# The 2014 Fields Medalists

Greetings readers! After a long break it seems only appropriate to open with a bang. So let us all delight in the awarding of the Fields Medal, the Nevanlinna Prize, the Chern medal and other prizes at the quadrennial (i.e., once in 4 years) ICM, the International Congress of Mathematicians. This year the ICM was held in Seoul, South Korea.

You can find out more about all the awards at the ICM site, http://www.icm2014.org or any number of news outlets:

http://www.mathunion.org/general/prizes/2014

http://www.nytimes.com/2014/08/13/science/top-math-prize-has-its-first-female-winner.html?_r=0

Especially important this year is the awarding of the Fields Medal to the remarkable Maryam Mirzakhani of Stanford, the first woman mathematician to receive this award. While there have been many extraordinary women mathematicians, the award has been given only to men so far making this recognition long overdue.

http://www.icm2014.org/en/awards/prizes/FieldsMedalist

The four winners this year from diverse areas are: Artur Avila (CNRS Paris), Manjul Bhargava (Princeton), Martin Hairer (Warwick, England) and Maryam Mirzakhani (Stanford).

And as a final shout out we should mention that Professor Mirzakhani is also a research collaborator of former OU faculty member, Kasra Rafi.

# 2010 Fields Medalists

As we first described here, every four years at the International Congress of Mathematicians up to four young mathematicians are given the Fields Medal.  It is the most famous award in mathematics — in part because you cannot be older then 40 to win it.  2010 is an ICM year and it is going on right now in Hyderabad, India.  So unless you are under 36 years old (which some of us are not), then you are out of luck.  Heck, even though Andrew Wiles proved Fermat’s Last Theorem when he was only 41 he was not given a Fields Medal!

As you might imagine, there is a lot of secrecy and excitement surrounding the Fields Medal in the days before the ICM.  But the wait is over, as the winners were announced a few days ago:

(No word on if Villani earned the “Most Dapper Mathematician” award.)

If you click on the links above you can read a bit about the winners and the math they’ve done.  If you go here, then you can read a more in-depth description of their work.

We should note that a number of other very important awards are also given at the ICM.  In addition, they gave out:

Indeed, just being invited as a speaker at a session of the ICM is considered a big honor!

If you’d like to read about a first person account of all the events at the ICM, then check out Timothy Gowers’s Blog.  He wrote a day-by-day account of his experiences at the ICM.

Even better, you can watch the awards yourself!

# Perelman, Part II

We failed to mention that all the press about Perelman might have overshadowed the tireless work of Dr. Stephen T. Colbert, DFA in this area of doughnut mathematics:

Check it out here.

# Grigori Perelman and the Poincaré Conjecture

In 1904, the famous mathematician Poincare made a reasonable conjecture:  If a 3-dimensional geometric object had the same homology groups as a 3-dimensional sphere and, and is compact, without boundary, and if every loop in the object can be squeezed down to a point (just like the 3-dimensional sphere), then the object must be the 3-dimensional sphere.

It’s a bit technical to describe homology, but beyond that Poincare’s conjecture is pretty simple to understand:  It’s the mathematical version of “If it looks like a duck, then it’s a duck.”

But, just like Fermat’s Last Theorem and the Riemann Hypothesis, being easy to state often means it’s hard to solve.  So much so that it was one of the problems with a \$1,000,000 bounty on its head as part of the Millennium Prize Problems!

Grigori Perelman

In 2002/2003 Grigori Perelman posted a sketch of a solution to Poincare’s Conjecture (and much more!) on the ArXiv using Ricci Flow (part of math having to do with differential equations).   Although Perelman was known to be a very good mathematician, his solution came out of the blue.  Although his work builds on the work of others, he made a number of important advances entirely on his own.

Terence Tao (who knows a thing or two about solving hard math problems) spoke about Perelman’s work and said:

They [the Millennium Prize Problems] are like these huge cliff walls, with no obvious hand holds. I have no idea how to get to the top. [Perelman’s proof of the Poincaré Conjecture] is a fantastic achievement, the most deserving of all of us here in my opinion. Most of the time in mathematics you look at something that’s already been done, take a problem and focus on that. But here, the sheer number of breakthroughs…well it’s amazing.

–Terence Tao

It took several years for the best mathematicians in the world to go through Perelman’s proofs carefully and check all the details.  At the 2006 International Congress of Mathematicians in Madrid, Spain it was declared that Perelman’s proof was correct.  At that time he was offered the Fields Medal (which is arguably the most famous prize in math and we’ve discussed it here).  Amazingly, Perelman declined the award!  He has since withdrawn from mathematics.  If and when he is offered his share of the \$1,000,000 Millennium Prize, it’s not clear that he’ll accept that either!

If you’d like to read more about Perelman and the Poincare Conjecture, or if you’re looking for a gift for someone who might be interested, then you should take a look at Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen, a new book about the human side of this story.  You can read a review of it here. Or, if you’d like to read more about the math, then check out The Poincare Conjecture: In Search of the Shape of the Universe by Donal O’Shea.

Or, you can just rap:

# The Abel Prize

The Abel Prize

The most famous prize in mathematics is the Fields Medal, which we already discussed in an earlier post.    Part of what makes it famous is it’s sometimes considered the “Nobel Prize” of mathematics.   And it doesn’t hurt that there are persistant unfounded rumours that there is no Nobel Prize in mathematics because a mathematician had an affair with Nobel’s wife.  Another part of what makes it so famous is that you have to be under 40 years of age to be eligible for the prize.

So what do people who have a long career in mathematics get?  One such prize started in 2002 and is called the Abel Prize (That’s the Abel of abelian groups). It’s awarded for “outstanding scientific work in the field of mathematics.”  It’s quickly been recognized as one of the most prestigious prizes in mathematics.

Gromov

A few weeks ago the 2009 winner of the Abel Prize was announced.  This year’s recipient is Mikhail Gromov for his work in geometry.  He is particularily known for his work in Geometric Group Theory.  This is an area of mathematics where people study groups by looking at related topological and geometric structures and using tools there instead (so turning algebra problems into geometry problems).  There is a strong OU connection since two of our faculty (him and him) along with several postdocs and grad students all are experts in this area of mathematics.

Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions to old problems. He has produced deep and original work throughout his career and remains remarkably creative. The work of Gromov will continue to be a source of inspiration for many future mathematical discoveries.

— The Abel Committee

To learn more about Gromov and his work you can read about him on Terence Tao’s Blog or read the official summary of his work by Vagn Lundsgaard Hansen.  Or you can watch what a topologist sees when you give him a doughnut:

Being Andy Miller

# You, too, can collaborate with Fields Medalists!

Timothy Gowers

First some background. The Fields Medal is the most famous prize in pure mathematics.  It is given to between two and four mathematicians during the International Congress of Mathematians (ICM).  The ICM is held only once every four years (or less often if there is a world war going on), so on average less than one Fields Medal per year is given out.  Perhaps most famously, in order to be eligible for the Fields Medal you cannot be over 40 years of age in the year when the prize is awarded (so some mathematicians are now out of luck).

Terence Tao

In 1998 Timothy Gowers received the Fields Medal for his research in functional analysis and combinatorics.  In 2006 Terence Tao received the medal for “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory“.

Both Gowers and Tao are most famous for their results in combinatorics having to do with finding arbitrary long arithmetic progressions (Remember, an arithmetic progression is a sequence of numbers which looks like a, a+b, a+2b, a+3b, a+4b, …, where a and b are two fixed numbers.  So 3,5,7,9,11, … is an arithmetic progression where a=3 and b=2).  In particular, there is the very famous Green-Tao Theorem which says that you can find an arithmetic progression as long as you want where all the numbers in the list are prime numbers.  Of course, just because you know a long arithmetic progression exists doesn’t mean it’s easy to find.  Even though by the Green-Tao theorem we know that arbitrary long arithmetic progressions of primes exist, so far the longest list people have actually found has only 25 numbers.

Why are we bringing this up just now?  Well, because both Timothy Gowers and Terence Tao have blogs, for one.  More than that, Gowers has suggested that people join him on his blog in an experiment to do mathematical research by having a public collaboration where anyone can join in.  The problem he would like to solve is give a combinatorial proof of the Hales-Jewett Theorem.  Actually, Gowers’s goal is less ambitious:

It is not the case that the aim of the project is to find a combinatorial proof of the density Hales-Jewett theorem when $k=3$. I would love it if that was the result, but the actual aim is more modest: it is either to prove that a certain approach to that theorem (which I shall soon explain) works, or to give a very convincing argument that that approach cannot work.

This open collaboration is going on right now on the blogs of Gowers and Tao (they’re friends and have common math interests, so Tao has joined in on Gowers’s experiment).  And it is open to everybody! So if you’d like to have a go at doing real research with  some of the best mathematicians in the world, then they’re ready to listen to what you have to say.  Unfortunately, since you’re a few days behind, you’ll have to read what people have already discussed and get caught up.  Fortunately, Gowers is providing summaries of the discussion so you don’t have to read the 100’s of posts that have already happened.  In any case, it’s cool to see real math research happening live in front of you.  Maybe this is the wave of the future in mathematics?

P.S. We’ve added links to Gowers’s and Tao’s blogs to the Blogroll.  Keep an eye out over there for other interesting blogs we find, or let us know if you know of one we should add to the list.

If you win one of these, then don't forget to mention the OU Math Blog at your award ceremony.