The class of automata groups contains several remarkable countable groups. Their study has led to the solution of a number of important problems in group theory. Its recent applications have extended to the fields of algebra, geometry, analysis and probability. Together with arithmetic and hyperbolic groups, automata groups dominate the modern landscape of theory of infinite groups. As examples of this construction we plan to present solutions to the following problems:

• Burnside problem. Infinite finitely generated torsion groups.
• Milnor problem. Constructions of groups of intermediate growth.
• Atiyah problem. Rationality of L2 Betti numbers.
• Day problem. New examples of amenable groups.
• Gromov problem. Groups without uniform exponential growth.

As survey papers on this topic one can mention:

• A. Zuk - Groupes engendrés par les automates - Séminaire Bourbaki, Astérisque 311, 2008.
• A. Zuk - Automata groups, Topics in noncommutative geometry, 165– 196, Clay Math. Proc., 16, Amer. Math. Soc., Providence, RI, 2012.
• A. Zuk - Analysis and Geometry on Groups, Discrete Geometric Analysis, Memoir MSJ, vol. 34, 2016, p. 113–157.
• A. Zuk - From partial differential equations to groups, in: Analysis on manifolds and graphs, Cambridge University Press, p. 368–381, 2020.

The lectures will take place at 12pm and 15pm (both Friday and Monday) at the lecture room 321 on the third floor of the Institute. There will be no online transmission.