• Date: 9/4/26
  • Dietmar Bisch, Vanderbilt University
  • Title: On the Jones index set of irreducible, hyperfinite subfactors
  • Abstract: Since Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Jones' rigidity theorem establishes all indices between 1 and 4, but above 4, there are many open questions.
    Caceres and I have recently shown that all indices of finite depth subfactors between 4 and 5 are also realized by new hyperfinite subfactors with Temperley-Lieb-Jones standard invariant. They are non-amenable, but have certain nice asymptotic commutativity properties.
    Our work leads to a conjecture and some results regarding Jones' index problem. The construction involves new families of commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors. It turns out that our subfactors are asymptotically commutative, a notion that I will explain. I will show how to construct subfactors that do not have this commutativity property.

• Date: 9/11/26
  • Michael Montgomery, Cascade Quantum
  • Title: Measure-free Koopman--von Neumann Dynamics and Noncommutative Geometry
  • Abstract: This talk is on recent joint work with Dimitrios Giannakis. The Koopman--von Neumann formulation of classical statistical dynamics maps the isometric evolution of probability densities in $L^1$ under the Liouville equation to a unitary evolution of quantum mechanical wavefunctions in $L^2$ generated by the antisymmetric part of the Liouville operator. This approach enables the use of Hilbert space techniques to model the statistical evolution of observables.
    However, being an $L^2$ method, it is not suitable for representing pointwise evolution along dynamical trajectories.
    Here, we propose a Koopman--von Neumann framework that replaces the $L^p$ spaces associated with a volume measure on the state space manifold, $X$, by a reproducing kernel Hilbert space (RKHS), $\mathcal H$, of continuous functions on $X$ that satisfy joint analyticity conditions with respect to the generator (vector field), $V$, of the dynamics and its RKHS adjoint. Through a series of dilations of the skew adjoint part of $V$ on $\mathcal H$ we build a a family of weak spectral triples $(\mathcal A_n, \mathfrak F, -i \mathcal D)$, $n \in \mathbb N$, wherein  $\mathcal A_n$ are non-abelian $*$-algebras generated by creation and annihilation operators and $-i \mathcal D$ plays the role of a Dirac operator (generally, with noncompact resolvent) inducing extended pseudometrics on the state spaces of $\mathcal A_n$. Using a state space embedding $\Psi \colon X \to S( \cup_n \mathcal A_n)$ the original dynamical system is recoverable from the spectral triple via the action of $e^{\mathcal D t}$. Furthermore, for large collections of examples, cutting down the weak spectral triples  $(\mathcal A_n, \mathfrak F, -i \mathcal D)$ to subrepresentations of $A_n$ yields true spectral triples with compact resolvent for $-i\mathcal D$.

• Date: 9/15/26 (Tuesday), 4:10pm-5:30pm, SC 1432
  • Greg Patchell, Vanderbilt University
  • Title: Isomorphism of infinite reduced free product C*-algebras
  • Abstract: In the 1990s, Dykema classified free products of approximately finite-dimensional von Neumann algebras. All such free products give rise to a direct sum of finite-dimensional algebras and interpolated free group factors. Consequently, free products in the category of von Neumann algebras is a very coarse operation. By contrast, reduced free products of C*-algebras often stay non-isomorphic, at least in known examples. The main known reason is the presence of K-theoretical obstructions. Recently, Hirshberg-Philips showed using Robert's classification of 1-dimensional NCCW complexes that infinite reduced free products of C*-algebras freely absorb algebras with trivial K-theory (such as C([0,1]) or the Jiang-Su algebra). In joint work with Curda, Hua, Pi, Shiner, and White, we use modern C*-classification techniques and a novel intertwining argument to recover Hirshberg-Philips in the case the C*-algebras are all nuclear and more generally show that such infinite reduced free products only see K-theory. As an application, we give an example of two non-amenable groups whose reduced group C*-algebras are isomorphic.

• Date: 9/25/26 Zoom talk at the special time 6pm-7:20pm
  • Zishuo Zhao, Tsinghua University
  • Title: Axiomatization of Levin-Wen wave function
  • Abstract: Levin-Wen model describes topological orders for a given unitary fusion category. An intrinsic characterization of the Levin–Wen ground-state wave functions without prior categorical symmetry nor Hamiltonian is an open question. We address this  in terms of six axioms on a family of wave functions associated with lattices in different scales. These conditions allow us to reconstruct the underlying unitary fusion category, and prove that the wave functions are the corresponding Levin–Wen ground states. By weakening some of the conditions, our approach provides a framework for studying lattice models before RG fixed points.

• Date: 10/2/26
  • Giovanni Ferrer, Vanderbilt University
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• Date: 10/9/26
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• Date: 10/16/26
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• Date: 10/23/26
  • No Meeting, Fall Break

• Date: 10/30/26
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• Date: 11/6/26
  • Thomas Sinclair, Purdue University
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• Date: 11/13/26 (no meeting tentatively)
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• Date: 11/20/26 (no meeting tentatively)
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• Date: 11/27/26
  • No Meeting, Thanksgiving Break

• Date: 12/4/26
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• End of Fall Semester.

Past NCGOA and Subfactor seminars