Schedule of the Workshop "Future directions for non-commutative geometry"

All talks are 45 minutes with 15 minutes for discussion.

Wednesday, December 17

9:30-10:30 Alain Connes: Geometry and the quantum
10:30-11:00 Coffee break
11:00-12:00 Ali Chamseddine: Quanta of geometry: Physical applications
12:00-14:00 Lunch break
14:00-15:00 Martin Zirnbauer: Bott periodicity and the "Periodic Table" of topological insulators and superconductors
15:00-16:00 Mairi Sakellariadou: Aspects of the bosonic spectral action
16:00-16:30 Tea and cake


(Underlined titles can be clicked for the video recording)

A universal twist (or "Drinfel'd Twist") based on a bi-algebra B consists in an element F of the second tensorial power of B that satisfies a certain cocycle condition. I will present a geometrical method to explicitly obtain such twists for a quite large class of examples where B underlies the universal enveloping algebra of a (generally non-Abelian) Lie algebra. I will also discuss the functional analytical aspects of the twists obtained, focusing on non-formal (e.g. "strict" in the sense of Marc Rieffel) universal deformation formulae for non-Abelian Lie group actions on Frechet or C*- algebras.


The goal of this talk is to explain how non-formal equivariant quantization gives rise to a theory of deformation of C*-algebras and to concrete examples of locally compact quantum groups, via the explicit construction of dual unitary 2-cocycles and of multiplicative unitaries. I will start to explain the general strategy and then give three classes of examples: the negatively curved Kählerian Lie groups, some p-adic groups and some non-geometric examples.


I shall first survey recent progress in understanding differential and conformal geometry of curved noncommutative tori. This is based on work of Connes-Tretkoff, Connes-Moscovici, and Fathizadeh and myself. Among other results I shall recall the computation of spectral invariants, including scalar curvature and noncommutative residues, for this class of spectral manifolds. In the second part of my talk I will show how to  compute the curvature of the determinant line bundle for a family of Dirac operators for noncommutative two tori (joint work with Fathi and Ghorbanpour). Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of canonical trace, defined on Connes' algebra of classical pseudodifferential (logarithmic) symbols for the noncommutative two torus, one can compute the curvature form of the determinant line bundle by computing the second variation \delta_w \delta_{\overline{w}} \log \det(\Delta).


Klaas Landsman: Operator algebras and topos theory

Independently inspired by constructive mathematics and by the foundations of quantum physics, since 2006 dozens of papers have appeared in which topos theory and operator algebras were related one way or the other. This development will be surveyed, with particular emphasis on the construction of a topos with associated internat commutative C*-algebra from a given non-commutative C*-algebra (defined, as usual, in the topos of sets that lies at the basis of classical mathematics). This, in turn, has suggested a new invariant for (unital) C*-algebras A, namely the poset C(A) of unital commutative C*-subalgebras of A. Like the K-Theory of A, this poset cannot be a complete invariant in general, but it is a near miss, and searches for some dressing of C(A) so as to make it complete are under way.


This talk is about the notion of Cartan subalgebras introduced by Renault, based on work of Kumjian. We explain how Cartan algebras build a bridge between dynamical systems and operator algebras, and why this notion might be interesting for the structure theory of C*-algebras as well.


Mairi Sakellariadou: Aspects of the bosonic spectral action

After a short introduction on noncommutative spectral geometry, I will first discuss some cosmological consequences of the spectral principle applied in almost commutative manifolds. In particular, using astrophysical data I will constrain one of the three cutoff momenta appearing in the asymptotic expansion of the (cutoff) bosonic spectral action. I will then investigate the role of the scalar fields within this context. Finally, to address the issues of renormalisability, ultraviolet completeness and spectral dimensions I will discuss an alternative definition of the bosonic spectral action based upon the zeta function regularisation.


Given a group action on a manifold, there is an associated class of operators represented as linear combinations of differential operators and shift operators along the orbits. Operators of this form appear in noncommutative geometry and mathematical physics when describing nonlocal phenomena.

In this talk, we show how methods of noncommutative geometry (KK-theory, cyclic cohomology, ...) can be successfully applied to study the index problem for operators associated with group actions. In particular, we describe the method of pseudodifferential uniformization, which enables one to reduce elliptic operators associated with group action to elliptic pseudodifferential operators (and solve the index problem using the celebrated Atiyah-Singer formula).


Given two compact manifolds with boundary X,Y, and a boundary preserving symplectomorphism \chi:T^*Y\setminus0\to T^*X\setminus0, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with \chi.

We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property.

Finally, we show how - in the spirit of a classical construction by A. Weinstein - a Fredholm operator of this type can be associated with \chi and a section of the Maslov bundle. If \dim Y>2 or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.

(Joint work with U. Battisti and S. Coriasco, Turin.)


Hermann Schulz-Baldes: Fredholm operators with symmetries for topological insulators

In solid state systems there are even and odd index theorems for invariants which are closely related to the (one-particle) Hamiltonians. If these Hamiltonians have discrete symmetries (such as time reversal) the corresponding Fredholm operators inherit symmetries that place them in the classifying spaces of Atiyah and Singer.


Christian Voigt: Clifford algebras, Fermions and categorification

Given the fundamental role of Dirac operators in K-homology and K-theory, it can be argued that "categorified Dirac operators" should be crucial for the understanding of elliptic cohomology. However, the actual construction of such operators seems a challenging open problem.

In this talk I shall try to illustrate that, in this context, already some very basic algebraic considerations lead to a rather rich structure when one tries to categorify. More precisely, I will describe a categorification of complex Clifford algebras arising from certain categories of twisted modules over fermionic vertex superalgebras. The product in the categorified Clifford algebra is closely related to fusion of surface defects in 3D topological field theory. I will include some background from the theory of unitary vertex algebras, and discuss how the String 2-group fits naturally into the picture.


Bora Yalkinoglu: Aspects of NT and NCG

After giving a brief overview over existing links between number theory and noncommutative geometry we'd like to propose some natural problems for further research.


(in collaboration with Hirshberg, Szabo, Winter, Wu)

Various noncommutative generalisations of dimension have been considered and studies in the past decades. In recent years certain new dimension concepts for noncommutative C*-algebras, called nuclear dimension and a related dimension concept for dynamical systems, called rokhlin dimension have been defined and studied. They play an important role in the classification programme. The theory is geared towards the class of nuclear C*-algebras and generalises the concept of covering dimension, in case of dynamical systems a type of equivariant covering dimension of topological spaces with a group action. There are interesting connections between coarse geometry and Rokhlin dimension. We will give an introduction to these concepts and survey some applications and connections between them.


Martin Zirnbauer: Bott periodicity and the "Periodic Table" of topological insulators and superconductors

Bott periodicity is said to be one the most surprising phenomena in topology. Perhaps even more surprising is its recent appearance in condensed matter physics. Invoking a twisted equivariant form of K-theory, Kitaev argued that symmetry-protected ground states of gapped free-fermion systems, also known as topological insulators and superconductors, organize into a sort of periodic table akin to Bott periodicity. In this talk, I will sketch the physical context and some new results of this ongoing story of mathematical physics.