• Date: 9/6/24
  • Rolando de Santiago, Cal State University Long Beach
  • Title: Bounding quantum chromatic numbers of quantum graphs.
  • Abstract: In this talk we will discuss extensions of the 4 fundamental products of graphs (cartesian, categorical, lexicographical, and strong products) to quantum graphs, and provide bounds on the resulting graphs akin to those for products of classical graphs. We will pay particular attention to the lexicographical product, discussing our notion of a quantum b-fold chromatic number as a tool for computing the quantum chromatic number of the lexicographical products.

• Date: 9/13/24
  • David Penneys, The Ohio State University
  • Title: Subfactor techniques in topological order
  • Abstract: Recently, operator algebras and subfactor theory have been used in multiple ways to analyze topologically ordered spin systems. We will discuss several of these methods and their applications.

• Date: 9/20/24
  • Adriana Fernandez I Quero, University of Iowa
  • Title: Connes rigidity conjecture for groups with infinite center
  • Abstract: In this paper we propose for study a natural version of Connes Rigidity Conjecture (1982) which involves property (T) groups with infinite center. Using methods at the rich intersection between von Neumann algebras and geometric group theory we provide several instances when this holds. This is joint work with Ionut Chifan, Denis Osin, and Hui Tan.

• Date: 9/27/24
  • Noah Snyder, Indiana University
  • Title: Braided trivalent categories and the Exceptional Series
  • Abstract: One major topic in planar algebras is to study "simple skein theories" which here "skein theory" means we are looking at planar algebras generated by certain simple diagrams (like trivalent vertices) and "simple" means the dimensions of the box spaces aren't too big. For example, questions of this kind were studied by Kazhdan-Wenzl (oriented skein theories generated by a 2-box), Wenzl-Tuba (unoriented braided skein theories), Bisch-Jones-Liu (shaded skein theories generated by a 2-box), and Kuperberg and Morrison-Peters-Snyder (skein theories generated by a trivalent vertex). In work joint with Thurston and joint in part with Morrison, we study simple braided trivalent skein theories. It turns out that for generic values of a twist parameter these skein theories are closely related to the conjectural exceptional family of Deligne-Vogel-Cvitanovic. It's also interesting to look at what happens when this twist parameter is a small root of unity, in which case we see examples in the G2, F4, and S_t families as well as a possible new example at a fifth root of unity.

• Date: 10/4/24
  • Bat-Od Battseren, Vanderbilt University
  • Title: M_d type approximation properties for locally compact groups
  • Abstract: M_d type approximation properties are group approximation properties that are studied in connection to Dixmier's similarity problem. These properties are known to be stable under measure equivalence, W*-equivalence, and von Neumann equivalence. In this talk, we will discuss how we can define these properties for locally compact second countable groups and show that lattices and their ambient group share the same properties.

• Date: 10/11/24
  • No Meeting, Fall Break

• Date: 10/18/24
  • Madeline Brandt, Vanderbilt University
  • Title: The virtual Euler characteristic for binary matroids
  • Abstract: Inspired by Kontsevich's graphic orbifold Euler characteristic, we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank r. Our main result provides a simple formula for the virtual Euler characteristic for the set of isomorphism classes of binary matroids of rank r. This is joint work with Juliette Bruce and Dan Corey.

• Date: 10/25/24
  • Darren Creutz, U.S. Naval Academy
  • Title: Word complexity of partially mixing symbolic dynamical systems
  • Abstract: A subshift is a closed shift-invariant subset of \mathcal{A}^{\mathbb{Z}} where \mathcal{A} is some finite set, the 'alphabet'. Endowing a subshift with a probability measure, there are then natural questions about how the asymptotic mixing properties of the system relate to quantitative aspects such as the number of distinct words p(q) of a given length q appearing anywhere in the subshift. In joint work with R. Pavlov, we established that weak mixing can manifest in systems with \lim p(q)/q = 1.5 but that any system with \limsup p(q)/q < 1.5 is isomorphic to a rotation on a compact abelian (adelic) group. Relatedly, I established that strong mixing can manifest in systems with p(q) < qf(q) for arbitrary f(q) \to \infty but that \liminf p(q)/q < \infty precludes strong mixing. I will present recent work, joint with Terry Adams, on the word complexity behavior of partially mixing systems. Specifically, we show the existence of partially mixing systems with \liminf p(q)/q = 2 and establish that, for rank-one systems, this is optimal.

• Date: 11/1/24
  • Julio Cáceres, Vanderbilt University
  • Title: New hyperfinite subfactors with infinite depth
  • Abstract: We will present new examples of irreducible, hyperfinite subfactors with trivial standard invariant and interesting Jones indices. These are obtained by constructing new finite dimensional commuting squares. We will use two graph planar algebra embedding theorems and the classification of small index subfactors to show that our commuting square subfactors cannot have finite depth. We also present a one-parameter family of commuting squares that, by a classification result of Kawahigashi, will also yield irreducible finite depth subfactors. This is joint work with Dietmar Bisch.

• Date: 11/8/24
  • Daniel Drimbe, University of Iowa
  • Title: Unique prime decomposition results for equivalence relations
  • Abstract: Let G be a lattice in a simple connected real Lie group with finite center. The seminar work of Zimmer shows that the orbit equivalence relation R of any free ergodic probability measure preserving action of G is prime, i.e. R cannot be isomorphic to a product of equivalence relations that have infinite orbits. In my talk I will show that a product of such equivalence relations satisfies a unique prime factorisation phenomenon. This is joint work with Cyril Houdayer.

• Date: 11/15/24
  • Talia Fernós, University of North Carolina, Greensboro
  • Title: AU-Acylindricity in Higher Rank, and its Accompanying (Imperfect) Semi-Simple Dictionary.
  • Abstract: AU-Acylindricity may be viewed as generalizing the type of action a lattice enjoys on its ambient space. In a recent joint work with S. Balasubramanya, we extend the theory of acylindrically hyperbolic groups to the higher rank setting, using the theory of S-arithmetic lattices in semi-simple linear groups as motivation. This leads to an (imperfect) dictionary between the classical theory of algebraic groups and isometric actions on finite products of delta-hyperbolic spaces. In this talk we will focus on the techniques of this recent work and connect them to both classes of groups mentioned above.

• Date: 11/22/24
  • Ionut Chifan, University of Iowa
  • Title:
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• Date: 11/29/24
  • No Meeting, Thanksgiving Break

• Date: 12/6/24
  • Title:
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• End of Fall Semester.

Past NCGOA and Subfactor seminars