Narrow Results

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

We study the spectral geometry of the quantum projective plane , a deformation of the complex projective plane ℂP², the simplest example of spin^c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0⁺-summable triple, equivariant under U_q(su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum.

We study a three-dimensional differential calculus on the standard Podleś quantum two-sphere , coming from the Woronowicz 4D₊ differential calculus on the quantum group SU_q(2). We use a frame bundle approach to give an explicit description of and its associated spin geometry in terms of a natural spectral triple over . We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.

We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3 + 1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal–Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables us to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level $\ell$ projective unitary positive-energy representations of any given loop group $\mathrm{\text{L}}G$. The construction is based on certain supersymmetric conformal field theory models associated with $\mathrm{\text{L}}G$ in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed product $C^{*}$-algebra $A{⋊}_{r}G$ was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], extending the construction by Belissard, Marcolli and Reihani in [Dynamical systems on spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the Kasparov product to make an ansatz for the Dirac operator. Supposing that the triple on $A$ is equivariant for an action of $G$, we show that the triple on $A{⋊}_{r}G$ is equivariant for the dual coaction of $G$. If moreover an equivariant real structure $J$ is given for the triple on $A$, we give constructions for two inequivalent real structures on the triple $A{⋊}_{r}G$. We compute the KO-dimension with respect to each real structure in terms of the KO-dimension of $J$ and show that the first- and the second-order conditions are preserved. Lastly, we characterize an equivariant orientation cycle on the triple on $A{⋊}_{r}G$ coming from an equivariant orientation cycle on the triple on $A$. We show, along the paper, that our constructions generalize the respective constructions of the equivariant spectral triple on the noncommutative $2$-torus.

The question whether an operator belongs to the domain of some singular trace is addressed, together with the dual question whether an operator does not belong to the domain of some singular trace. We show that the answers are positive in general, namely for any (compact, infinite rank) positive operator A we exhibit two singular traces, the first being zero and the second being infinite on A. However, if we assume that the singular traces are genrated by a "regular" operator, the answers change, namely such traces alway vanish on trace-class, non singularly traceable operators and are always infinite on non trace-class, non singularly traceable operators. These results are achieved on a general semifinite factor and make use of a new characterization of singular traceability (cf. [7]).

Starting with a vertex-weighted pointed graph $(\mathrm{\Gamma },\mu ,v_0)$, we form the free loop algebra $S_0$ defined in Hartglass–Penneys’ article on canonical ${\mathrm{\text{C}}}^{\ast }$-algebras associated to a planar algebra. Under mild conditions, ${{𝒮}}_0$ is a non-nuclear simple ${\mathrm{\text{C}}}^{\ast }$-algebra with unique tracial state. There is a canonical polynomial subalgebra $A\subset {{𝒮}}_0$ together with a Dirac number operator $N$ such that $(A,L^{2}A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify $({{𝒮}}_0,A,N)$ yields a compact quantum metric space in the sense of Rieffel.

We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our ${\mathrm{\text{C}}}^{\ast }$-algebras are non-nuclear, we adjust the Lip-norm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces.

As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) ${\mathrm{\text{C}}}^{\ast }$-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS ${\mathrm{\text{C}}}^{\ast }$-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance.

We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle .

This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk concentrates on two main aspects: the relation of Stark numbers to the geometry of noncommutative tori with real multiplication, and the shadows of modular forms on the noncommutative boundary of modular curves, that is, the moduli space of noncommutative tori.

The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action.

The idea here is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein–Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one.

The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators on Hilbert spaces, the notion of noncommutative residue, Dixmier trace, pseudodifferential operators etc. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus.