In this study we plan to settle the Krein-Milman theorem for the space of sofic representations of a countable group. This is related to an open problem by Sorin Popa, asking if such a space always has extreme points. The space of sofic representations can be embedded as a convex, non-compact set in a Banach space. Lack of compactness, and the difficulty of constructing sofic representations is what makes this problem challenging.
Another objective of this project is to advance the use of non-standard analysis, in particular of the Loeb measure space construction, in order to provide a systematic investigation of soficity. A group is sofic if and only if it is a subgroup of the universal sofic group. The Loeb space provides a new picture of the universal sofic group that will be used to better understand its structure. These new methods will provide tools for showing the soficity of new wide classes of groups or hint at obstructions to be used in order to ultimately construct a non-sofic group.
Project coordinator: Liviu Păunescu
Research team (in alphabetical order):
Ovidiu Preda
Florin Rădulescu
Andrei Sipoș
Submitting for publication of at least two articles in specialised journals on the subject of the project proposal: sofic objects, von Neumann algebras, ultraproducts, convex spaces.
J. Bassi; F. Rădulescu: Separable boundaries for non-hyperbolic groups, Journal of Operator Theory, Volume 87, Issue 2, Spring 2022 pp. 461-470. arXiv:2107.01236
J. Bassi; F. Rădulescu: An example of a non-amenable dynamical system which is boundary amenable, accepted for publication in Proceedings of the American Mathematical Society, arXiv:2108.13936
O. Preda; M. Stanciu: Vaisman theorem for lcK spaces, accepted for publication in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2022). arXiv:2109:01000
R. Munteanu; L. Păunescu: On Krein-Milman theorem for the space of sofic representations, arXiv:2107.01236
L. Păunescu; A. Sipos: A proof-theoretic metatheorem for tracial von Neumann algebras
J. Bassi; F. Rădulescu: A mixing property for the action of SL(3, Z) × SL(3, Z) on the Stone-Cech boundary of SL(3, Z), arXiv:2111.13885
G. Arzhantseva; L. Păunescu, Action traces and stability in permutations.
L. Păunescu; F. Rădulescu, Expected value of words evaluated in unitaries and permutations.
O. Preda, Vaisman theorem for lcK spaces, Workshop for Young Researchers in Mathematics, 10th edition, May 20-21, 2021.
F. Rădulescu, Separable boundaries for non-hyperbolic groups, Conference on Operator Algebras and related topics. In memory of Vaughan Jones, Istanbul Center of Mathematics, June 8-10, 2021.
L. Păunescu, The convex space of sofic representations, Conferința Cercetării Științifice din Academia Română, November 23, 2021.
L. Păunescu, The convex space of sofic representations, Workshop for Young Researchers in Mathematics, 11th edition, Bucharest, May 19-20, 2021.
L. Păunescu, Convex space of sofic representations, The 28th International conference in Operator Theory, Timisoara, June 28th - July 1st, 2022.
F. Rădulescu, Connes embedding problem for discrete groups, The 28th International conference in Operator Theory, Timisoara, June 28th - July 1st, 2022.
F. Rădulescu, Connes embedding problem for discrete groups, The mathematics of quantum entanglement via nonlocal games, University of Copenhagen, August 15-19, 2022.
L. Păunescu, Recent results on P-stability, Pure Mathematics Colloquium talks at Southampton, University of Southampton, February 5, 2021.
L. Păunescu, Constraint metric approximation and constraint stability, Stability and Testability, Institute for Advanced Study, Princeton, March 10, 2021.
L. Păunescu, Recent results on P-stability, Subfactor Seminar, Vanderbilt University, March 26, 2021.
L. Păunescu, Connes' conjecture: from Operator Algebra to Turing Machines, Bilkent University, Department of Mathematics, Ankara, Turkey, November 10, 2021,
L. Păunescu, The convex space of sofic representations, Institute of Mathematics of the Romanian Academy, series of talks, 2021-2022.
L. Păunescu, Connes' embedding conjecture and complexity theory, Institute of Mathematics of the Romanian Academy, series of talks, 2021-2022.
F. Rădulescu, Dimension formulae of Gelfand-Graev, Jones and their relation to automorphic forms and temperdness of quasiregular representations, Geometric Group Theory Seminar, Wien University, June 2, 2022.