QIC 890 - Approximate Representation Theory of Groups and Nonlocal Games

Quantum Information and Computing (QIC) 890 - Approximate Representation Theory of Groups and Nonlocal Games is held with Pure Math (PMATH) 945

Description:

A finitely-presented group is a group defined by a finite set of generators and relations. In elementary terms, a representation of a finitely-presented group is an assignment of
matrices to generators such that the matrices satisfy the relations of the group. Representation theory is a central topic in mathematics, and there's a lot to be said about representations of finitely-presented groups. But what happens if we have an assignment of matrices to generators which only approximately satisfies the defining relations? For instance, is such an assignment always close to an actual representation of the group? This is a basic question of approximate representation theory, an active research area with natural applications to quantum information.

In this course, we'll cover some of the basics of approximate representation theory, as well as some recent research results, including the Gowers-Hatami stability theorem and the existence of a group which is non-appoximable in the Frobenius norm. We'll also cover some applications to quantum information, and in particular to robust self-testing in nonlocal games.  

Sample topics:

• Approximate representations
• Ulam stability
• Amenable groups and Kazhdan's stability argument
• Approximation properties of groups: sofic, hyperlinear, and MF groups
• Stability of relations
• Gowers-Hatami theorem and stability of finite groups
• Property T and k-Kazhdan groups
• De Chiffre-Glebsky-Lubotzky-Thom: non-approximable group in the Frobenius norm
• Hyperlinear profile
• Applications to quantum information: self-testing in non-local games, robustness, and entanglement functions
• Stability results from non-local games: the linearity test and the low-degree test