[Une version française de cette page est également disponible.]

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).

See my professional Web site.

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 (`michelraynaudmathu-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.

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):

- Version 9 (dvi or ps (compressed); English)
- Version 11 (dvi or ps (compressed); English)
- Version 7 (dvi or ps (compressed); English)
- Version 4 (dvi or ps (compressed); English)

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.

- 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).

- 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).

- 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
, which I wrote for the readers of the sci.math newsgroup (revised 2000/12/07; dvi, pdf or ps; English).`p`-adic numbers

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*.

- A quick survey of the
**RSA cryptosystem**(dvi or ps; French) - Introduction to
**riemannian geometry and general relativity**(dvi or ps (compressed); French) - First introduction to
**algebraic geometry**(dvi or ps (compressed); French) - A brief tour of
**set theory**(dvi or ps (compressed); French) - A proof of the
**Banach-Tarski paradox**(dvi, pdf or ps (compressed); French) - Examples in
**algebraic number theory**(dvi or ps (compressed); French) - A little more about
**set theory**(dvi or ps (compressed); French) - 99 questions for young genii (dvi or ps (compressed); French) (this one contains a
*lot*of mistakes) - The list of
**Hilbert's 23 problems**(dvi or ps (compressed); French) - The answers of the 1995
**entrance exam to the École Normale Supérieure of Lyon**(dvi or ps (compressed); French)

- The first
**Dynkin diagrams**(ps). - Various tables of
**nonabelian simple groups**(html). - Some variants of the
**axiom of choice**(html; French).

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).

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.