• 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
  • Title: Higher quantum symmetries for commutative C*-algebras
  • Abstract: A compact Hausdorff space T has a familiar group of symmetries: its homeomorphism group. From the perspective of Morita theory however, the commutative C*-algebra C(T) associated to T admits higher quantum symmetries. In this talk, we will introduce the higher Morita category of commutative C*-algebras, C*-algebras, C*-correspondences, and adjointable maps. We will then describe these higher quantum symmetries for C(T) in terms of the cohomology groups of the underlying space T.

• Date: 10/9/26
  • Sayan Das, Vanderbilt University
  • Title: Geometry of Directed Landscape
  • Abstract: The directed landscape is a random metric model on the plane that arises as the scaling limit of classical metric models in the KPZ universality class. In this talk, I will discuss the model, some interesting properties of it, and discuss a large deviation principle for the random metric. Based on a joint work with Duncan Dauvergne and Balint Virag. 

• Date: 10/16/26
  • Yanyu Wang, Vanderbilt University
  • Title:
  • Abstract:

• Date: 10/23/26
  • No Meeting, Fall Break

• Date: 10/30/26
  • Darren Creutz, Vanderbilt University
  • Title: A Hierarchy Amongst Low Complexity Dynamical Systems
  • Abstract: The simplest nontrivial dynamical system is a rotation of the circle by an irrational angle. In joint work with R. Pavlov (JEMS, preliminary acceptance), we show that all systems of sufficiently low complexity (to be defined at the talk) share this sort of algebraic structure: all are isomorphic to rotations on compact abelian groups which are specific p-adic Lie groups by lattices.

    I will discuss that result and several related (TAMS '26 solo, Forum Math '23 joint w/ Pavlov, ETDS '23 solo, [in preparation] joint w/ T. Adams) shedding light on what appears to be a hierarchy amongst low complexity dynamical systems relating quantitative bounds on word complexity with qualitative structural statements about such systems.

• Date: 11/6/26
  • Thomas Sinclair, Purdue University
  • Title: The pentagonal system and the lifting property
  • Abstract: The Smith—Ward problem asks whether every 3-dimensional operator system has the lifting property. This was resolved in the negative in July 2026 by Scherer. In this talk I will present an alternate counterexample, namely the operator subsystem of C[Z/5Z] generated by a single, nontrivial character. I'll explain how a concrete construction of a non-liftable map surprisingly uses a construction from an undergraduate research project I had supervised. This is joint work with Andy Qiu. Link to the preprint: The regular pentagonal system and the Smith--Ward problem: https://hexagonmath.org/2610.00567v1

• Date: 11/13/26
  • Julio Caceres, Vanderbilt University
  • Title: What becomes of continuous dimension in a Subfactor?
  • Abstract: In a II_1 factor the trace gives projections a continuous dimension: every value in [0,1] occurs. Replace the scalars by a subfactor N ⊂ M, and the conditional expectation E_N(p) becomes a "dimension of p relative to N," now taking values in N. Which positive contractions of N arise this way? In finite index, the Pimsner–Popa inequality rules out most of them, and the question reduces to Popa's relative dimension set Λ(M,N), the scalars α with E_N(p) = α1. Popa computed Λ(M,N) in 1989 except on a middle interval that opens up once the index exceeds 4, and that interval has remained open since.

    I will explain two results. First, in joint work with Dietmar Bisch, we show that if N ⊂ M has uniform infinite index, then every positive contraction of N is the expectation of a projection. In particular this holds for crossed products by infinite groups and for L(H) ⊂ L(G) with [G:H] = ∞. Second, for every extremal inclusion of finite index d > 4, the whole middle interval lies in Λ(M,N). Together, these show that Λ(M,N) depends only on the index for every irreducible inclusion of II₁ factors.

• Date: 11/16/26 (Monday), 4:10pm-5:30pm, SC 1313
  • Michael Brannan, University of Waterloo
  • Title: Unital channels and the operator space spanned by the generators of free unitary quantum groups
  • Abstract: The Birkhoff-von Neumann theorem tells us that any bistochastic matrix can be decomposed into a convex combination of permutation matrices. Since bistochastic matrices are precisely the unital classical information channels, it is natural to ask about the convex structure of unital quantum channels acting on $M_n$. I'll discuss various attempts at formulating a quantum channel version of the Birkhoff-von Neumann, including our recent attempt based on non-commutative (NC) convexity: is every unital channel on $M_n$ an NC convex combination of quantum automorphisms of $M_n$? We will exhibit examples which show that the answer to this question is no in general, and relate them to the notion of a factorizable channel studied by Anantharaman-Delaroche, Haagerup, Musat, and others. We will also connect these results to the problem of determining the structure of the operator space spanned by the generators of the free unitary quantum groups.

    This is joint work with Jason Crann and Alec Gow.

• Date: 11/27/26
  • No Meeting, Thanksgiving Break

• Date: 12/3/26 (Thursday), Special Subfactor Seminar, 2:45pm-3:45pm, SC 5211
  • Vern Paulsen, University of Waterloo
  • Title: Open problems about quantum channels and k-positive maps
  • Abstract: A quantum channel is called mixed unitary if it is a convex combination of unitary conjugations. But if one knows that a channel is mixed unitary, how does one find the fewest number of unitaries needed to represent it? We give an estimate in terms of the Choi rank of the channel, but it is not known if the estimate is saturated. A second set of questions concerns k-positive maps. These maps appear in QI as the Stormer duals of the k-entanglement breaking maps. Currently it is unknown how large the cb-norm of a unital k-positive map on a matrix algebras can be. This question is also related to problems about exactness and the lifting problem for finite dimensional operator systems. If time permits we will also discuss a problem about the confusability graphs of quantum channels.

• Date: 12/4/26
  • Vern Paulsen, University of Waterloo
  • Title: Amenable traces and the joint numerical radius
  • Abstract: We provide necessary and sufficient characterizations of the existence of an amenable trace on a C*-algebra in terms of the joint free numerical radius of tuples of unitaries, isometries, and partial isometries in the algebra. We apply these results to obtain new obstructions to various lifting properties. Using this joint numerical radius instead of earlier norm inequalities leads to more direct proofs of some known results. We also include a survey of some other results about the joint numerical radius.

• End of Fall Semester.

Past NCGOA and Subfactor seminars