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.
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 $$S = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...$$ \emph{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 \emph{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 \emph{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 \emph{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.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: \emph{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 \emph{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 \emph{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 \emph{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.The Abel Prize is named after the Norwegian mathematician Niels Henrik Abel, 1802-1829.
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
- You can read Tim Gower's overview of some of Deligne's work on the Abel Prize website.
- You can read about previous Abel Prize laureates on Plus and also listen to an interview with Ragni Piene, the chair of the Abel Prize committee.
- You can read more about complex numbers and the Riemann hypothesis on Plus.
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)