Groups and Operator Algebras seminar: Cecelia Higgins

Title: Complexity of finite Borel asymptotic dimension

Speaker: Cecelia Higgins (UCLA, https://www.math.ucla.edu/~ceceliahiggins/ )

Abstract: A Borel graph is hyperfinite if it can be written as a countable increasing union of Borel graphs with finite components. It is a major open problem in descriptive set theory to determine the complexity of the set of hyperfinite Borel graphs. In a recent paper, Conley, Jackson, Marks, Seward, and Tucker-Drob introduce the notion of Borel asymptotic dimension, a definable version of Gromov's classical notion of asymptotic dimension that strengthens hyperfiniteness and implies several nice Borel combinatorial properties. We show that the set of Borel graphs having finite Borel asymptotic dimension is $\mathbf{\Sigma}^1_2$-complete. This is joint work with Jan Grebik.