A schoolboy from St. Petersburg received the Grand Award for the study “Fast algorithm for calculating the commutator length in a free group”

    Danil Fialkovsky, a student of the 11th grade of school number 564, received a prize in mathematics at the World Student Science and Engineering Achievement Competition (Intel ISEF). Danila won the battle with 1,700 participants from 75 countries.

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    A prize in the $ 500 dollars section of mathematics confirmed the level of St. Petersburg mathematical education, writes the Laboratory for Continuing Mathematical Education , where Danila is studying. Petersburg was represented at the competition by seven students of the Laboratory . The works of schoolchildren were evaluated by world-famous scientists, Nobel Prize winners, and representatives of the largest universities in the world.

    Danila Fialkovsky is studying for 8 or more hours a day, is engaged in summer mathematics schools and has attended dozens of special courses in matanalysis, algebra and topology. His name was 29th on the list of Intel ISEF winners and medalists studying at the Laboratory for Continuing Mathematical Education.

    Danila presented the study “Fast algorithm for calculating the commutator length in a free group”:
    Groups are one of the central concepts of modern algebra. The study of the commutator length of elements in various groups is carried out in various fields 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 with homomorphism the commutator length does not increase. Therefore, the commutator length of the elements of a free group is of particular interest.

    We propose a fast algorithm for calculating the commutator length of an element from the commutant of a free group. This algorithm is based on the already existing Bardakov algorithm, which, unlike what we have proposed, does not give an explicit representation of an element as a product of commutators. In addition, our algorithm is faster.

    Also, based on my algorithm, a program can be written that allows you to read the commutator length of an element and get an explicit representation as a product of commutators faster and more efficiently than any existing one.

    Danila Fialkovsky
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    Photo from the award ceremony, which took place on May 15 in Pittsburgh, USA
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