The Riemann Hypothesis

Ameya Pitale

Our own Dr. Ameya Pitale will be speaking on

Wednesday, March 2nd at 5pm in PHSC 1105

on the famous Riemann Hypothesis.

 

The Riemann zeta function is defined for any complex number, s, using an infinite series as:

Well, technically, this is not right since even your friend in Calc III knows (or at least should know!) that this series doesn’t converge if s is less than 1.  The above function is the Riemann Zeta function for s greater than 1, and then you have to use some complex analysis to define it for all other s values.

The Riemann hypothesis is then the conjecture that the only non-stupid zeros of the the function are when the real part of s is 1/2.  According to Wikipedia, 10,000,000,000,000 zeros have so far been found and all of them are where they should be.  If you prove or disprove the Riemann hypothesis, then besides being as famous as Bat Boy (see below), you’ll earn yourself a million bucks.

Here’s a cool video which shows you what we mean:

Dr. Pitale will tell us all about the math involved and why the Riemann Zeta function is so darned important:

 

I am sure that students have often wondered while studying sequence and series in a Calculus 3 class as to the need and usefulness of the material.  I would like to start with some very simple questions whose answers lead naturally to understanding certain series. These series are special cases of a very famous function in number theory and all of math – the Riemann Zeta function. This function is full of rich information about prime numbers and studying it gives us deep insights into some very difficult questions in number theory. And if you are really smart and very very hard working, then it can also lead you to a million dollars.

– Dr. Pitale’s abstract

 

 

Or, xkcd’s take on it:

Thanks, but no thanks.

As we discussed here, the Clay Institute offered Perelman the $1,000,000 prize for solving one of their Millennium Problems.  Dr. Kornelson gave a link in the comments to that post to an article which claimed he planned to turn down the prize.

Now it’s official.

A couple of days ago the Clay Institute posted an announcement on their webpage officially announcing Dr. Perelman’s decision to decline the prize.  See here for more information.  The Clay Institute also says:

In the fall of 2010, CMI will make an announcement of how the prize money will be used to benefit mathematics.

– Clay Institute

P.S. to the Clay Institute:  OU MathBlog HQ could use a new computer:

A 24 carat gold Macbook

Will They Send Him One of Those Big Novelty Checks?

Poincare

You might recall that some time ago we talked about the interesting story of Perelman and his proof of the Poincare Conjecture (actually, he proved the Thurston Geometrization Conjecture which implies the Poincare Conjecture).  Perelman was offered (and declined) the most famous award in math, the Fields Medal.

The Poincare Conjecture was also one of the seven Clay Institute Millennium Prize Problems.  Those are seven of the most interesting and difficult open problems in math research as of May 24th, 2000.  The Clay Institute offered

to anyone who could solve any of these problems.  Of course nobody (except maybe Perelman!) had any idea that the solution to one of these amazingly difficult problems was just around the bend.  As of yesterday, Perelman was officially offered the $1,000,000.  No word yet on if he’ll accept it or not.

In the meanwhile, here is a a great lecture by Dr. Curtis McMullen of Harvard University explaining the math around the Poincare Conjecture.

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: