## Abel Prize for Yakov Sinai

March 27, 2014

When a tool maker designs a drill she does not expect it to also be useful as a hammer or a torch. You might be able to use it as either at a pinch but it would not be very effective. If you want a universal tool that can do lots of different things you need to design it with those things in mind from the start and the result may be just as complex as a box of individual tools.

The strangest thing is that this is not the case with mathematical tools. A concept or method used to solve one mathematical problem often turns out to be just as useful in solving others that are completely different and seemingly unrelated. To give a simple example, the number pi was first defined to quantify the ratio of the circumference to diameter of a circle, yet it appears in a whole host of mathematical and physical equations that have nothing to do with circles. The same is true for a few other special numbers such as e, the base of the natural logarithms. Why do the same few numbers keep coming up in mathematics instead of different ones for every problem? Other examples abound. Special functions, groups, algorithms and many more mathematical structures prove useful over and over again. Why does mathematics have this natural universality?

People who don’t know mathematics well think that mathematicians invent the methods they use in the same way that people invent stories or fashions. Most mathematicians say that this is not the case. Their experience of developing new mathematics feels more like a process of discovery rather than invention. It is as if the mathematical structures were already there before any human was aware of them. You can also invent mathematical structures like the game of chess, but chess is not regarded as important in mathematics because it is not useful for solving unrelated problems. It does not have universality and it is this universality that distinguishes the interesting mathematics from the uninteresting.

Why then is such universality so easy to find in mathematics without trying to look for it? Why does it even extend into physics where deep mathematical ideas originally used to solve problems in pure mathematics turn out to be important for understanding the laws of nature (complex numbers, differential geometry, topology, group theory etc.) ? It is even more surprising to pure mathematicians when theory developed by physicists turns to be useful in mathematics, yet even string theory has already proved useful for solving a whole host of mathematical problems that seemed otherwise intractable. It remains a huge and deep mystery why this happens.

Yesterday the Abel committee in Norway announced that it was awarding its annual prize of 6 million Kroner to Yakov Sinai for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics.  Sinai’s work covers complex dynamical non-linear systems with many variables. Naively our prior expectation for such systems would be that they are going to behave in complex unpredictable ways and the only thing they are likely to have in common is going to be randomness, but that is not what happens. Systems in statistical physics have entropy and temperature. These are macroscopic emergent quantities that follow derived laws which are common to different systems irrespective of the microscopic description of the dynamics. They have phase transitions and near these transitions you get critical phenomena that obey universal laws.

Sinai looked in particular at chaotic systems of non-linear dynamical equations where more universal emergent behavior is found and described in terms of certain Feigenbaum constants. Another area he worked in was algorithmic complexity of binary sequences and dynamical systems. All this work is highly applicable to practical problems but it is also important as a tool for understanding universality and why it arises. Perhaps one day by building on the work of Sinai we will learn much more about the unity of mathematics, the laws of physics and why these things are so beautifully connected by universality.

Congratulations to Yokov Sinai for this well deserved award which will raise the profile of such important work.

## Book review: Love and Math

October 20, 2013

“There is a secret world out there.” This is the beginning of Edward Frenkels book about his mathematics and his story of how he fell in love with it. Popular books about mathematics are rare compared to areas of science such as particle physics, cosmology or even biology. It is hard to write a mathematics book that will appeal to the masses. You cant really play the trick of skipping all the equations or the details because these are really the essence of what makes mathematics so beautiful to those who master it.

Even rarer are such books written by the people who are at the bleeding edge of current mathematical research. There are some great maths books by Marcus de Sautoy, Ian Stewart, Simon Singh etc. , While some of these authors are maths professors their popular books cover stories of mathematical problems solved by others. I have always found that the most engaging books in popular science are the ones written by those who were closest to the discoveries themselves and this book is an excellent example.

In “Love and Math” Frenkel recounts his voyage of discovery with details of the maths and the equally fascinating story of his passage through the education system of Russia in the 1980s where he faced ridiculous obstacles placed in his way simply because his family name is Jewish. Despite glowing exam results from high school he was not permitted to attend Moscow University and has to settle for another college more geared to industrial engineering . Luckily such difficulties were compensated for by a system of informal mentoring by some of Russia’s greatest mathematicians that supported the most promising young students like Frenkel.

The tale of his progress from school to Harvard professor is interwoven with potted lessons in group theory as he had to learn it to solve the problems posed by his mentors. These are aimed at non-experts. For someone like myself who is already familiar with the standard methods but not with all the recent progress this is light and enjoyable reading right up to the final chapters where he described his work with Ed Witten on geometric Lamglands. I cant say how a complete novice would find it but young math students would surely find inspiration and useful knowledge here and others can skip the details and enjoy the human side of the story.

The book ends with a chapter about his controversial short film “Rites of Love and Math.” This is said to have made Frenkel something of a sex symbol among mathematcians, certainly a new idea. Unfortunately the film is not available through the inline rental services I use so I cant tell you any more about it. Here is the trailer from his youtube site.

## Abel Prize 2013 goes to Pierre Deligne, and Milner Prize to Alexandre Polyakov

March 20, 2013

The Abel prize in mathematics for 2013 has been awarded to Pierre Deligne for his work on algebraic geometry which has been applied to number theory and representation theory. This is research that is at the heart of some of the most exciting mathematics of our time with deep implications that could extend out from pure mathematics to physics.

Deligne is from Belgium and works at IAS Princeton.

I obviously can’t beat the commentary from Tim Gowers who once again spoke at the announcement about what the achievement means, so see his blog if you are interested in what it is all about.

Update: Also today the fundamental Physics Prize went to Polyakov, another worthy choice.

Update: Some bloggers such as Strassler and Woit seem uncertain this morning about whether Polyakov got the prize. He did. They played a strange trick on the audience watching the live webcast from CERN by running a 20 minute film just before the final award. They did not have broadcast rights for the film so they had to stop the webcast. After that the webcast resumed but you had to refresh your browser at the right moment to get it back. The final award to Polyakov was immediately after the film so many people would have missed it. I saw most of it and can confirm that Polyakov was the only one who finished the night with two balls (so to speak). To make matters worse there does not seem to have been a press announcement yet so it is not being reported in mainstream news, but that will surely change this morning. As bloggers we are grateful to Milner for this chance to be ahead of the MSM again.

I would have done a screen grab to get a picture of Polyakov but CERN have recently changed their copyright terms so that we cannot show images from CERN without satisfying certain conditions. This contrasts sharply with US government rules which ensure that any images or video taken from US research organisations are public domain without conditions.

## Shaw Prizes for Enrico Costa, Gerald Fishman, Jules Hoffmann, Ruslan Medzhitov, Bruce Beutler, Demetrios Christodoulou and Richard Hamilton

June 7, 2011

Today seven scientists are up to \$500,000 minus tax richer for having won this years Shaw Prizes.

## Astronomy

First up are Enrico Costa and Gerald J Fishman for leading the NASA mission that resolved the origin of gamma ray bursts. It does not seem to many years ago since gamma-ray bursts were regarded as one of the great unsolved mysteries of science. They had first been detected in 1967 by the Vela satellites which had been placed in orbit by the US military to check that the USSR was not detonating nuclear weapons in contravention of the 1963 partial test ban treaty. Nuclear explosions would send gamma rays into space where the satellites would detect them. Instead they observed gamma ray bursts coming from space.

From 1973 when their existence was declassified until 1997, these events were so mysterious that astronomers could not even tell if they came from nearby in our galaxy or billions of years away across the universe. NASA launched the BeppoSAX satellite to try to resolve the question, In 1997 it observed a powerful gamma ray burst which left an afterglow long enough for Earth based telescopes to lock onto its location just 8 hours later. Now they could see that it came from a very distant galaxy.

The gamma rays are so bright at that distance that it is inconceivable that they are being radiated equally in all directions in such a short space of time. The amount of energy that would have to be concentrated into a small volume is juts not possible. It is thought that they come from energetic supernovae with a rapidly rotating remnant that focuses the gamma rays into a tight beam. we only see the burst for the small fraction of events where we happen to lie in the direction of the ray.

## Life Science and Medicine

Next were Jules A Hoffmann, Ruslan M Medzhitov and Bruce Beutler for uncovering the biological mechanisms for innate immunity. When an animal or plant is infected it deploys a number of mechanisms to defend itself. One of the first is the innate immune system, thought to be one of the earliest mechanisms to evolve because it is so widespread across diverse forms of life. In plants it remains the dominant immune system, but advanced animals have developed more effective systems of adaptive immunity that can change to attack specific viruses or other contagents.

Understanding all forms of immunity is vital to medicine because it provides the knowledge needed to find drugs that help us fight diseases.

## Mathematics

Finally, Demetrios Christodoulou and Richard S Hamilton won the mathematics prize for work on differential manifolds with implications for general relativity and the Poincaré conjecture.

When Grigori Perelman famously turned down the Fields medal and the million dollar Clay prize for resolving the Poincaré conjecture, he said that his reason was that other mathematicians such as Richard Hamilton has contributed just as much to the proof. He need not have been so concerned since Hamilton has now himself been recognized with a lucrative award.

It was Hamilton who discovered the theory of Ricci flow on differential manifolds that lead Perelman to his proof of the Thurston geometrization conjecture that was known to imply the truth of the Poincaré conjecture, a mathematical problem that had remained unsolved for a hundred years.

Demetrios Christodoulou is a mathematical physicist who worked for his doctorate at Princeton under the direction of John Wheeler. He is known for his extraordinarily difficult proof of the unsurprising fact that flat empty Minkowski space is stable under the action of nonlinear gravitational dynamics as described by general relativity.

## Abel Prize 2011 for John Milnor

March 23, 2011

The 2011 Abel prize in mathematics has to John Milnor for work on geometry, algebra and topology.

His discovery of exotic smooth spheres in 7 dimensions changed the landscape of mathematics. He went on to solve the problem in all dimensions. This was just a small part of his total contribution to mathematics described this morning at the prize announcement in Norway.

A talk on his work was delivered by Fields medal winner Timothy Gower, you should read his blog for a lot more detail.

## A Christmas Puzzle

December 26, 2010

I was given “The Big Book of Brain Games” by Ivan Moscovich for Xmas. Most are too easy but here is a nice one (number 331):

Construct a square from four identical linkages hinged at the corners. Such a figure is capable of moving on its hinges to become a rhombus. How many linkages of the same length must be added to make the square rigid? The linkages must be in the same plane as the square and each one can be connected only at the hinges.

My best solution so far has 43 extra linkages which must be far too many.

Update 28-Dec-2010: Lubos has given a nice solution with no overlapping links which requires only 31 extra edges or 29 if you allow the links to cross. However I have found out that this is still not the best solution for the case where overlaps are allowed! so keep trying.

Final Update: Since posting this puzzle I have learnt that a version of it was posed in Martin Gardner’s SciAm column in 1963. His version required that the bracing links do not overlap. Seven readers sent in the solution with 23 added links shown below.

Erich Friedman considered the case where links can cross in 2000 and posted results on his Math Magic website. His best solution had 17 extra links. However, someone later informed him that Andrei Khodulyov had found a solution some time ago with just 15 extra links.

Well done to all those who posted solutions here and over at The Reference Frame.

## 2010 Fields Medals, Lindenstrauss, Ngo, Smirnov, Villani: Video

August 19, 2010

In case you missed the 2010 Fields Medal awards this morning here it is on video. The four prize winners were Ngô Bảo Châu, Cedric Villani, Elon Lindenstrauss and Stanislav Smirnov.

After the Fields Medal awards three more prizes were handed out: The Nevanlinna Prize to Daniel Spielman, The Gauss Prize to Yves Meyer and the Chern Medal to Louis Niremberg.

For a commentary on the ceremony from somebody who was there and on the Fields Medal committee, try Gowers’s Weblog .