Category Archives: Blog

Entropy and the spectral action

Today my new preprint with Ali Chamseddine and Alain Connes appeared on “entropy and the spectral action”. In this paper we compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple and show that it is given by a spectral action of the spectral triple for a specific universal function. We find a surprising relation between this function and the Riemann zeta function.

In the paper we pass from the one-particle level of spectral triples \((\mathcal A, \mathcal H,D)\) to a fermionic second-quantized level according to the following dictionary:

algebra \(\mathcal A\) Semigroup of inner perturbations acting on \(\sigma_t\)
Hilbert space \( \mathcal H\) Clifford algebra \({\rm Cliff}_{\mathbb C}(\mathcal H_{\mathbb R} )\)
Self-adjoint operator \(D\) One-parameter group \(\sigma_t\) of automorphisms on \( {\rm Cliff}_{\mathbb C}(\mathcal H_{\mathbb R} )\)

There is then a unique KMS\({}_\beta\)-state associated to the \(C^*\)-dynamical system \( ( {\rm Cliff}_{\mathbb C}(\mathcal H_{\mathbb R} ), \sigma_t )\). We show that the von Neumann information theoretic entropy of this state is equal to the spectral action \({\rm Trace}( \mathcal E(e^{-\beta D}))\) where \(\mathcal E(x) = \log (x+1) – \frac {x \log x}{x+1}\) (note that the latter expression is the entropy of the partition of the unit interval in two intervals with ratio of size \(x\)). The function \( \mathcal E(e^{-x})\) looks like:

In the second part of the paper we analyze the structure of this function and show that it is a Laplace transform. Hence we can exploit heat kernel techniques that might be available for the heat kernel \({\rm Trace} (e^{-t D^2}) \). We then establish that the coefficients \(\gamma(a)\) of \(t^a\) in an asymptotic heat expansion for the entropy/spectral action are given by \(\gamma(a) = \frac{1-2^{-2a}}{a} \pi^{-a} \xi (2a) \) in terms of the Riemann xi function. We have listed a few of these values below:

What this table of values also shows is that the functional equation gives a duality between the coefficients of the high energy expansion in even dimension with the coefficients of the low energy expansion in the odd dimensional case.



Workshop on Noncommutative Geometry and Index Theory for Group Actions and Singular Spaces, Texas

Noncommutative Geometry and Index Theory

for Group Actions and Singular Spaces

Monday, May 21st – Friday, May 25th, 2018

Texas A&M University

This weeklong workshop will be an opportunity for young researchers in noncommutative geometry to share their ideas with their peers and with some of the leaders of the field. Apart from lectures by young researchers there will be mini-courses by Pierre Albin, Nigel Higson and Shmuel Weinberger. Topics will include the analysis and geometry of hypoelliptic operators, group actions on aspherical manifolds, index theory on singular spaces, and secondary invariants.

Some funding is available for participants, and there are still opportunities for contributed talks. For further information, see the link below.

Mathematical models for noncommutativity in physics and quantum spacetime

Within the European COST network “Qspace” (Quantum structure of spacetime) the meeting is devoted to the mathematical aspects of assumed and studied models. Our aim is to gather experts from different corners of the field to share their approaches and present recent results within a very broad scope of topics that very rarely come together.

Speakers: Joakim Arnlind, Fabien Besnard, Pierre Bielavski, Ludwik Dąbrowski, Daniele Guido, Jens Hoppe, Giovanni Landi, Shahn Majid, Alexander Schenkel, Harold Steinacker.

Supported by COST (Qspace), Banach Center and Templeton Foundation.


Masterclass Applications of the UCT for C*-algebras


Applications of the UCT for C*-algebras

University of Copenhagen
October 2nd to October 6th, 2017

This masterclass will focus on KK-theory and the Universal Coefficient Theorem, with a special regard towards applications to the structure and classification theory of nuclear C*-algebras. Further topics, such as equivariant extensions of KK-theory for C*-algebras over topological spaces or with group actions, will also be explored.


Last lecture series Alain Connes @ CdF

Last Thursday the last lecture series of Alain Connes at Collège de France has started. It will treat two topics, entitled:

There was an overwhelming interest for the first lecture, which has also been recorded and can be viewed here.

Postdoc Quantum Geometry in Nijmegen

A postdoctoral position in quantum gravity and quantum geometry will be available at the Institute for Mathematics, Astrophysics and Particle Physics (IMAPP) of the Radboud University, starting in autumn of 2017. The successful applicant will become part of the research program “Quantum gravity and the search for quantum spacetime”, a collaboration of high-energy theorists and mathematical physicists, sponsored by the Dutch Foundation for Fundamental Research on Matter (FOM).

Applicants should have a background in quantum gravity or related areas, and an active interest in contributing to the consortium’s research in nonperturbative quantum gravity, which focuses on causal dynamical triangulations, renormalization group methods and noncommutative geometry. The appointment will be for two years. The deadline for applications is December 15, 2016. See for more information and how to apply.

KK-theory, Gauge Theory and Topological Phases, School+Workshop

KK-theory, Gauge Theory and Topological Phases School – Workshop
from 27 Feb 2017 through 10 Mar 2017

Lorentz Center Leiden

Scientific organizers:

Alan Carey (Canberra, Australia)
Steve Rosenberg (Boston, MA, USA)
Walter van Suijlekom (Nijmegen, The Netherlands)

This is a school and workshop on new developments in Kasparov theory (also referred to as KK-theory) motivated by applications to gauge theory and topological phases of matter.

The school is intended for PhD-students and postdocs working on KK-theory and/or its applications to physics. This includes young scientists who work on gauge theory or on topological phases with a strong mathematical background, and who want to learn the novel approach to these fields that KK-theory has in stock. Moreover, it is an opportunity for more senior people who are in related fields and want to learn KK-theory.

The workshop is intended for scientists working in the field and also in topological phases of matter, and particularly for the participants of the preceding school. It will attract leading experts from all parts of the world, but will also function as a platform for talented young scientists.


Noncommutative Geometry and Higher Structures (Perugia)

Joint Meetings on

Noncommutative Geometry and Higher Structures

Università di Perugia, 25-29 July 2016


Conference website

The aim of the workshop is to bring together people working on noncommutative geometry, deformation theory and related fields, to promote new collaborations and interaction between senior scientists and students/junior researchers, and give to young mathematicians some perspectives on who is doing what in this field.The conference will take place in the mathematics department of the University of Perugia, located in Via Luigi Vanvitelli 1, Perugia.

The scientific activities of the conference will start at 2:30 PM on Monday 25 July and finish at 12:30 AM on Friday 29. To register, please send an email with your name and affiliation to Nicola Ciccoli (no registration fee is required).


Organizing committee:

Scientific committee:

Speakers include:

  • Iakovos Androulidakis (Univ. of Athens)
  • Paolo Antonini (SISSA – Trieste)
  • Serguei Barannikov (Univ. Diderot-Paris 7)
  • Damien Broka (Penn State)
  • Oleksandr Iena (Univ. Luxembourg)
  • Niek de Kleijn (Univ. Copenhagen)
  • Niels Kowalzig (Univ. Sapienza di Roma)
  • Giovanni Landi (Università di Trieste)
  • Camille Laurent-Gengoux (Univ. Paul-Verlaine)
  • Luigi Lunardon (Univ. Sapienza di Roma)
  • Marco Manetti (Univ. Sapienza di Roma)
  • Francesco Meazzini (Univ. Sapienza di Roma)
  • Valerio Melani (Univ. Pierre et Marie Curie Paris)
  • Chiara Pagani (Georg-August-Univ. Göttingen)
  • Francois Petit (Luxembourg Univ.)
  • Martin Schlichenmaier (Luxembourg Univ.)
  • Mathieu Stienon (Penn State University)
  • Alfonso Tortorella (Università di Firenze)
  • Ping Xu (Penn State University)