometry that is most interesting to read if you have
already read my magistère dissertation (see above), and that also bears some relation
with noncommutative algebraic geometry.
- Why the plane
cannot be written as a union of closed disks of positive radius.
- A simple
explanation of what an “ample invertible sheaf” is.
- Concerning the
axiom of regularity.
- Elementary remarks
on the plane projections of Lobachevskian geometry.
Didactic
Graduate level
- A (50-page long) introduction to the theory of categories
(dvi or ps
(compressed); English), and a former version of the same (dvi or ps
(compressed); English).
- Some notes on game theory (note that I wrote this
before I read Conway's works on the subject) (dvi or ps (compressed);
English).
Undergraduate level
- A proof of the rational linear independence of the square roots
of the square-free integers (dvi or ps (compressed); French).
- A text on the mathematical aspects of cartography
(pdf or ps
(compressed); French).
Beginners
- A page about pitfalls in probability theory (html; French).
- An introduction to infinity in mathematics
(supersedes or complements the next item) which I gave as a talk in
the lycée Saint Louis (dvi, pdf or ps;
French).
- An introduction to ordinal numbers, which was
published in the journal “VIRUS” of the lycée Louis
le Grand (html or
ps; French).
- An introduction to p-adic numbers,
which I wrote for the readers of the sci.math newsgroup (revised 2000/12/07; dvi, pdf or ps; English).
Old stuff
What follows was written a long time ago (before I entered the ENS,
essentially), and may be hopelessly wrong, out of date, or anything
like that: caveat sumptor.
Miscellaneous
- The first Dynkin diagrams (ps).
- Various tables of nonabelian simple groups (html).
- Some variants of the axiom of choice (html; French).
Khôlles
If you don't know about the French educational system, the above word
means “glue” (with a very strange spelling —
something like “ghloo”) and refers to an oral
interrogation. Anyway, I used to be a “khôlleur”
and you can find the list of exercices I gave on the French version of this page (they are all in
French, naturally).
DEUG MIAS
The same comment as above applies here. DEUG
refers to the first two years of University (i.e. undergraduate years)
in France; MIAS means “Mathematics, Informatics (i.e. computer
science) and Applications to Sciences”. Since september 2000, I
teach in DEUG MIAS. I have written a few complete solutions to
various exercices: see the French version
of this page.