Group actions in geometric/arithmetic Combinatorics

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This project seeks to further and combine results and tools from the theory of growth in finite groups with state of the art methods of arithmetic and geometric combinatorics. This is a modern area of research at the crossroads of pure mathematics, with connections to computer science and coding and complexity theory, unified by the general theme of pseudorandomness. A significant progress in this area began in the 2000s after foundational work of Helfgott, followed by Bourgain, Gamburd, Sarnak, and others. The growth phenomenon appears to be inherently connected with the renown Sum-Product conjecture of Erdos and Szemerédi, towards which there has been a lot of progress in the past 15 years. More specifically the project aims to look at specific groups families, such as those of upper-triangular matrices, uncover and categorise the structures therein that pose obstruction to growth and establish quantitative estimates for growth in their absence. The nature of these obstructions much depends on the field, where the matrix elements come from: analysing various scenarios to this effect is a specific novel feature of this project. Partially this scope of questions furthers the earlier results by Breuillard, Green and Tao, Gill and Helfgot, Murphy and Petridis and others. Growth in groups, and especially the concept of energy arising in its study are immediately related to geometric incidence theory estimates, arising in connection of these groups' action son homogeneous spaces. This constitutes the other thread of the project, currently focusing on the Mobius hyperbolae. The aim, in particular, is to improve on earlier results due to to Bourgain, Solymosi and Tardos, Shkredov and others by using a special set of tools both from growth in groups and geometric incidence theory.