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Texas A&M University
Mathematics

Noncommutative Geometry Seminar

Fall 2021

 

Date:September 17, 2021
Time:2:00pm
Location:ZOOM
Speaker:Giles Gardam , Muenster
Title:Kaplansky’s conjectures
Abstract:The Kadison–Kaplansky conjecture states that the reduced C*-algebra of a torsion-free discrete group has no idempotents other than 0 and 1. It holds for groups satisfying the Baum–Connes conjecture. If we restrict focus to group algebras, there are stronger conjectures attributed to Kaplansky on zero divisors and units. I will discuss these conjectures and my counterexample to the unit conjecture.

Date:September 24, 2021
Time:2:00pm
Location:ZOOM
Speaker:Marc Rieffel , University of California at Berkeley
Title:Dirac Operators for Matrix Algebras Converging to Coadjoint Orbits
Abstract:In the high-energy physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as C*-metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces like coadjoint orbits), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. I will sketch a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. As Connes showed us, from Dirac operators we may obtain C*-metrics. Our unified construction enables us to prove our main theorem, whose content is that, for the C*-metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance. This is a long story, but I will sketch how it works.

Date:October 6, 2021
Time:2:00pm
Location:Zoom
Speaker:Jesus Sanchez Jr, Penn State
Title:The Geometry of Mehler's Kernel
Abstract:Mehler's Kernel made its first appearance in Index Theory through the work of Ezra Getzler in his computation of the index of a Dirac Operator. The appearance of Mehler's Kernel in this approach is through the introduction of a symbol calculus which refines the usual symbol calculus of differential operators and smoothing operators. In a different approach to computing the index of a Dirac Operator, Nicole Berline and Michele Vergne study heat flow on the principal Spin bundle and show that the corresponding Index density arises naturally by studying the local geometry in this setting. What we will show is that we can extend the insight of Berline and Vergne to fully recover Mehler's Kernel and give geometric insight into the curvature terms appearing within the kernel. This will give a more unified treatment of these two seemingly different proofs of the local index theorem for Dirac Operators.

Date:October 8, 2021
Time:2:00pm
Location:ZOOm
Speaker:Benjamin Steinberg, City College of New York and CUNY Graduate Center
Title:Cartan pairs of algebras
Abstract:In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting “nicely” inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence). In recent years, Matsumoto and Matui exploited that for directed graphs satisfying Condition (L), the corresponding graph C*-algebra and its commutative subalgebra of functions on the path space of the graph form a Cartan pair to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type. The key point was translating these dynamical conditions into groupoid language. Since the Leavitt path algebra associated to a graph is the “Steinberg” algebra of the same groupoid, this led people to wonder about whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the path space. The answer is yes and it turns out in the algebraic setting, one doesn’t even need Condition (L). Initially work was focused on recovering an ample groupoid from the pair consisting of its “Steinberg” algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture. This is joint work with Becky Armstrong, Gilles G. de Castro, Lisa Orloff Clark, Kristin Courtney, Ying-Fen Lin, Kathryn McCormick, Jacqui Ramagge and Aidan Sims

Date:October 20, 2021
Time:3:00pm
Location:BLOC 302
Speaker:Simone Cecchini, University of Gottingen
Title:Distance estimates in the spin setting and the positive mass theorem
Abstract:The positive mass theorem states that a complete asymptotically Euclidean manifold of nonnegative scalar curvature has nonnegative ADM mass. It relates quantities that are defined using geometric information localized in the Euclidean ends (the ADM mass) with global geometric information on the ambient manifold (the nonnegativity of the scalar curvature). It is natural to ask whether the positive mass theorem can be ``localized’’, that is, whether the nonnegativity of the ADM mass of a single asymptotically Euclidean end can be deduced by the nonnegativity of the scalar curvature in a suitable neighborhood of E.

I will present the following localized version of the positive mass theorem in the spin setting. Let E be an asymptotically Euclidean end in a connected Riemannian spin manifold (M,g). If E has negative ADM-mass, then there exists a constant R > 0, depending only on the geometry of E, such that M must either become incomplete or have a point of negative scalar curvature in the R-neighborhood around E in M. This gives a quantitative answer, for spin manifolds, to Schoen and Yau's question on the positive mass theorem with arbitrary ends. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions <8 assuming Schwarzschild asymptotics on the end E. I will also present explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E. The bounds obtained are reminiscent of Gromov's metric inequalities with scalar curvature. This is joint work with Rudolf Zeidler.

Date:October 22, 2021
Time:2:00pm
Location:ZOOM
Speaker:Inna Zakharevich, Cornell University
Title:Characteristic polynomials and traces
Abstract:In this talk we give a description of a lift of the Dennis trace and the characteristic polynomial to TR using the framework of bicategories and bicategorical traces. The goal of this construction is to demonstrate that TR is the natural home of the characteristic polynomial, and to give a natural and clean demonstration of this fact. The overarching goal of this project is to show that most traces have natural interpretations in terms of bicategories; time permitting, we will show that the Reidemeister trace associated to a self-map of a finite CW complex is another example of such a construction.

Date:October 29, 2021
Time:2:00pm
Location:ZOOM
Speaker:Marc Rieffel , University of California at Berkeley
Title:Dirac Operators for Matrix Algebras Converging to Coadjoint Orbits
Abstract:In the high-energy physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as C*-metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces like coadjoint orbits), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. I will sketch a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. As Connes showed us, from Dirac operators we may obtain C*-metrics. Our unified construction enables us to prove our main theorem, whose content is that, for the C*-metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance. This is a long story, but I will sketch how it works.

Date:November 5, 2021
Time:2:00pm
Location:ZOOM
Speaker:Ernesto Lupercio, Cinvestav, Mexico
Title:On the moduli space for Quantum Toric Varieties
Abstract:Just as the quantum torus generalizes the ordinary commutative torus, quantum toric varieties generalize ordinary ones. But unlike ordinary ones, quantum torics form moduli spaces with a rich structure. In this talk I will describe these spaces and some examples of its compactifications. Joint work with Katzarkov, Meersseman, and Verjosky.

Date:November 10, 2021
Time:2:00pm
Location:ZOOM
Speaker:Bo Zhu, University of Minnesota
Title:Geometry of positive scalar curvature on complete three-dimensional manifolds
Abstract:One of the basic questions related to positive scalar curvature is how the positive scalar curvature controls the size of geometry, many concepts and conjectures have been introduced by Gromov and many questions remain conjectural level. In this talk, we will discuss the interplay of geometry and positive scalar curvature on a complete manifold. Particularly, we will talk about volume growth of geodesic balls, an estimate of integral of scalar curvature and Uryson width on complete manifolds with nonnegative Ricci curvature and strictly positive scalar curvature based on my recent progress.

Date:November 12, 2021
Time:2:00pm
Location:ZOOM
Speaker:Dustin Clausen, University of Copenhagen
Title:New foundations for functional analysis
Abstract:I will discuss joint work with Peter Scholze in which we propose replacements for the standard categories of topological vector spaces and complete locally convex topological vector spaces, called condensed vector spaces and p-liquid vector spaces. We claim that these replacements are much more suitable for performing algebraic constructions, while still housing all standard examples. I will explain why and give some examples.

Date:November 19, 2021
Time:2:00pm
Location:ZOOM
Speaker:Gilles de Castro , Federal University of Santa Catarina
Title:C*-algebras and Leavitt path algebras for labelled graphs
Abstract:Since the work of Cuntz and Krieger in 1980, there has been an interest in studying C*–algebras associated with subshifts and combinatorial objects, such as 0-1 matrices and graphs. Bates and Pask introduced a class of algebras associated with labelled graphs that generalised several of the previously mentioned algebras. A couple of decades before the work of Cuntz and Krieger, Leavitt studied a class of rings that do not satisfy the IBN. In the early 2000s, Ara et al. studied a purely algebraic analogue of the Cuntz-Krieger algebras, and they observed that when considering the matrix consisting only of 1’s (which in the C*-algebra are the Cuntz algebras), we obtain a subclass of Leavitt rings. Motivated by this work, Abrams and Pino introduced an algebraic analogue of C*-algebras of graphs, which they called Leavitt path algebras. These algebras share several properties and a way of proving this is by considering them as groupoid algebras/C*-algebras. In this talk, I will present the C*-algebras of labelled graphs and their algebraic counterpart and I will explain how we can obtain groupoid models for these algebras. I will also talk about the approach using partial actions and give some conditions on partial actions of the free group such that they can be modelled using labelled graphs.