What's new:
What was new in 2006? 2005? 2004? 2003?
2002? 2001? 2000? 1999?
Feb 22, 2007
Feb 16, 2007
- Solved:
The puzzle described in “Squares, cubes, and smooth
functions” was already solved by Henri Joris (Arch. Math. (Basel) 39 (1982), no
3., 269-277). Thanks to Catalin
Bedea for the reference. I have
updated the article to reflect a proof of the puzzle inspired by that
paper.
Feb 13, 2007
Feb 7, 2007
- Uploaded:
“What is good
mathematics?”, submitted, Bull. Amer. Math. Soc. This (my first attempt at writing
meta-mathematics!) is a solicited article on the (possibly dangerous!)
topic of what constitutes good mathematics. My answer is a bit complicated: there
are many possible metrics to measure quality mathematics, but the relative
importance of these metrics depends on the state and nature of the field,
and evolves over time. Also, one
has to see how a piece of mathematics fits into the context of a larger
mathematical story. I illustrate
these points with the story of Szemeredi’s theorem.
- Uploaded:
The short story “Squares,
cubes, and smooth functions”.
This is a description of a cute little problem I heard from Brian
Conrad a while back: if f: R -> [0,+infty) is such that f^2 and f^3 is
smooth, is f itself smooth? I don’t
know the answer, but it ought to be decidable. I don’t foresee any major applications
of this little puzzle, but it was amusing enough to be worth sharing.
Jan 26, 2007
- Congratulations
to Tim Flannery
for being selected as 2007 Australian of
the Year. (I had a lot of fun
attending that event, even if a short and harmless conversation I had
there managed to make it to the local paper :-).)
Jan 21, 2007
- Uploaded:
“A note on the
Freiman and Balog-Szemeredi-Gowers theorems in finite fields”,
with Ben Green,
submitted, J. Aust. Math.
Soc. Here we improve the bounds
on the Freiman and Balog-Szemeredi-Gowers theorems (which describe the
behaviour of sets with small sumset, or sets with many additive
quadruples) in the model setting of the finite field geometry F_2^n. By using an iterative approach to
gradually improve the doubling constant or additive energy, until the sets
involved are “flat” enough for Fourier techniques to be
effective, we obtain improved exponential (but still non-polynomial,
unfortunately) bounds; more precisely, for sets A of doubling K, we obtain
a large intersection with a subspace of cardinality K^{-O(sqrt(K))} |A|,
and similarly for sets A with |A|^3/K additive quadruples we obtain a
large intersection with a subspace of cardinality K^{-O(K)} |A|. In a sequel to this paper we shall use
these results to obtain an asymptotic for the best way to cover a set of
small doubling in F_2^n by a subspace.
Jan 17, 2007
Jan 11, 2007
- Uploaded:
The short story “Ornstein’s
counterexample to an L^1 maximal inequality”. An old maximal inequality of Stein
asserts that the maximal operator sup_n P^n f is bounded on L^p for 1 <
p <= infinity whenever P is a self-adjoint Markov operator. It is natural to ask for the weak (1,1)
analogue of this result, but an example of Ornstein shows that it
fails. Here we present
Ornstein’s example, which is an interesting Markov process created
by designing the percolation probabilities in order to “delay”
the appearance certain portions of mass into a region until other portions
have “diffused”.
Jan 9, 2007
- Uploaded:
The short story “Wolff’s
proof of the corona theorem”.
These are my notes on Wolff’s proof of the celebrated Corona theorem of
Carleson, based on Gamelin’s exposition. I replaced some of the complex-analytic
arguments with real-analytic ones, and observed that by using
Uchiyama’s constructive version of the Fefferman-Stein
decomposition, the proof can in fact be made entirely constructive (e.g.
no appeal to Hahn-Banach is necessary).
But in essential details, the argument is Wolff’s.