Schoolchild from St. Petersburg received the Grand Award for the study "Fast algorithm for calculating the switch length in a free group"

Pupil of the 11th grade of school number 564, Danila Fialkovsky, received an award in mathematics at the World Contest of Scientific and Engineering Achievements of Schoolchildren (Intel ISEF). Danila won the battle against 1700 participants from 75 countries of the world.



The $ 500 dollars prize in the mathematics section confirmed the level of Petersburg mathematics education, writes the Laboratory of Continuing Mathematical Education , where Danila is engaged. Petersburg was represented at the competition by seven students of the Laboratory . The works of schoolchildren were evaluated by world-renowned scientists, Nobel Prize laureates, representatives of the largest universities in the world.

Danila Fialkovsky is studying for 8 or more hours a day, he studies at summer mathematics schools and has attended dozens of special courses in mathematical analysis, algebra and topology. His name became the 29th in the list of winners and runners-up of Intel ISEF, studying at the Laboratory of Continuing Mathematical Education.
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Danila presented the study “Fast algorithm for calculating the switch length in a free group”:

Groups are one of the central concepts of modern algebra. The study of the commutator length of elements in different groups is carried out in various areas of mathematics. In particular, information on the commutator length of elements of algebraic groups is used in algebraic K-theory. Researches in this area were conducted by such scientists as C.Edmunds, R. Golstein, E. Turner, M. Culler, L. Comerford, D. Calegari, V. Bardakov and others.

It is well known that any group can be represented as a factor in a group of a free group, and under a homomorphism the commutator length does not increase. Therefore, the commutator length of elements of a free group is of particular interest.

We propose a fast algorithm for calculating the switch length of an element from the commutant of a free group. This algorithm relies on the already existing Bardakov algorithm, which, unlike the one we have proposed, does not explicitly represent an element as a product of commutators. In addition, our algorithm is faster.

Also, on the basis of my algorithm, a program can be written that makes it possible to read the switching length of an element and get an explicit representation of the product of switches faster and more efficiently than any of the existing ones.

Danila Fialkovsky


Photos from the awards held on May 15 in Pittsburgh, USA