The Abel Prize 2013

by Marianne Freiberger

This year's Abel Prize has been awarded to the Belgian mathematician Pierre Deligne for "seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields". The Abel Prize was established in 2003 in memory of the Norwegian mathematician Niels Henrik Abel. It's awarded annually by the Norwegian Academy of Science and Letters and makes up for the fact that there isn't a Nobel Prize in mathematics.

Deligne

Pierre Deligne.

Deligne has been honoured for a whole body of work, but there is one result that is particularly striking. It has its origin in the work of the legendary 18th century mathematician Leonhard Euler. Euler was considering sums of the form

$S_ n=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{n}$

and

$T_ n=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+... +\frac{1}{n^2}.$

What happens when you let $n$ get larger and the larger, that is when you add more and more terms to those sums? In the first case the sum $S_ n$ gets arbitrarily large as you increase $n$: you can make it as large as you like simply by adding more terms. Mathematicians say that the infinite sum

  \[ S = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+... \]    

diverges. In the second case the sum $T_ n$ also gets larger and larger – but this time it never goes beyond the number $\pi ^2/6.$ You can make $T_ n$ get arbitrarily close to $\pi ^2/6$ by increasing $n$, but never any bigger. The infinite sum

  \[ T=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+ ... = 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+... \]    

is said to converge to $\pi ^2/6$.

Thus, introducing the power two in the denominators of the fractions has made a huge difference. What about introducing other powers? It turns out that the series

  \[ \zeta (s)=1+\frac{1}{2^ s}+\frac{1}{3^ s}+\frac{1}{4^ s}+.... \]    

converges for any power $s$ that is greater than 1. This turns $\zeta (s)$ into a function of $s$. For every $s>1$ the value of $\zeta (s)$ is the number the corresponding sum converges to.

But Euler made another remarkable discovery. In 1737 he proved that the infinite sum $\zeta (s)$ could be written as an infinite product

  \[ z(s) = \frac{1}{(1 - 1/2^ s)(1 - 1/3^ s)(1 - 1/5^ s)(1 - 1/7^ s)...}. \]    

This is quite an intriguing result: what started out as an expression involving all the natural numbers 1, 2, 3, ... turned into an expression involving only the prime numbers.

Riemann

Bernhard Riemann, 1826-1866.

More than a hundred years after Euler, the German mathematician Bernhard Riemann considered Euler's so-called zeta function again, making another remarkable connection and shedding light on a problem that has intrigued mathematicians for centuries: just how are the prime numbers distributed among the other numbers?

Euler’s zeta function takes real numbers $s$ (greater than 1) as input and returns real numbers $\zeta (s)$ as output. Riemann considered an extended version of the zeta function which involved a slightly more exciting set of numbers: complex numbers. You build them by pretending that the equation

$x^2= -1$

has a solution (which of course it hasn’t in the real numbers) and calling that solution $i.$ The complex numbers are numbers of the form

  \[ a+ib \]    

where $a$ and $b$ are real numbers. You can define addition, subtraction, multiplication and division, for example, you add two complex numbers $a+ib$ and $c+id$ like this:

  \[ (a+ib) + (c+id) = (a+c)+i(b+d) \]    

and you multiply them like this

  \[ (a+ib) \times (c+id) = ac+iad+ibc+i^2bd = ac-bd+i(ad+bc) \]    

(recalling that $i^2=-1.$)

What’s more, there is also a geometric interpretation of complex numbers: you can identify each complex number $a+ib$ with the point on the plane that has coordinates $(a,b).$ The ordinary real numbers sit inside the complex numbers – they are of the form $a+i0$ – and they correspond to points on the plane whose second coordinate is 0. Riemann’s zeta function takes complex numbers as input and returns complex numbers as output and it agrees with Euler’s function for real numbers $s$ with $s>1.$

With remarkable insight Riemann recognised that his function could give you information about the primes. The prime number theorem states that the number of primes you find below a given number $n$ is approximately equal to $n/\ln {n}.$ Since this is only an estimate, the obvious question is by how much the formula misses the correct value: you want to know the error term. Riemann realised that the complex numbers $z$ for which $\zeta (z)=0$ could give you information about that error term. He conjectured that all these so-called zeroes (or at least the interesting ones) are of the form

  \[ \frac{1}{2}+ib, \]    

in other words they correspond to points on the plane whose first coordinate is $1/2.$ If this hypothesis is true, then this puts tight bounds on how large that error term can be.

To this day Riemann’s famous hypothesis has not been proved: it presents one of the hardest unsolved problems in mathematics, some would argue the hardest one.

So far all this has been about numbers, but Deligne's work has been in a field called algebraic geometry. Where is the connection? Tim Gowers has written an excellent overview of this, so we'll borrow some of his insight to give a cursory glance.

That algebra is connected to geometry is something we are all familiar with from school. If you give the points on the plane coordinates $(x, y)$, then the equation

  \[ x^2+y^2=1 \]    

defines a circle: it consists of all the points that are at distance 1 from the point $(0,0)$. So algebra can define geometric shapes.

Abel

The Abel Prize is named after the Norwegian mathematician Niels Henrik Abel, 1802-1829.

In this example the numbers $x$ and $y$ we considered as coordinates were just ordinary real numbers. But there are other sets of numbers too. We’ve already met the complex ones, but you can also consider the hours on a clock, starting at 0 and going all the way around to 11. They are ordinary natural numbers with the important difference that when adding hours you start again from the beginning when you’ve gone once around the clock. You could do this for any other number too. For example, in a three-hour day you have 2+1=0, 2+2=1 and so on. The numbers from 0 to 2 with this form of addition form what is called the field $F_3$. Unlike the real, natural or complex numbers, this field contains only a finite amount of numbers.

Now you can ask yourself what happens to the circle if you are only allowing $x$ and $y$ to be members of $F_3$: which numbers $x$ and $y$ are members of $F_3$ and at the same time satisfy the equation

  \[ x^2+y^2=1? \]    

It turns out that our "circle" now only contains four points: $(1,0), (0,1), (2,0)$ and $(0,2).$

Now you can build a bigger field from $F_3$ in a similar way to how you built the complex numbers. The equation

  \[ x^2=2 \]    

has no solution in $F_3$ but let’s nevertheless give that non-existing solution a name, say $q.$ It plays the same role as $i$ in the complex numbers: we look at numbers of the form

  \[ a+qb \]    

where $a$ and $b$ are elements of $F_3.$ As for the complex numbers there is a notion of addition, subtraction, multiplication and division for this new field. But as our original field $F_3$ this new field contains only a finite amount of numbers: it contains $3^2=9$ of them. Using a similar approach (adding solutions to particular equations) you can construct a whole sequence of fields

  \[ F_3, F_{3^2}, F_{3^3}, F_{3^4}, ... \]    

where the subscripts indicate the number of elements of each field. And you can do this not just for the number three but for any prime number $p$ to get a sequence

  \[ F_ p, F_{p^2}, F_{p^3}, F_{p^4}, ... \]    

For each of these sequences you can ask how many points the "circle" given by our equation

  \[ x^2+y^2=1 \]    

contains. This will give you a sequence of numbers:

$a_1$ (the number of points in the circle for $F_ p$), $a_2$ (the number of points in the circle for $F_{p^2}$), $a_3$ (the number of points in the circle for $F_{p^3}$), and so on.

And there is more room for play. Rather than just considering the equation of a circle

  \[ x^2+y^2=1 \]    

you can consider any set of equations (to be precise, any set of polynomial equations). Again, given a sequence of fields

  \[ F_ p, F_{p^2}, F_{p^3}, F_{p^4}, ... \]    

you get a sequence of numbers

  \[ b_1, b_2, b_3, b_4,... \]    

each number giving the number of solutions that exist for the set of equations in the corresponding field.

And now we are slowly approaching the connection to the Riemann hypothesis. Riemann’s zeta function, if the hypothesis is true, gives us information about the error terms associated to the prime number theorem. Now given a sequence of numbers coming from a sequence of fields as described above you can form a function $Z(x):$

  \[ Z(x) = \exp {\left(b_1x + \frac{b_2x^2}{2}+\frac{b_3x^3}{3}+\frac{b_4x^4}{4}+...\right).} \]    

In the 1940s the mathematician André Weil made a series of conjectures about functions derived in this way which came to have a huge influence in mathematics. Some of these are the exact analogue of things that were already known about Riemann’s zeta function. And one of them, concerning the zeroes of the functions in question, is an analogue of the Riemann hypothesis. Functions such as $Z(x)$ above are also called zeta functions.

And there is more to the analogy. Like the Riemann zeta function these new zeta functions give information about error estimates. As an example, suppose you have a natural number $n$ and want to know in how many ways you can write it as a sum of 24 squares. You’re looking for sets of 24 numbers $x_1, x_2, x_3, ..., x_{24}$ so that

  \[ n=x_1^2+ x_2^2+....+x_{24}^2. \]    

How many such sets are there? It turns out that there is an approximate formula in terms of the number $n$. But it’s only an estimate so again you would like to know information about the error is for each $n$.

In 1916 the legendary Indian mathematician Srinivasa Ramanujan suggested bounds for that error term. It later turned out that Ramanujan was right: that's a consequence of Weil's analogue of the Riemann hypothesis.

In 1974 Deligne proved Weil's analogue of the Riemann hypothesis in what the Abel Prize citation describes as a "real tour de force". And the citation continues, "Deligne's powerful concepts, ideas, results and methods continue to influence the development of algebraic geometry, as well as mathematics as a whole." The original Riemann hypothesis, though, remains tantalisingly open.


Further reading

Comments

Query

How is (2,0) a solution ??
Am I missing something about multiplying the nos. of F3 ??
Please reply @ amandeepji@gmail.com

RE: Query

It must be a mistake. The only two points that satisfy the equation are (0, 1) and (1, 0). (0, 0), (0, 2), (1, 1), (1, 2), (2, 0), (2, 1) and (2, 2) are the other points from the field, and none of them satisfy the equation.

2^2+0^2=4+0=1 (mod 3)

2^2+0^2=4+0=1 (mod 3)