Spring 2020

- Date:
**2/21/20****Lauren Ruth**, Vanderbilt University- Title: The Baum-Connes correspondence for the pure braid group on 4 strands
- Abstract: We calculate the left-hand side and the right-hand side of the Baum-Connes correspondence for the pure braid group on 4 strands, each side relying on different techniques. Our long-term goal is to elucidate Schick's abstract proof of the correspondence for braid groups by explicitly describing the assembly map. This is joint work with Sara Azzali, Sarah Browne, Maria Paula Gomez Aparicio, and Hang Wang.

- Date:
**4/3/20****Ben Hayes**, University of Virginia- Title: CAT(0)-spaces associated to II_{1}-factors.
- Abstract: I will discuss recent joint work with Lewis Bowen and Frank Lin. In it, we consider a natural metric satisfying the CAT(0) condition (a certain natural negative curvature condition) on a space of operators affiliated to a tracial von Neumann algebra (a version of this space appeared in previous work of Andruchow-Larotonda). We also investigate how the geometric properties of this CAT(0) space reflect algebraic/analytic properties of the underlying von Neumann algebra.

- Date:
**4/10/20****Thomas Sinclair**, Purdue University- Title: Maximal Rigid Subalgebras of Deformations
- Abstract: We show that any diffuse subalgebra which is rigid with respect to a mixing s-malleable deformation is contained in a subalgebra which is uniquely maximal with respect to being rigid. In particular, the algebra generated by any family of rigid subalgebras that intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members was the motivation for this work, showing for example that if $G$ is a countable group with positive first L$^2$-Betti number cannot be generated by two property (T) subalgebras with diffuse intersection. This is joint with Dan Hoff, Ben Hayes, and Rolando de Santiago.

- Date:
**4/17/20****Ellen Weld**, Purdue University- Title: Connective Bieberbach groups
- Abstract: Bieberbach groups are discrete subgroups of isometries of $\mathbf{R}^n$. These groups describe the symmetries of a crystal, are the fundamental groups of compact flat Riemannian manifolds, and have been extensively studied in math, physics, and chemistry. Connectivity is a geometric property of $C^*$-algebras that is equivalent to the unsuspension of $E$-theory of Connes and Higson. In this talk, we discuss when a Bieberbach group is connective and provide necessary and sufficient conditions for determining connectivity. This is joint work with Marius Dadarlat.

- Date:
**4/24/20****Corey Jones**, The Ohio State University- Title: Rank finiteness for braided fusion categories
- Abstract: Fusion categories arise as finite systems of bifinite bimodules of II1 factors, and play an important role in the theory of subfactors and algebraic quantum field theory. The number of (isomorphism classes) of simple objects in a fusion category is called its rank. It is natural to ask whether there are only finitely many fusion categories for each rank. While this question is wide open in general, we show that there are only finitely many braided fusion categories of each rank, interpolating the classical result for symmetric fusion categories and the more recent result for modular categories due to Bruillard, Ng, Rowell and Wang. Based on joint work with Scott Morrison, Dmitri Nikshych and Eric Rowell.

- Date:
**5/1/20****Srivatsav Kunnawalkam Elayavalli**, Vanderbilt University- Title: On Jung-type behavior in II_1 factors.
- Abstract: In mathematics, one of the central philosophies is to try and understand structural properties of an object by looking at how it embeds into familiar and rich objects. In this talk, we will look at separable II_1 factors and try to understand their structure by looking at the space of its embeddings into ultraproduct von Neumann algebras, like R^w for instance. The main motivation is a theorem of Jung that says that if any two embeddings (provided they exist) of a tracial von Neumann algebra N into R^w are unitarily conjugate, then N is amenable. In recent joint works with S. Atkinson and I. Goldbring, we have been able to generalize this theorem in very interesting directions. We will look at some of these results in this talk.

- Date:
**5/8/20****Vaughan Jones**, Vanderbilt University- Title: Bergman space zero sets, modular forms, von Neumann algebras and ordered groups
- Abstract: $A^2_{\alpha}$ will denote the weighted $L^2$ Bergman space. Given a subset $S$ of the open unit disc we define $\Omega(S)$ to be the infimum of $\{s| \exists f \in A^2_{s-2}, f\neq 0, \mbox{ having $S$ as its zero set} \}$. By classical results on Hardy space there are sets $S$ for which $\Omega(S)=1$. Using von Neumann dimension techniques and cusp forms we give examples of $S$ where $1<\Omega(S)<\infty$. By using a left order on certain Fuchsian groups we are able to calculate $\Omega(S)$ exactly if $\Omega (S)$ is the orbit of a Fuchsian group. This technique also allows us to derive in a new way well known results on zeros of cusp forms and indeed calculate the whole algebra of modular forms.

- Date:
**5/15/20****Spencer Dowdall**, Vanderbilt University- Title:
- Abstract:

- Date:
**5/22/20****Zhengwei Liu**, Harvard University- Title: Fusion bialgebras and Fourier analysis--new analytic obstructions of categorification.
- Abstract: In recent work joint with Jinsong Wu and Sebastien Palcoux, we introduce fusion bialgebras and their duals and systematically study their quantum Fourier analysis, inspired by quantum Fourier analysis on subfactors, such as quantum analogues of Hausdorff-Young inequality, Young's inequality, sum-set estimates and uncertainty principles. As an application, we discover new analytic obstructions on the unitary categorification of fusion rings. In particular, the Schur product property holds on the Grothendieck ring of a unitary fusion category, but not on a fusion ring, which turns out to be a surprisingly efficient analytic obstruction of unitary categorification.

- End of Spring Semester.