Gage Hoefer: Junior Fellows Seminar: The Hypergraph isomorphism game and quantum groups

Speaker
Gage Hoefer, Dartmouth College

Abstract
In this talk, we will develop an algebraic and operational framework for quantum isomorphisms of hypergraphs, using tools from compact quantum group theory and non-local games. We will introduce a new synchronous version of the hypergraph isomorphism game whose game algebra uniformly encodes multiple notions of quantum isomorphisms of hypergraphs. For graphs, we show that the *-algebra of the hypergraph isomorphism game is a quotient of the *-algebra of the graph isomorphism game. We further prove that the hypergraph game algebra forms a bi-Galois extension over the quantum automorphism groups of the underlying hypergraphs. This allows us to deduce that the algebraic notion of a quantum isomorphism of hypergraphs coincides with the operational one coming from the existence of perfect quantum strategies. Viewing games themselves as hypergraphs, we analyze isomorphisms and the transfer of strategies within this setting. Time permitting, I’ll discuss how we construct a *-algebra whose representation theory characterizes distinct classes of quantum isomorphisms between non-local games. This is based on joint work with Georgios Baziotis and Alexandros Chatzinikolaou.