On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

Is life a thermal horizon ?

This talk aims at questioning the vanishing of Unruh temperature for an inertial observer in Minkovski spacetime with finite lifetime, arguing that in the non eternal case the existence of a causal horizon is not linked to the non-vanishing of the acceleration. This is illustrated by a previous result, the diamonds temperature, that adapts the algebraic approach of Unruh effect to the finite case.Comment: Proceedings of the conference DICE 2006, Piombino september 200

Carnot-Caratheodory metric and gauge fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Caratheodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes's distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator ? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Caratheodory distance dh defined by A. In this paper we precise this link, showing that the equality of d and dh strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).Comment: published version with additional figures to make the proof more readable. Typos corrected in this ultimate versio

Line element in quantum gravity: the examples of DSR and noncommutative geometry

We question the notion of line element in some quantum spaces that are expected to play a role in quantum gravity, namely non-commutative deformations of Minkowski spaces. We recall how the implementation of the Leibniz rule forbids to see some of the infinitesimal deformed Poincare transformations as good candidates for Noether symmetries. Then we recall the more fundamental view on the line element proposed in noncommutative geometry, and re-interprete at this light some previous results on Connes' distance formula.Comment: some references added. Proceedings of the Second Workshop on Quantum Gravity and Noncommutative Geometry, Universidade Lusofona, Lisbon 22-24 September 200

Connes distance and optimal transport

We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states

Museums and the making of textile histories: Past, present, and future

Many different types of museums collect, document, and preserve textiles, interpreting them through temporary and semi-permanent exhibitions, publications, and web- site interventions – sometimes independently, sometimes as part of a broader histo- ry of art and design, science and technology, social history and anthropology, local history or world cultures (for example, see the range and approaches in major fash- ion capitals such as London, Paris, Milan, New York with a long tradition of textile production as well as consumption, and in manufacturing cities such as Krefeld, Lyon, Manchester). Nonetheless, textile-focused events seldom receive great public attention or crit- ical acclaim, with the possible exceptions of innovative temporary exhibitions such as Jean-Paul Leclercq, “Jouer la Lumière” (Paris, Les Arts Décoratifs, 2001); Thomas P. Campbell, “Tapestry in the Renaissance: Art and Magnificence” (New York, The Metropolitan Museum of Art, 2002); Amelia Peck et al., “Interwoven Globe. The Worldwide Textile Trade, 1500-1800” (New York, The Metropolitan Museum of Art, 2013-2014); John Styles, “Threads of Feeling” (London, The Foundling Hospital, 2010-2011; Colonial Williamsburg, 2014).1 The aims of this debate are to draw on the different cultural experiences and disciplinary backgrounds of participants: – To generate discussion over the role of museums in making and representing tex- tile histories. Museums are not only depositories of textile objects, but also write or make both public and academic history through displays and publications. But how does their work relate to university research and dissemination, feed such research, or react to it? How might interactions between museums and universities in different regions and cultures be developed in the future? – To consider where innovative museum work is being undertaken (locally, region- ally, nationally, internationally), wherein lies its innovation, and how it might suggest directions for the future (in collecting, interpretation, etc.). By interpreta- tion, I mean any analogue or digital explanation that contextualizes the objects on display. – To suggest that the most dynamic study of objects from 1500 to the present is no longer limited to art historians – indeed, that the focus in art history on textiles that belong within a well-established tradition of connoisseurship (in which tap- estries and high-end commissions for wall-hangings dominate) is being challenged by the adoption of a more inclusive approach among historians, design historians, and historians of material culture. [Lesley Miller] EAN: 978-2-917902-31-