**Scientific results:**

- The scientific results achieved over the first year of the work of the Laboratory are deeper and broader than those we envisioned and expected. In particular, we have provided a solution to the classical problem of group theory, on which leading specialists have been working for decades. The employees of the laboratory «Modern Algebra and Its Applications» have been actively conducting scientific research, seminars, organized visits by foreign colleagues to work together and wrote academic articles. The team of the Laboratory have solved a problem formulated by Anna G. Ershler, namely on building a finitely-generated fractal solvable group. A fractal group is a group with unbounded iterated identity. The built group is a step-3 solvable. It had been known that such groups did not exist in the class of metabelian group. The results were published in the journal «Works of Steklov Institute of Mathematics» under the title «An example of a finitely-generated fractal solvable group» 307 (2019), this issue of the journal was devoted to the memory of Igor R. Shafarevich. New results have been obtained in the domain of group localization: we introduced several new classes of localizations (idempotent monads) on a category of groups (1) right exact localizations, (2) free;y defined localizations, (3) localizations defined by equations, (4) epi-preserving localizations; (5) all localizations. Their properties were studied. It has been proven that (1), (2), (3), (4), (5) is a strictly increasing sequence of classes. Employees of the Laboratory continue studying localizations. The main objective of this study is the problem of preserving the nilpotency class during localization. As of now we have managed to prove that for the case of localizations of special type (so-called right-exact loalizations). We presented a hypothesis on the multiplicity of an irreducible representation of the H subgroup contained in an irreducible presentation of the group G under condition that the reduced G and Н groups match.
- In the work «Commutative Lie Algebras And Commutative Cohomology In Characteristic 2» we introduced and studied commutative cohomologies of Lie algebras over a field of characteristic 2. We also introduced a structure of a ring (similar to the classical case) of cohomology. We compared these two cohomologies to the classical ones and also computed rings of commutative cohomologies in a one-dimensional, two-dimensional Lie algebra and the Heisenberg algebra. We also set new problems for further study. In the work «Bipartite Graphs As Polynomials And Polynomials As Bipartite Graphs», for each finite bipartite graph we built a polynomial of one variable with coefficients in the natural numbers, additionally, for each polynomial of one variable with coefficient in the natural numbers we built a finite bipartite graph. Moreover, we demonstrated that this is a one-on-one relationship. At the same time, it turned out that the Winskel product of bipartite graphs corresponds to the product of the polynomials (in the category of Petri nets). A similar correspondence was conducted for finite bipartite directed graphs, in which the correspondence is set with a polynomial of two variables with coefficients in the natural numbers. Therefore, in the set of directed bipartite graphs we introduced a Zariski topology and reviewed such notions as the vicinity of a graph, its primality (as the primality of the corresponding ideal) etc.
- The efforts of the members of our research team have helped to solve major open problems and built new theories. Young employees of the Laboratory have performed their work at a high level. The Laboratory has become a center of informative lectures and fruitful academic discussions. New results have been obtained in the theory of limits in in the categories of free corepresentations, in particular, new descriptions of fr-codes that encode functors in categories of groups. The achieved results were published in the article «Limits, standard complexes and fr-codes».
- The employees of the Laboratory have achieved new results in the domain of localizations of groups. We introduced several classes of localizations in categories of groups and study their properties and relations between them. The most interesting class among the reviewed localizations is the class of localizations matching their zero derived functor. Such localizations are called exact and it is proven that an exact localization L preserves the class of nilpotent groups, as well as the fact that for any finite p-group G the G →LG morphism is an epimorphism. Our research also demonstrates that some known localizations (Baumslag’s P-localization in relation to the set of prime numbers P, Bousfield’s HR localizations, Levine’s localization, Levine–Cha’s Z localization) are exact. Finally, our research demonstrates some particular case of the hypothesis of Farjoun’s conjecture on Nikolov–Segal maps.
- Members of the academic team of the Laboratory have found a series of 3-manifolds that are pairwise homologically non-equivalent, however, their groups have isomorphic completions. The invariants distinguishing these manifolds are build behind the first limit ordinal, are transfinite and are produced form transfinite factors on members of the lower central series of HZ localizations in the sense of Bousfield’s fundamental groups of 3-manifolds.
- Currently the academic team of the Laboratory continues to actively research the fr language, the research in the field of the theory of representations of real-valued and p-adic groups, the theory of automorphic forms, as well as finite-dimensional representations of algebraic groups with involution, the research in the field of the theory of homotopies and the theory of group rings, new relation between them, the research of HZ localization of a free group, localizations and completions of some other groups.

**Education and career development: **

Since 2018, when the Laboratory was created, the team has been organizing two seminars on a regular basis, «Languages and spaces» and «Modern algebra and applications» with the participation of the world’s prominent researchers in the domain of mathematics and philosophy. Additionally, we conduct lectures on the topic of our research that are open to everybody.

Over the course of the existence of the Laboratory three employees have obtained the degree of Candidate of Sciences in Physics and Mathematics, one employee received a master’s degree diploma under the supervision of a senior research fellow of the Laboratory and one employee has completed a one-month internship at the Indian Institute of Technology Bombay (permanent place of work of the leading scientist Dipendra Prasad).