Large groups and closures of actions

The topic of the project are so-called `large groups` (or `infinite-dimensional groups`), their unitary representations, and related harmonic analysis. The term `large groups` is informal and includes the following types of groups: groups of matrices of infinite order (the most important are groups of matrices with real or complex elements, but matrices over finite fields and p-adic fields also are interesting and these directions are independent), `infinite symmetric groups` (various groups of permutations of infinite sets), groups of transformations of measure spaces, the group of diffeomorphisms of the circle. There are also some other types of large groups, which are less popular,but in my context, are important for general development (as oligomorphic groups and groups of transformations of non-locally finite trees). Such groups appeared in different domains of pure mathematics in the process of their natural development (as new aims or as tools for solution of old problems). On the other hand, even earlier some large groups became a tool of quantum field theory (Karl Friederichs, Irving Segal, Felix Berezin, Miguel Virasoro); recent research on `harmonic analysis` on large groups (Grigory Olshanski, Anatoly Vershik, Sergey Kerov, Alexei Borodin et al.) contained new standpoints for theory of `random matrices`, which are important in modern physical theories. The theory of large groups was a new entity and not a `generalization` in the usual mathematical sense. Some of their properties seem impossible from the point of view of classical theories. The list of `Large groups` looks like a collection of disjoint objects but the theory is united by deep analogues, parallel technologies and mutual applications. I worked in `large groups` for a long time and in 1980-1996 and 2010-2023 this was the main domain of my activity. The most interesting problem of the project is construction of counterparts of the Fourier transform. Partially this is known (existing results give counterparts of `Harish-Chandra spherical transform`) but there is a hope to extend this to a wider context; in particular, this could be a way to special functions in an infinite number of variables. I intend to investigate classifications of unitary representations, descriptions of categories of double cosets, closures of large groups in representations and in actions on measure spaces. I intend to extend the theory to some supergroups. I also intend to investigate descent to finite dimensional level and to apply an infinte-dimensional point to usual groups of matrices of finite order and to finite groups of permutation. I also intend to search for applications to stochastic processes, combinatorics, and ergodic theory.