Purdue Operator Algebras Seminar
Spring 2021
Organizer: Thomas Sinclair. email: tsincla(at)purdue(dot)edu
Tuesdays, 1:30-2:30pm, eastern
- Date: 2/9/21
- Therese-Marie Landry (UC Riverside)
- Title: Metric Convergence of Spectral Triples on the Sierpinski Gasket and Other Fractal Curves
- Abstract: Many important physical processes can be described by differential equations. The solutions of such equations are often formulated in terms of operators on smooth manifolds. A natural question is to determine whether differential structures defined on fractals can be realized as a metric limit of differential structures on their approximating finite graphs. One of the fundamental tools of noncommutative geometry is Alain Connes' spectral triple. Because spectral triples generalize differential structure, they open up promising avenues for extending analytic methods from mathematical physics to fractal spaces. The Gromov-Hausdorff distance is an important tool of Riemannian geometry, and building on the earlier work of Marc Rieffel, Frederic Latremoliere introduced a generalization of the Gromov-Hausdorff distance that was recently extended to spectral triples. The Sierpinski gasket can be viewed as a piecewise \(C^1\)-fractal curve, which is a class of fractals first formulated by Michel Lapidus and Jonathan Sarhad for their work on spectral triples that recover the geodesic distance on these spaces. In this talk, we will motivate and describe how their framework was adapted to our setting to yield approximation sequences suitable for metric approximation of spectral triples on piecewise \(C^1\)-fractal curves.
- Zoom ID: 913 9257 6733, Passcode: 441923
- Date: 2/23/21
- Daniel Drimbe (KU Leuven)
- Title: New examples of W* and C*-superrigid groups
- Abstract: Classes of von Neumann algebras that are very studied are due to the work of Murray and von Neumann. They associated in a natural way a von Neumann algebra L(G) to every countable discrete group G. The problem of classifying L(G) in terms of G is often very difficult since these algebras tend to have only a “faded memory” of the underlying group. A good illustration of this phenomenon is a remarkable theorem of Connes which asserts that that all icc amenable groups give rise to isomorphic von Neumann algebras. The non-amenable case is much more complex and reveals a striking rigidity phenomenon; many examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa’s deformation/rigidity theory. The most extreme form of rigidity is when G is W*-superrigid, meaning that L(G) completely remembers the underlying group G. There have been discovered only two types of group theoretic constructions that lead to W*-superrigid groups: some classes of generalized wreath products groups with abelian base (Ioana-Popa-Vaes '10, Berbec-Vaes '12) and amalgamated free products (Chifan-Ioana '17).
In this talk I will introduce several new constructions of W*-superrigid groups which include direct product groups, semidirect products with non-amenable core and HNN-extensions. I will also present some applications of these results to C*-algebras by presenting new examples of groups that are completely remembered by their reduced C*-algebra. This is based on a joint work with Ionut Chifan and Alec Diaz-Arias.
- Zoom ID: 998 5685 4583, Passcode: 361654
- Date: 3/16/21
- Pieter Spaas (UCLA)
- Title: Stable decompositions and rigidity for product equivalence relations
- Abstract: After discussing the motivation behind the talk and some necessary preliminaries, we will consider the ''stabilization'' of a countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group. Using a new local characterization of the Schmidt property, we show that this always gives rise to a so-called stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. We will also discuss some new structural results for product equivalence relations, which we will obtain using von Neumann algebraic techniques.
- Zoom ID: 973 9904 1618, Passcode: 795314
- Date: 3/23/21
- Sam Harris (Texas A& M)
- Title: Non-local games and entanglement
- Abstract: An increasing number of tasks or protocols that are carried out today rely heavily on the existence of entanglement, which is one of the fundamental concepts coming from quantum mechanics. Since the 1960s, non-local games (or interactive provers) have been used as a starting point for experiments to demonstrate certain forms of entanglement. Along the way, non-local games have had a significant impact in areas of mathematics such as operator algebras and computational complexity. In this talk, we'll look at the history of non-local games, and we will focus on some fascinating examples arising from finite, undirected graphs. As time allows, we will look at recent extensions of these games to quantum graphs.
- Zoom ID: 984 0855 6229, Passcode: 971324
- Date: 4/6/21
- Sven Raum (Stockholm University)
- Title: Right-angled Hecke operator algebras
- Abstract: With every Coxeter system one can associate a family of algebras considered as deformation of its group algebra. These so-called Hecke algebras, are classical objects of study in combinatorics and representation theory. Complex Hecke algebras admit a natural *-structure and a natural *-representation on a Hilbert space. Taking the norm- and SOT-closure in such representation, one obtains Hecke operator algebras, which have recently seen increased attention.
In this talk, I will motivate and introduce Hecke operator algebras, focusing on the case of right-angled Coxeter systems. This case is particularly interesting from an operator algebraic perspective, thanks to its description by iterated amalgamated free products. I will survey known results on the structure of Hecke operator algebras, before I describe recent joint work with Adam Skalski on the factor decomposition of right-angled Hecke von Neumann algebras as well as the K-theory of right-angled Hecke C*-algebras. Applications to representation theory will be described too.
- Zoom ID: 928 2648 1749, Passcode: 224396