# Celebrations in Sweden and Norway

### Celebrations for Endre, Jean and Terry

Anders Bjorner presents the 2012 Crafoord Prize in Mathematics

I am in Sweden for two weeks to work with colleagues and to take part in two celebrations. Jean Bourgain and Terence Tao are the 2012 laureates of the Crafoord Prize in mathematics which was awarded  last Tuesday at Lund. Along with them the 2012 Crafoord Prize in Astronomy was awarded to Reinhand Genzel and Andrea Ghez.  I took part in the symposium entitled “From chaos to harmony” to celebrate the event.

Next Friday the Swedish Royal academy will celebrate with a mini-symposium in honor of the 2012 Abel prize winner Endre Szeméredi. (Here are the slides of my future talk looking at and around the Szeméredi-Trotter theorem. Please alert me of mistakes if you see them.) The Abel prize symposium and ceremony in Oslo are  Tuesday (Today! see the picture above) and Wednesday of this week.

Congratulations again to Jean, Terry and Endre for richly deserved awards.

### Crafoord days at Lund

Owing to the passing of Count Carl Johan Bernadotte af Wisborg, H.M. King Carl XVI Gustaf was unable to attend the Crafoord Days 2012. The prizes were presented by Margareta Nilsson, daughter of the Donors, Holge and Anna-Greta Crafoord. Ms Nilsson’s kind hospitality, deep devotion to science, culture and other noble social causes, and moving childhood memories shared at the dinner,  have led Reinhard Genzel in his moving speech on behalf of the winners in Astronomy to refer to Margareta Nilsson by the words: “You were our King these past two days!”.

The one day symposium itself was very interesting, and so were the four prize lectures on Tuesday morning. In a few days The videos of the two days’ lectures will be are posted here. Here are the slides of my talk on analysis of Boolean functions, featuring, among other things a far-reaching conjectural extension of a recent theorem by Hamed Hatami.

### Gothenburg and Stockholm

From Lund I continued to a short visit of Gothenburg hosted by Jeff Steif with whom I share much interest in noise sensitivity and many other things. I then continued to Stockholm where I visit Anders Björner who is a long-time collaborator and friend since the mid eighties. For me this is perhaps the twelfth visit to Stockholm and it is always great to be here.

We will celebrate on this blog these exciting events with a rerun of the classic, much-acclaimed piece by Christine Björner on the Golden room and the golden mountain.

Speakers at Crafoord symposium, (from right to left) Carlos Kenig, Ben Green, Jean Bourgain, Terry Tao, me and Michael Christ. Copyright: Crafoord foundation.

Update(Oct 2014): Here is a picture of me and  Jean at IHES 1988

# Roth’s Theorem: Tom Sanders Reaches the Logarithmic Barrier

I missed Tom by a few minutes at Mittag-Leffler Institute a year and a half ago

Suppose that $R_n$ is a subset of $\{1,2,\dots, n \}$ of maximum cardinality not containing an arithmetic progression of length 3. Let $g(n)=n/|R_n|$.

Roth proved that $g(n) \ge log logn$. Szemeredi and Heath-Brown improved it to $g(n) \ge log^cn$ for some 0″ src=”http://l.wordpress.com/latex.php?latex=c%3E0&bg=ffffff&fg=000000&s=0&#8243; alt=”c>0″ /> (Szemeredi’s argument gave $c=1/4$.) Jean Bourgain improved the bound in 1999 to $c=1/2$ and in 2008 to $c=2/3$ (up to lower order terms).

Erdös and Turan who posed the problem in 1936 described a set not containing an arithmetic progression of size $n^c$.  Salem and Spencer improved this bound to $g(n) \le e^{logn/ loglogn}$. Behrend’s upper bound from 1946 is of the form $g(n) \le e^{C\sqrt {\log n}}$. A small improvement was achieved recently by Elkin and is discussed here.  (Look also at the remarks following that post.)

In an earlier post we asked: How does $g(n)$ behave? Since we do not really know, will it help talking about it? Can we somehow look beyond the horizon and try to guess what the truth is? (I still don’t know if softly discussing this or other mathematical problems is a fruitful idea, but it can be enjoyable.)

We even had a poll collecting people’s predictions about $g(n)$.  Somewhat surprisingly 18.18% of answerers predicted that $g(n)$ behaves like $(\log n)^c$ for some $c<1$. Be the answer as it may be, reaching  the logarithmic barrier was considered extremely difficult.

A couple of months ago Tom Sanders was able to refine Bourgain’s argument and proved that $g(n) \ge (\log n)^{3/4}$. Very recently Tom have managed to reach the logarithmic barrier and to prove that

$g(n) \ge (\log n)/(\log \log n)^{5}.$

Quoting from his paper: “There are two main new ingredients in the present work: the first is a way of transforming sumsets introduced by Nets Katz and Paul Koester in 2008, and the second is a result on the $L_p$-invariance of convolutions due to Ernie Croot and Olof Sisask (2010).”

This is a truly remarkable result.

# Budapest, Seattle, New Haven

Here we continue the previous post on Summer 2010 events in Reverse chronological order.

## Happy birthday Srac

In the first week of August we celebrated Endre Szemeredi’s birthday. This was a very impressive conference. Panni, Endre’s wife, assisted by her four daughters, organized a remarkable exhibition by mathematicians who are also artists. Panni also organized tours and activities for accompanying people which my wife told me were great. A few mathematicians chose to attend the tours rather than the lectures. In Hungary, Endre has a nickname “Srac” which means “kid”, and, to my amazement, people (including young people) really call him Srac.

Szemeredi Kati and Zsuzsa

### After-dinner speech. The distinction between surprising and amazing, and the question of do we age when it comes to our emotions and inner soul experiences

Being asked to give the after-dinner speech for Endre in the festive boat dinner ranks second in my life among cases where I was chosen for a job for which so many others were much, much more deserving and qualified. Continue reading