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We all learned with immense sorrow that Vaughan Jones died on Sunday September 6th.
I met him in the late seventies when he was officially a student of André Haefliger but contacted me as a thesis advisor which I became at a non-official level. I had done in my work on factors the classification of periodic automorphisms of the hyperfinite factor and Vaughan Jones undertook the task of classifying the subfactors of finite index of the hyperfinite factor among which the fixed points of the periodic automorphisms give interesting examples.
By generalizing an iterative construction which I had introduced he was first able to show that the indices of subfactors form the union of a discrete set with a continuum exactly as in conformal field theory. But his genius discovery was when he understood the link between his theory of subfactors and knot theory which is the geometry of knots in three space!
This is really a fantastic discovery that led to a new invariant of knots : the Jones polynomials!
This discovery was afterwards dressed using functional integrals but the real breakthrough is indisputably due to Vaughan Jones.
To me his discovery is one of the great jewels of the unity of mathematics where a seemingly remote problem such as the classification of subfactors of finite index turned out to be deeply related to a fundamental geometric problem!
For this reason I do not hesitate to affirm that Vaughan Jones’ discovery is one of pure genius and that his work has all characteristic features that grant it immortality.
The Noncommutative Geometry Seminar
While noncommutative geometry is entering into its fifth decade, we are sure some of this blog’s readers were thinking would be appropriate to have a volume of articles by experts looking at its main developments in the past 40 years and looking ahead.The book, Advances in Noncommutative Geometry, dedicated to Alain Connes’ 70th birthday, published by Springer late last year, certainly fills this gap! Please take a look by checking Springer’s book website linked above. See also further down the page for a brief introduction. (Posted by Caterina Consani and Masoud Khalkhali).
From the foreword to the book:
“Deeply rooted in the modern theory of operator algebras and inspired by two of the most influential mathematical discoveries of the 20th century, the foundations of quantum mechanics and the index theory, Connes’ vision of noncommutative geometry echoes the astonishing anticipation of Riemann that ”it is quite conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry” and accordingly ”we must seek
the foundation of its metric relations outside it, in binding forces which act upon it. The radically new paradigm of space proposed by Connes in order to achieve such a desideratum is that of a spectral triple, encoding the (generally non-commuting) spatial coordinates in an algebra of operators in Hilbert space, and its metric structure in an analogue of the Fermion propagator viewed as “line-element.”
For the analytic treatment of such spaces, Connes devised the quantized calculus, whose infinitesimals are the compact operators, and where the role of the integral is assumed by the Dixmier trace. On the differential-topological side, Connes has invented a far-reaching generalization of de Rham’s theory, cyclic cohomology which, in conjunction with KK-theory, provides the key tool for a vast extension of index theory to the realm of noncommutative spaces. Besides the wealth of examples of noncommutative spaces coming from physics (including space-time itself with its fine structure), from discrete groups, Lie groups (and smooth groupoids), with their rich K-theory, a whole class of new spaces can be handled by the methods of noncommutative geometry and in turn lead to the continual enrichment of its toolkit. They arise from a general principle, which first emerged in the case of foliations. It states that difficult quotients such as spaces of leaves are best understood using, instead of the usual commutative function algebra, the noncommutative convolution algebra of the associated equivalence relation.
An important such new space is the space of adele classes of a global field that Connes has introduced to give a geometric interpretation of the Riemann–Weil explicit formulas as a trace formula. The set of points of the simplest Grothendieck toposes are typically noncommutative spaces in the above sense and the adele class space itself, for the field of rationals, turns out to be the set of points of the scaling site, a Grothendieck topos which provides the missing algebro-geometric structure as a structure sheaf of tropical nature.
The pertinence and potency of the new concepts and methods are concretely illustrated in the contributions which make up this volume. They cover a broad spectrum of topics and applications, shedding light on the fruitful interactions between noncommutative geometry and a multitude of areas of contemporary research, such as operator algebras, K-theory, cyclic homology, index theory, spectral theory, geometry of groupoids and in particular of foliations. Some of these contributions stand out as groundbreaking forays into more seemingly remote areas, namely high energy physics, algebraic geometry, and number theory.”
Sir Michael Atiyah was considered one of the world’s foremost mathematicians. He is best known for his work in algebraic topology and the codevelopment of a branch of mathematics called topological 𝐾-theory, together with the Atiyah–Singer index theorem, for which he received the Fields Medal (1966). He also received the Abel Prize (2004) along with Isadore M. Singer for their discovery and proof of the index theorem, bringing together topology, geometry, and analysis, and for their outstanding role in building new bridges between mathematics and theoretical physics. Indeed, his work has helped theoretical physicists to advance their understanding of quantum field theory and general relativity. (Continued here.)
The goal of the workshop is thus that of coordinating these diverse research areas and with a common aim that is deeply related to the Riemann Hypothesis (RH).
The Noncommutative Geometry approach to RH has unveiled the exotic nature of the adele class space of the rationals, based on the theory of Grothendieck toposes on one hand and tropical geometry on the other. The search of a suitable Riemann-Roch formula on the square of the Scaling Site seems to require the development of algebraic geometry in characteristic one and of refinements of the tropical Riemann-Roch theory as developed so far and its connections to game and potential theory.
On the other hand, the theory of Segal’s Gamma-rings provides a natural extension of the theory of rings and of semirings: the clear advantage of working with Gamma-rings is that this category forms the natural groundwork where cyclic homology is rooted (this means that de Rham theory is here naturally available). In fact, much more holds, since the simplicial version of modules over Gamma-rings (ie the natural set-up for homological algebra in this context) forms the core of the local structure of algebraic K-theory. In arithmetic geometry, the new Gamma-ring arising as the stalk of the structure sheaf of the Arakelov compactification of Spec Z at the archimedean place, is intimately related to hyperbolic geometry and the Gromov norm. The theory of Gamma-rings in fact culminates with the development of a new theory of ABSOLUTE ALGEBRAIC GEOMETRY