Saturday, July 18, 2015

Uffe Haagerup

Uffe Haagerup was a wonderful man, with a perfect kindness and openness of mind, and a mathematician of incredible power and insight. 
His whole career is a succession of amazing achievements and of decisive and extremely influential contributions to the field of operator algebras, C*-algebras and von Neumann algebras. 
His first work (1973-80) concerned the theory of weights and more generally the modular theory of Tomita-Takesaki. Uffe  Haagerup began by solving a key open question, showing that semi-finite weights are indeed supremum of families of normal states. This allowed him to develop the standard form of von Neumann algebras, a basic result used over and over since then.  I remember vividly his first appearance in the field of operator algebras and the striking elegance, clarity and strength of his contributions.
He was the first to introduce the Lp-spaces associated to a type III von Neumann algebra. These new spaces have since then received a lot of attention both from operator algebraists and specialists of Banach spaces and operator spaces.
In a remarkable paper published in Inventiones at the end of the seventies, Haagerup was able to analyse the operator norm in the C*-algebra of the free group, to control it by suitable Sobolev norms in spite of the exponential growth of the group and to prove in particular that in spite of its lack of nuclearity the reduced C*-algebra has the Banach space approximation property. This turned out in the long run to be a breakthrough of major importance. It was extended by Haagerup and his collaborators to any discrete subgroup of a simple real rank one Lie-group. One corollary is the property RD of rapid decay which allows one to define the analogue of the Harish-Chandra algebra of smooth elements in the general context of Gromov hyperbolic groups.  This highly non-trivial result of Haagerup turned out to be the technical key in the proof of the Novikov conjecture for such groups. Moreover the study of discrete groups with the Haagerup property continues to play a major role in geometric group theory.  The associated approximation property for the corresponding factors of type II_1 (called the Haagerup property) also played a major role in the solution by Popa of the long-standing problem of factors N non-isomorphic to M_2(N), and of exhibiting a factor with trivial fundamental group. 
The next fundamental contribution of Haagerup is to the classification of injective factors (1983-87). In my work on the subject I had left one case completely open (in 76) and the classification was thus incomplete. After several years of extremely hard work Haagerup was able to settle this question by proving that there is up to isomorphism only one hyperfinite factor of type III_1. This is a wonderful achievement.
Then Haagerup turned to the theory of subfactors, and, once again, was able to make fundamental contributions such as his construction of new irreducible subfactors of small index > 4, and outclass the best specialists of the subject created by Vaughan Jones at the beginning of the eighties. 
Another key contribution of Haagerup is to Voiculescu's Free Probability and to random matrices which he was able to apply very successfully to the theory of C*-algebras and of factors of type II1. For instance he proved that, in all II1 factors fulfilling the approximate embedding property (which is true in all known cases) every operator T with a non-trivial Brown measure has a non-trivial closed invariant subspace affiliated with the von Neumann algebra generated by T. 
Uffe continued at the same relentless pace to produce amazing contributions and the few mentioned above only give a glimpse of his most impressive collection of breakthrough achievements.
Uffe Haagerup was a marvelous mathematician, well-known to operator algebraists, and to the general community of analysts.  From my own perspective an analyst is characterized by the ability of having ”direct access to the infinite” and Uffe Haagerup possessed that quality to perfection. His disparition is a great loss for all of us. 

Daniel Kastler

Daniel Kastler played for many many years a key role as a leading Mathematical Physicist in developing Algebraic Quantum Field Theory. He laid the foundations of the subject as the famous
"Haag-Kastler" axioms in his joint paper with Rudolf Haag in 1964. He gathered around him, in Bandol, a whole international school of mathematicians and physicists.  With the devoted help of his beloved wife Lisle, he created a perfect atmosphere for lively discussions ranging from technical points to philosophical issues, where the quest for truth was the only goal and "arguments of authority" were banished. I remember my first encounter with Daniel and Lisle, in Seattle at the Battelle institute where I had been invited in 1971, as a young mathematician.  Being a beginner I would have expected to be treated with condescendance by the "pros" of the field and was amazed and in fact enthralled by the warmth, the openness of mind which I found in this group, and with a clear maximum at "Daniel", a charming man combining a German care for exactness of details with a Mediterranean tendency for story telling. Daniel had a deeply romantic personality which came as a precious antidote to the harshness of the technical work which is unavoidable when doing research. Discussing with him, one was free to dream.  He had a great sense of "metaphors" and one of his preferred one concerned the despise with which particle physicists sometimes treat the "Standard Model" which some consider just as an effective theory. What Daniel used to say is that: "the Standard Model is in fact a Shakespearian Prince disguised as a beggar, but with diamonds in his pockets!" And Daniel was also a remarkably skilled "technician" who was able, while in his seventies, to perform with utmost care the most complicated calculations needed to give solid backing to an abstract idea.

Over the years I have understood that the special warmth and openness of mind which I had met in the field already in 1971, and which I have appreciated all my life, owed a lot to the presence of Daniel among the leaders. His disparition is a great loss for our field.

Two great losses

It is with incommensurable sadness that we learned of the death of two great figures of the fields of operator algebras and mathematical physics.
Daniel Kastler died on July 4-th in his house in Bandol.
Uffe Haagerup died on July 5-th in a tragic accident while swimming near his summer house in Denmark.
I will write on each of them separately.

Friday, January 2, 2015

QUANTA OF GEOMETRY

This is a short update on the post called "particles in quantum gravity",
there were interesting comments and rather than answering them in the
blog i just want to point to a long and detailed talk which I gave in the
Hausdorff Institute in Bonn in December and which is now available on YouTube.

In any case this is a good occasion to wish you all a

                      HAPPY NEW YEAR 2015!

Friday, December 12, 2014

Sunday, November 9, 2014

PARTICLES IN QUANTUM GRAVITY

The purpose of this post is to explain a recent discovery that we did with my two physicists collaborators Ali Chamseddine and Slava Mukhanov. We wrote a long paper Geometry and the Quantum: Basics which we put on the arXiv, but somehow I feel the urge to explain the result in non-technical terms. 
The subject is the notion of particle in Quantum Gravity. In particle physics there is a well accepted notion of particle which is the same as that of irreducible representation of the Poincaré group. It is thus natural to expect that the notion of particle in Quantum Gravity will involve irreducible representations in Hilbert space, and the question is "of what?". 
What we have found is a candidate answer which is a degree 4 analogue of the Heisenberg canonical commutation relation [p,q]=ih. The degree 4 is related to the dimension of space-time. The role of the operator p is now played by the Dirac operator D. The role of q is played by the Feynman slash of real fields, so that one applies the same recipe to spatial variables as one does to momentum variables. The equation is then of the form E(Z[D,Z]^4)=\gamma where \gamma is the chirality and where the E of an operator is its projection on the commutant of the gamma matrices used to define the Feynman slash. 
Our main results then are that: 

1) Every spin 4-manifold M (smooth compact connected)
appears as an irreducible representation of our two-sided equation. 
2) The algebra generated by the slashed fields is the algebra of functions on M
with values in A=M_2(H)\oplus M_4(C), which is exactly the slightly noncommutative
algebra needed to produce gravity coupled to the Standard Model minimally
extended to an asymptotically free theory. 
3) The only constraint on the Riemannian metric of the 4-manifold is that its volume 
is quantized, which means that it is an integer (larger than 4) in Planck  units. 

The result 1) is a consequence of deep results in immersion theory going back to the work of Smale, and also to geometric results on the construction of 4-manifolds as ramified covers of the 4-sphere, where the optimal result is a result of Iori and Piergallini asserting that one can always assume that the ramification occurs over smooth surfaces and with 5 layers in the ramified cover. The dimension 4 appears as the critical dimension because finding a given manifold as an irreducible representation requires finding two maps to the sphere such that their singular sets do not intersect. In dimension n the singular sets can have (as a virtue of complex analysis) dimension as low as n-2 (but no less) and thus a general position argument works if (n-2)+(n-2) is less than n, while n=4 is the critical value. 

The result 2) is a consequence of the classification of Clifford algebras. When working in dimension 4, the sphere lives in five dimensional Euclidean space and to write its equation as the sum of squares of the five coordinates one needs 5 gamma matrices. The two Clifford algebras Cliff(+,+,+,+,+) and Cliff(-,-,-,-,-) are respectively M_2(H)+ M_2(H) and M_4(C). Thus taking an irreducible representation of each of them yields respectively M_2(H) and M_4(C).

The result 3) comes from the index formula  in noncommutative geometry. One shows that the degree 4 equation implies that the volume of the manifold (which is defined as the leading term of the Weyl asymptotics of the eigenvalues of the Dirac operator) is the sum of two Fredholm indices and is thus an integer. It relies heavily on the cyclic cohomology index formula and the determination of the Hochschild class of the Chern character. 

The great advantage of 3) is that, since the volume is quantized, the huge cosmological term which dominates the spectral action is now quantized and no longer interferes with the equations of motion which as a result of our many years collaboration with Ali Chamseddine gives back the Einstein equations coupled with the Standard Model. 

The big plus of 2) is that we finally understand the meaning of the strange choice of algebras that seems to be privileged by nature: it is the simplest way of replacing a number of coordinates by a single operator. Moreover as the result of our collaboration with Walter van Suijlekom, we found that the slight extension of the SM to a Pati-Salam model given by the algebra M_2(H)\oplus M_4(C) greatly improves things from the mathematical standpoint while moreover making the model asymptotically free! (see Beyond the spectral standard model, emergence of Pati-Salam unification.)

To get a mental picture of the meaning of 1), I will try an image which came gradually while we were 
working on the problem of realizing all spin 4-manifolds with arbitrarily large quantized volume as a 
solution to the equation. 

"The Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis."