[Turkmath:2863] Langlands hakkinda guzel bir yazi

[Baris Kendirli]

The Canadian mathematician Robert Langlands has won the 2018 Abel Prize —
one of mathematics’ most-prestigious awards — for discovering surprising
and far-ranging connections between algebra, number theory and analysis,
the Norwegian Academy of Science and Letters announced on 20 March.

At 81, he is still an active member of the Institute for Advanced Study
(IAS) in Princeton, New Jersey, where he occupies the office that was once
Albert Einstein’s.

The mathematician outlined what became known as the Langlands programme in
1967 and carried out parts of it himself. The programme is a sort of
Rosetta stone that allows researchers to translate between different fields
of mathematics. That way, a problem that seems unsolvable in one language
can become more approachable in the other. And this connection reveals two
seemingly different concepts to be two aspects of a deeper truth.

Other researchers have gone on to greatly expand the scope of the
programme. At least three mathematicians have won Fields Medals for
confirming small parts of the grand scheme. Over time, researchers realized
that some older problems of mathematics were actually special cases of the
extended programme. One, called the Weil conjectures, was solved by the
Belgian mathematician Pierre Deligne, who received the 2013 Abel Prize for
that work. Another was a problem cracked in the 1990s by British number
theorist Andrew Wiles
<https://www.nature.com/news/fermat-s-last-theorem-earns-andrew-wiles-the-abel-prize-1.19552>
 and a coauthor: that work led them to solve Fermat’s last theorem, earning
Wiles the Abel Prize in 2016.

The span of the connections was so broad — earning the description ‘grand
unified theory of maths’ — that they often baffled Langlands himself. “It’s
almost like you are an archaeologist and you dig up a stone in the desert —
and it turns out to be the top of a pyramid,” says mathematical physicist
Robbert Dijkgraaf, who heads the IAS.

The Abel Prize is modelled after the Nobel prize and has been given out
annually since 2003. It carries an award of 6 million kroner (US$777,000).

Langlands outlined the first version of the programme in 1967, when he was
a young mathematician visiting the IAS. His starting point was the theory
of algebraic equations (such as the quadratic, or second-degree, equations
that children learn in school). In the 1800s, French mathematician Évariste
Galois discovered that, in general, equations of higher degree can be
solved only partially.

But Galois also showed that solutions to such equations must be linked by
symmetry. For example, the solutions to *x*5 = 1 are five points on a
circle when plotted onto a graph comprised of real numbers along one axis
and imaginary numbers on the other. He showed that even when such equations
cannot be solved, he could still glean a great deal of information about
the solutions from studying such symmetries.

Inspired by subsequent developments in Galois’s theory, Langlands’ approach
allowed researchers to translate algebra problems into the ‘language’ of
harmonic analysis, the branch of mathematics that breaks complex waveforms
down into simpler, sinusoidal building blocks.

In the 1980s, Vladimir Drinfel’d, a Ukrainian-born mathematician now at the
University of Chicago in Illinois, and others proposed a similar connection
between geometry and harmonic analysis. Although this idea seemed to be
only loosely inspired by the Langlands programme, mathematicians
subsequently found stronger evidence that the two fields are connected.
(Drinfel’d received a Fields Medal in 1990.)

This geometric Langlands programme encompassed an older conjecture that
also related certain equations to harmonic analysis, and which was
confirmed in Wiles' proof of Fermat's last theorem, a problem in number
theory that had been unsolved for more than 300 years. “It was a great
pleasure for me, but also a great surprise,” Langlands wrote in 2007, when
Wiles incorporated some of his work into their proof.

The field that blossomed from the Langlands programme has become so broad
that Langlands has said that he does not fully understand all of the work
that goes on in it, and in particular, some of the implications that the
geometric version might have in physics. His IAS colleague Edward Witten, a
theoretical physicist and a winner of the 1990 Fields Medal who
investigated those connections in the 2000s, has said, “I personally only
understand a tiny bit of the Langlands programme.”
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