# Trails in a Non Commutative Land

Titles, Asbtracts and Slides

## SISSA - Trieste, May 18-20, 2011

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## Jean Bellissard

### Transverse geometry on tiling spaces

Abstract: this mini-course will be divided into three equal parts of one hour each.

- In a first part the notion of Delone set or tiling which are repetitive with finite local complexity will be introduced and studied. The Hull and the transversal will be described. Associated with such a tiling are a Bratteli diagram (Combinatorial description), a groupoid (Algebraic description) and the Anderson-Putnam complex (Geometric description).

References:

- J. Bellissard, D. Herrmann, M. Zarrouati, Hull of Aperiodic Solids and Gap Labelling Theorems (pdf) in Directions in Mathematical Quasicrystals, CRM Monograph Series, Volume 13, 207-259, (2000), M.B. Baake and R.V. Moody Eds., AMS Providence.
- J. Bellissard, R. Benedetti, J.-M. Gambaudo, Spaces of Tilings, Finite Telescopic Approximations and Gap-labelling, Commun. Math. Phys., 261, (2006), 1-41.
- J. Bellissard, A. Julien, J. Savinien, Tiling Groupoids And Bratteli Diagrams, arXiv: 0911.0080 [math.OA] Annales Henri Poincar

- The notion of metric on a tiling space will be introduced and discussed. The transversal will become an ultrametric Cantor set. The notion of Michon graph, and its relation with the Bratteli diagram will be introduced. Using the concept of spectral triple, the transversal, and more generally an ultrametric Cantor set will be described as a (noncommutative) Riemannian manifold. The analog of the Laplace Beltrami operator will be introduced. Some properties of this operator will be listed.

Reference:- J. Pearson, J. Bellissard, Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets arXiv: 0802.1336v1 [math.OA] J. Noncommut. Geo., 3, (2009), 447-480

- Extending the spectral triple to the algebra of the groupoid is a problem that will be discussed as an extension of the notion of spectral metric space and of dynamical systems on it. A spectral metric space is the noncommutative analog of a compact metric space, while the dynamic is represented by one homeomorphisms. It will be shown that this extension is possible if and only this homeomorphism is conjugate to an isometry. In the tiling space this homeomorphism is hyperbolic and does not respect the metric. To overcome this difficulty, we will propose to use a method initiated by Connes and Moscovici in 1994, namely the use of the metric bundle over a manifold, on which all diffeomorphism becomes an isometry.

Reference:- Bellissard J., Marcolli M., Reihani K. Dynamical Systems on Spectral Metric Spaces, submitted to Journal of Noncommutative Geometry arXiv: 1008.4617 (August 30 2010)

## Fabio Cipriani

### Noncommutative Potential Theory

The discussion is intended to show how is possible to develop aspects of potential theory on noncommutative C*-algebars with trace, based on symmetric Markovian semigroups or Dirichlet forms. This includes the canonical differential calculus associated to Dirichlet forms, the study of finite energy states and their potentials, multipliers of the Dirhclet algebras and, in particular, the generalization of the Deny embeddings concerning the distribution of energy i.e. the measure class of the carré du champ.

## Gaetano Fiore

### On the relation between quantum mechanics with a magnetic field on **R**^{n} and
on a torus **T**^{n}

We consider a scalar charged quantum particle on **R**^{n} subject
to a background U(1) gauge potential A. We clarify in the general
setting how the requirement that the wavefunctions be `quasiperiodic'
under translations in a lattice L leads to a periodic field strength
B=dA with integral fluxes and to the analogous theory on the torus
**T**^{n}=**R**^{n}/L:
the wavefunctions defined on **R**^{n}
play the role of sections
of the associated hermitean line bundle E on
**T**^{n}, since E can be
described as a quotient, beside by local trivializations. The
covariant derivatives corresponding to a constant B generate a Lie
algebra g_{Q} and together with the periodic functions the
′algebra of
observables′ O_{Q}.
The non-abelian part of g_{Q} is a Heisenberg Lie
algebra with the electric charge operator Q as the central generator;
the corresponding Lie group G_{Q} acts on the Hilbert space as the
translation group up to phase factors. Also the space of sections of E
is mapped into itself by a g in G_{Q}.
We identify the socalled magnetic
translation group as a subgroup of the observables′ group
Y_{Q} and
determine the unitary irreducible representations of O_{Q},
Y_{Q}
corresponding to integer charges. We also clarify how in the n=2m case
a holomorphic structure and Theta functions arise on the associated
complex torus. These results apply equally well to the physics of
charged scalar particles both on **T**^{n} and on
**R**^{n} in the presence of
periodic magnetic field B and scalar potential.

## Laszlo Erdös

### Quantum Brownian motion

Einstein's kinetic theory of the Brownian motion, based upon light water molecules continuously bombarding a heavy pollen, provided an explanation of diffusion from the Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from the Schrödinger equation. The first step in this program is to verify the linear Boltzmann equation as a certain scaling limit of a Schrödinger equation with random potential. In the second step, one considers a longer time scale that corresponds to infinitely many Boltzmann collisions. The intuition is that the Boltzmann equation then converges to a diffusive equation similarly to the central limit theorem for Markov processes with sufficient mixing. In these lecture notes we present the mathematical tools to rigorously justify this intuition. The new material relies on joint papers with H.-T. Yau and M. Salmhofer.

## Giovanni Landi

### Gauge fields over non-commutative manifolds

Starting from monopoles and instantons as connections on bundles over spheres, we arrive to very natural deformations of spaces and bundles. The ′noncommutative′ manifolds and vector bundles that one obtains have very interesting and rich geometrical structures that can be described with natural tools.

## Pier Alberto Marchetti

### Quantum logic and non-commutative geometry

We propose a general scheme for the "logic" of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non Commutative Geometry. It involves Baire*-algebras, the non-commutative version of measurable functions, arising as envelope of the C*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of the state of "initial conditions" in the standard (W*-)algebraic approach to classical systems.

## Gianluca Panati

### Geometric derivation of the TKNN equations

I will present a rigorous geometric derivation of the TKNN equations. Each equation is interpreted as a relationship between the Chern numbers of two relevant vector bundles over the two-dimensional torus, associated respectively to the Harper and to the Hofstadter operator. More generally, we prove that an analogous equation holds true for any smooth orthogonal projector affiliated to the rational rotation C*-algebra, alias the algebra of the (rational) noncommutative torus. Finally, by assuming a "noncommutative geometry viewpoint", we rewrite the generalized TKNN equation in a form which is valid also in the case of irrational value of the deformation parameter, which physically corresponds to the magnetic flux per unit cell in a model for the Quantum Hall effect. The talk is based on joint work with Giuseppe De Nittis and partially with Frederic Faure.

## Andrea Posilicano

### On the many Dirichlet laplacians on a non-convex polygon

By Birman and Skvortsov it is known that if Ω is a plane curvilinear
polygon with *n* non-convex corners then the Laplace operator with
domain H^{2}(Ω)∩ H^{1}_{0}(Ω) is a closed symmetric operator
with deficiency indices *(n,n)*. Here, by providing all self-adjoint
extensions of such an operator, we determine the set of self-adjoint
non-Friedrichs Dirichlet Laplacians on Ω and, by a corresponding Krein-type resolvent formula, show that any element in this set is the norm resolvent limit of a suitable sequence of Friedrichs-Dirichlet Laplacians with *n* point interactions.

## Jean Luc Sauvageot

### K-theory and homology on fractal spaces

The idea is to use the tools provided by noncommutative geometry to investigate the geometrical structure of fractal spaces, of which the prototype is the Sierpinsky Gasket in the plane.

We shall explain how the geometry is determined by an energy functional which is a Dirichlet form; and what are the associated Dirichlet space, and tangent module of "square integrable vector fields".

From this starting point, we shall propose three ways to get a hint on the geometry of the space:

- The first one is the natural Fredholm module associated with the differential calculus in the tangent space.
- The second one is the Riesz duality between vector fields and differential forms. We shall propose a theory of differential one-forms and their integrals along paths in the Gasket, in order to get a homological approach of fractal spaces.
- The last topic will consist in proposing an alternative way to get Fredholm modules by "deconstructing the Gasket", and show how the theory of A. Connes' Dixmier traces allows to recover the whole geometry from those Fredholm modules, through trace formulas or residue formulas.

All this stuff is from joint works with Fabio Cipriani, Daniele Guido e Tommaso Isola

## Alessandro Zampini

### A class of Laplacians on the quantum SU(2) and the standard Podles′ sphere

In this talk I shall describe a formulation to introduce Hodge duality operators, and the corresponding Laplacians, on the quantum group SU(2), in terms of the spectral properties of the antisymmetriser operators associated to the specific differential calculi. A class of them will be then projected on the induced exterior algebras over the Podles′ quantum sphere.