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.