The award of the Abel-2020 awarded for random walks

Norwegian
Academy of Sciences has announced the winners of the award Belevskoe 2020. They were
Israeli mathematician Hillel Furstenberg
and American mathematician, Grigory Margulis. The prize is awarded “for innovative use
methods of probability theory and dynamics of group theory, number theory and
combinatorics”. About it reports
press release on the awards website.

Hillel of Furstenberg and Gregory
Margulis has developed a method of random walks for the study of such mathematical
objects like groups and graphs. Random walks are called sequence
(chain) of random steps, where each step does not depend on the previous one. Example
random walks can be Brownian motion — the trajectory of the particles
pollen on the water surface, which dramatically changes the speed and direction of movement.

This probabilistic approach allowed
to solve the many open problems of group theory, number theory, combinatorics and
graph theory. For example, it failed to prove that among the Prime numbers
there are arbitrarily long arithmetic progressions (but all such
progression end). The work of Furstenberg and Margulis, as the Chairman
the prize Committee, show how effective can be the use of
methods some areas of mathematics other areas of mathematics. They “broke the traditional
the wall between pure and applied mathematics,” says Hans Munthe-Kaas.

Among the works received
mathematicians also have important applied results, for example, expanding graphs or
the expanders. Suppose you decide to connect 50 computers to the network,
so each computer can reach each other. In this case
it is desirable that the path was as short as possible, as possible, and to herself
the network was reliable (a roll of a single computer does not violate the connectivity).If
to connect every computer with each, it would require too many connections, if you connect
each computer with one “Central” system is unreliable. Count-expander
is the best option for the connection that is not any very large group
computers on the network are quite well connected. Grigory Margulis
proposed a method of constructing such graphs.

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