Wednesday, September 1, 2021

Global noncommutative geometry seminar (Americas) resumes this Friday, Sept 3, with a talk by Alain Connes

 After a very successful first year, Global Noncommutative Geometry Seminar (Americas) starts again this Friday at 3 PM (Toronto time), 9 PM (Paris time) with a talk by Alain Connes. Title: On the Notion of Space.

Here is the Zoom link: Join Zoom Meeting

Meeting ID: 951 5490 4208

Passcode: 446996

Added  Sept 7: Talk can be watched  here

For abstracts and details of other  forthcoming talks in Global Noncommutative Geometry Seminar please check the seminar website: Global NCG Seminar

Sunday, August 22, 2021

Online International Conference: Cyclic Cohomology at 40: achievements and future prospects

The year 2021 will mark the 40th anniversary of the discovery and development of Cyclic Cohomology Theory by Alain Connes. Its further flourishing fostered by many mathematicians including Dan Quillen, Jean Louis Loday, Boris Tsygan, Boris Feigin, Joachim Cuntz, Ib Madsen, Max Karoubi, and many others has brought a remarkable development of the subject in many distinct directions, in particular in global analysis, representation theory and index theory, algebraic K-theory and cyclotomic trace maps, arithmetic geometry and mathematical physics. The goal of this meeting will be to bring top experts from around the world to lecture on the state of the art in the field, to look towards the future and discuss the most pressing current open problems in the field.

 For Conference Poster Click Here

Registration is free. To register please visit the Fields Institute Conference Page

Tuesday, June 1, 2021

Noncommutative Geometry meets Topological Recursion (registration now open)

We plan to run the workshop in hybrid form with a considerable number of participants present in Münster and video transmission to the outside world.

This workshop intends to be a first meeting point for specialists and young researchers active in noncommutative geometry, free probability, and topological recursion. In the two first areas, one often wants to compute expectation values of a large class of noncommutative observables in random ensembles of (several) matrices of size N, in the large N limit. The motivations come from the study of various models of 2d quantum gravity via random spectral triples, or from the problem of identifying interesting factors via approximations by matrix models. Topological recursion and its generalisations provide a priori universal recipes to make and to compactly organise such computations, not only for the leading order in N, but also to all orders of expansion in 1/N. In this way connections are established to domains like enumerative geometry, tropical geometry, mirror symmetry, topological and more generally low-dimensional quantum field theories where topological recursion has also been applied.

Concretely, the last 10 years have witnessed the developement

  • of analytic techniques to establish the existence of large-N asymptotic expansions,
  • of applications of the topological recursion to a growing class of matrix models which now include some of direct interest in the study of random spectral triples and in noncommutative probability,
  • and of connections between the combinatorics of free probability (i.e. higher order free cumulants) and the topological recursion together with symplectic transformations acting on it.

The workshop will explore the consequences of these remarkable algebraic structures axiomatised in topological recursion for problems in noncommutative geometry and free probability. Knowledge will also flow in the other direction, as the very nature of topological recursion hints at connections to (noncommutative) algebraic geometry and to Hopf algebraic structures and Connes-Kreimer renormalisation.

Mathematical models and phenomena under consideration are common to all these fields, and we wish to unite the strength of probabilistic/asymptotic, algebraic/geometric and combinatorial approaches for the benefit of all the communities involved. This interaction should in fine lead to a better geometric understanding, more powerful computational tools, and new results.


Introduction to topological recursion

Elba Garcia-Failde (IRIF, Université Paris Diderot)

Noncommutative geometry and spectral triples

Walter van Suijlekom (IMAPP, Radboud University Nijmegen)

Free probability and higher order freeness

Roland Speicher (Fachrichtung Mathematik, Universität des Saarlandes)

(Mixed) topological recursion and the two-matrix model

Bertrand Eynard (Institut de Physique Théorique, Paris-Saclay / IHÉS)


Gaëtan Borot (Institut für Mathematik & Institut für Physik, Humboldt-Universität zu Berlin)

Masoud Khalkhali (Department of Mathematics, University of Western Ontario)

Hannah Markwig (Fachbereich Mathematik, Eberhard Karls Universität Tübingen)

Jörg Schürmann (Mathematisches Institut, WWU Münster)

Raimar Wulkenhaar (Mathematisches Institut, WWU Münster)

For Registration please check the 

Conference Website

Tuesday, May 25, 2021

Summer school on K-Theory and Representation Theory


An online summer school on K-Theory and Representation Theory, organized by N. Higson, R. Plymen and H. Sengun and sponsored by the London Mathematical Society and the Isaac Newton Institute, will take place 19-23 July 2021.

There will be four lecture series:

– Basic representation theory of p-adic Lie groups, by Anne-Marie Aubert

– Basic representation theory of real Lie groups, by Peter Hochs

– Basics of group C*-algebras and K-theory, by Bram Mesland

– Dirac operators and representation theory, by Hang Wang

The lectures will be aimed towards graduate students, but all are welcome to attend. Registration and further information are at

Sunday, February 21, 2021

SHELL BEACH - the search for the final theory, (guest post by Jesper Grimstrup)

My book SHELL BEACH - the search for the final theory is now out. In the book, which is written for a general audience, I discuss topics such as noncommutative geometry, loop quantum gravity and the question of a final theory - mixed with a dose of whitewater kayaking and massive (intellectual) waterfalls

Best wishes, Jesper Grimstrup

Friday, December 25, 2020

Fields Academy Graduate Courses

 Starting in January 2021, the Fields Institute in Toronto is going to run various advanced graduate courses in collaboration with Ontario universities. Students from all over the world can either take these courses, or just audit them. For details  see  here


The Fields Academy has an undergraduate training component as well. For details  check


Monday, November 2, 2020

The Metric Nature of Matter, Guest Post byJohannes Aastrup and Jesper Grimstrup

In a recent series of papers [1-3], we have discovered a connection between non-perturbative quantum field theory and the (noncommutative) geometry of configuration spaces. In this blog-post, we would like to outline our findings in a non-technical manner intended for readers, who are familiar with the basics of noncommutative geometry and high-energy physics.

The noncommutative standard model

Chamseddine and Connes and co-workers have shown that the standard model of particle physics coupled to general relativity can be formulated in terms of an almost-commutative spectral triple [4-8]. Their seminal work, which renders key aspects of the standard model in a completely new light, raises two fundamental questions:

  1. where does the almost-commutative algebra, that underlies Chamseddine’s and Connes’ work, originate from? Is there a deep reason why Nature chose this algebraic structure?
  2. what role should quantum field theory play in this approach to fundamental physics? As it stands now, perturbative quantum field theory is applied to Chamseddine’s and Connes’ construction as something like an afterthought and only to the standard model part: gravity is not quantized.

Embedded within this last question lies also the question of whether gravity should be quantized. 

In our recent work, we propose a novel answer to these two questions. It turns out that a geometrical construction over a configuration space of gauge connections gives rise to a non-perturbative quantum field theory on a curved background — including both bosonic and fermionic degrees of freedom — which produces an almost-commutative algebra similar to that of Chamseddine and Connes. Thus, what we propose is that the standard model of particle physics originates from a geometrical framework intimately related to non-perturbative quantum field theory. 

For another possible answer to the first question see also the work of Connes, Mukhanov, and Chamseddine [9,10]. 

The geometry of moving stuff around

The starting point of our construction is an algebra called the HD algebra [11,12], which is generated by parallel transports along flows of vector fields in a three-dimensional manifold. That is, the HD algebra encodes how tensor-degrees of freedom are moved around in space. The HD algebra, which comes with a high degree of canonicity, is a non-commutative algebra of functions over a configuration space of gauge connections.

The next step is to formulate an infinite-dimensional Bott-Dirac operator on the configuration space of gauge connections. The construction of this operator, which resembles an infinite-dimensional Bott-Dirac operator that Kasparov and Higson constructed in 2001 [13], is severely restricted by the requirement of gauge-covariance.

A key feature of the Bott-Dirac operator is that its square produces the Hamilton operator of a Yang-Mills theory coupled to a fermionic sector as well as a topological Yang-Mills term together with higher-order terms — all on a curved background. Furthermore, the Bott-Dirac operator and its interaction with the HD algebra reproduce the canonical commutation relations of a quantized gauge and fermionic fields. 

Thus, the formulation of a noncommutative geometry on a configuration space of gauge connections gives rise to the basic building blocks of a non-perturbative Yang-Mills-Dirac theory on a curved background. 

The origin of fermionic quantum field theory

One interesting feature of this construction is the role fermions play. The Bott-Dirac operator requires an infinite-dimensional Clifford algebra — just as an ordinary Dirac operator requires a finite-dimensional Clifford algebra — and it is this Clifford algebra that gives us the CAR algebra and the fermionic Fock space. Furthermore, when we compute the square of the Bott-Dirac operator then a certain commutator turns up, which gives us precisely the Dirac Hamilton operator. Thus, the Dirac Hamiltonian can be understood as a quantum fluctuation of the bosonic theory.

All this shows that the fermionic degrees of freedom play an intrinsically geometrical role in this framework; they are intimately related to the geometry of the underlying configuration space. 

A link to the standard model

In a semiclassical limit, the HD-algebra will give rise to a matrix algebra. If we choose a configuration space of spin connections then the HD-algebra will give us a three-by-three matrix algebra in a classical limit. The construction of the Bott-Dirac operator gives, however, rise to additional structure, which means that in a classical limit we find an almost-commutative algebra with a matrix factor given by: 

M_3(C) + M_3(C) + M_2(C)

which looks surprisingly similar to the matrix factor, which Chamseddine and Connes have found in the standard model [4,5]:

C + H + M_3(C)

where H is the quaternions. Could there be a connection here? More analysis is required to determine whether this link can be substantiated, but for now, we are encouraged by our findings.

Dynamical gauge-covariant regularisation

A key question is whether a Hilbert space representation of the HD algebra and the Bott-Dirac operator exists. We have shown that such a representation (strongly continuous, separable) exists in the special case where the gauge-symmetry is broken [3], but we do not have a proof in the gauge-covariant case. We are, however, confident that such representations do exist.

A key feature of the representation, which we have found, is a UV-regularisation in the form of a Sobolev norm that dampens degrees of freedom beyond a certain scale. In order to obtain a gauge-covariant representation, it is natural to make this Sobolev norm gauge-covariant. This is possible but in doing so we change the entire construction. 

Why is that? Well, normally a UV-regularisation is a computational artifact, that should ultimately be removed. This is the origin of renormalization theory. There are of course several reasons why a regularisation must be removed, one being that there are countless ways of regularising and no way to choose between them. But this is no longer the case with a gauge-covariant regularisation because it will have a time-evolution, i.e. it will be dynamical. This is what we call a dynamical gauge-covariant regularisation [1].

What we find is that the UV-regularisation should be understood not as a computational artifact but as a physical feature: the UV-regularisation is part of the metric data of the configuration space. One thing that makes this possible is that from a perturbative QFT perspective this UV-regularisation will simply look like higher-order derivative terms. The non-locality, that the dynamical regularisation introduces, will only be seen non-perturbatively.

Note that the concept of a dynamical regularisation is almost inevitable in a non-perturbative gauge theory since any Hilbert space representation will likely require some kind of UV-regularisation. The requirement that this regularisation is gauge-covariant automatically makes it dynamical.

It is widely believed that a theory of quantum gravity will give rise to a Planck-scale screening, which in turn will impact the Lorentz symmetry and the causal structure of space-time. If, however, the Planck-scale screening originates from a framework of non-perturbative quantum field theory, as we suggest, then it would make a theory of quantum gravity obsolete. The regime, where such a theory would otherwise reign, would be screened off.  We find this idea intriguing: perhaps the reason why a theory of quantum gravity has eluded theoretical physicists for so long is that it does not exist?

We believe that the idea of a dynamical UV-regularisation deserves much attention. After all, there exist very few concrete ideas as to how a Planck-scale screening may arise in Nature. This is one such idea.

Open questions

In our recent work, we have shown that essentially all the key building blocks of modern high-energy physics — bosonic and fermionic quantum gauge theory, gravity, an almost-commutative algebraic structure — emerges from a simple geometrical framework on a configuration space. This work raises many questions, some of the most important ones are:

  • does a gauge-covariant Hilbert space representation exist?
  • what does the construction look like when we use the Levi-Civita connection on the configuration space (we know that the Levi-Civita connection exists).
  • what does the time-evolution of the UV-regularisation look like?
  • we assume that the non-locality will be restricted to scales beyond the Planck scale. What happens to causality? The Lorentz symmetry?
  • can the conjectured connection to the standard model be substantiated? In particular, does it offer an answer to the question of why there are three particle generations?

With this, we end this blog-post; 

Johannes Aastrup 

Jesper Møller Grimstrup


[1] J. Aastrup and J. M. Grimstrup, “The Metric Nature of Matter” arXiv: 2008.09356.

[2] J. Aastrup and J. M. Grimstrup, “Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces," arXiv:1910.01841.

[3] J. Aastrup and J. M. Grimstrup, “Representations of the Quantum Holonomy-Diffeomorphism Algebra," arXiv:1709.02943.

[4] A. Connes, “Gravity coupled with matter and the foundation of noncommutative geometry," Commun. Math. Phys. 182 (1996) 155.

[5] A. H. Chamseddine and A. Connes, “Universal formula for noncommutative geometry actions: Unification of gravity and the standard model," Phys. Rev. Lett. 77 (1996) 4868.

[6] A. Connes, ”Noncommutative geometry and the standard model with neutrino mixing,'' JHEP 11 (2006), 081.

[7] A. H. Chamseddine, A. Connes and M. Marcolli, ”Gravity and the standard model with neutrino mixing,’' Adv. Theor. Math. Phys. 11 (2007) no.6, 991-1089.

[8] A. H. Chamseddine and A. Connes, “Why the Standard Model,’' J. Geom. Phys. 58 (2008), 38-47.

[9] A. H. Chamseddine, A. Connes and V. Mukhanov, “Geometry and the Quantum: Basics,’' JHEP 12 (2014), 098.

[10] A. H. Chamseddine, A. Connes and V. Mukhanov, “Quanta of Geometry: Noncommutative Aspects,’' Phys. Rev. Lett. 114 (2015) no.9, 091302.

[11]  J. Aastrup and J. M. Grimstrup, “C*-algebras of Holonomy- Diffeomorphisms and Quantum Gravity II”, J. Geom. Phys. 99 (2016) 10. 

[12]  J. Aastrup and J. M. Grimstrup, “The quantum holonomy-diffeomorphism algebra and quantum gravity,’' Int. J. Mod. Phys. A 31 (2016) no.10, 1650048.

[13] N. Higson and G. Kasparov, "E-theory and KK-theory for groups which act properly and isometrically on Hilbert space", Inventiones Mathematicae, vol. 144, issue 1, pp. 23-74.