Recorded 27 February 2025. Jesse Peterson of Vanderbilt University presents "A hierarchy of Haagerup-type approximation properties" at IPAM's Free Entropy Theory and Random Matrices Workshop. Abstract: The Haagerup property for groups and von Neumann algebras is a well-studied approximation property, allowing for certain deformability phenomena to extend beyond the amenable realm and into the realm of free group factors. We introduce successive weakenings of the Haagerup property, indexed by the ordinal numbers. We show that for each countable ordinal α , the α -Haagerup property, like the Haagerup property itself, is an invariant of the group von Neumann algebra and passes to von Neumann subalgebras. For each countable ordinal α we construct countable groups that have the α -Haagerup property but do not have the β -Haagerup property for any β α . This gives a new proof of Ozawa's theorem that there is no universal separable II1 factor. This is joint work with Fabian Salinas. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/free-entropy-theory-and-random-matrices/