On the occasion of the PhD defense of Malte Leimbach (on December 1, 16:30 in the Aula) we will organize a small NCG Symposium on Tuesday December 2 (in the Hilbert space, Huygensgebouw). This will include coffee/tea, lunch and a small reception at the end of the day. The schedule is:

Titles and abstracts:

Francesca Arici: The trouble with essentially normal

Quadratic algebras offer a natural setting for exploring quantum spaces and deformations, particularly within the framework of quantum groups as envisioned in Manin’s program for noncommutative geometry. In this talk, I will present recent developments on extending various operations on quadratic algebras to their C*-algebraic counterparts, with a focus on Veronese powers. This operation plays a pivotal role in addressing the question of essential normality of d-tuples of operators.

Ian Koot: Relative Positions of Half-sided Modular Inclusions and Emergent Spacetime Symmetries.

In Algebraic Quantum Field Theory, the representation of the spacetime symmetries acts on the von Neumann algebras of observables in a way that is dictated by the geometry of the space-time regions the algebras are associated to. In this way, the geometry of a space-time region dictates some of the algebraic properties of the observable algebras associated to that region. The most well-known example of this is a result called Borchers’ theorem, which states that if the representation of the spacetime symmetries has an analyticity property called ‘positive energy’, then the modular group given by the Tomita-Takesaki Theory of the von Neumann algebra associated to the Rindler wedge has fixed commutation relations with the spacetime symmetries (it acts like Lorentz boosts). Surprisingly, in one-dimensional spacetime this result has a converse: the space-time symmetry can be recovered purely from the modular data of the algebra associated to the right half-line, through a construction known as ‘half-sided modular inclusion’. We discuss these concepts in the simplified setting of standard subspaces, from which von Neumann algebras can be constructed through Weyl algebras. We discuss a recent result characterising how multiple half-sided modular inclusions can interact, and thereby will be able to show that Borchers’ theorem does not have a converse in higher dimensions, essentially because the induced space-time translations can not be guaranteed to commute

Yuezhao Li: The spectral localiser via E-theory

The topological invariant of a topological insulator is given by
the index of a Fredholm operator constructed from the Hamiltonian.
Equivalently, this is the index pairing between the K-theory class of the
Hamiltonian and the spectral triple given by the position operator. Hermann
Schulz-Baldes and Terry Loring have introduced the spectral localiser as a
means to compute the index pairing. These are invertible matrices supported
on a finite-dimensional spectral subspace of the spectral triple. In the
odd case, we interpret the spectral localiser as an index pairings in
E-theory. Namely, the image of an odd K-theory class under the asymptotic
morphism generated by an odd spectral triple. This allows for speaking of
spectral truncation of index pairings in the framework of bivariant
K-theory. This is based on joint work with Bram Mesland.

Adam Rennie: Spectral flow formulae for unitaries in 1+Schatten

Abstract: We present new formulae for spectral flow of “Schatten” unitaries.
They take the form “integral of exact form” over the Banach-Lie group of such
unitaries. There are some peculiar possibilities opened for spectral flow
of self-adjoint Fredholm operators, as well as applications to Levinson’s
theorem in scattering theory.

Joint work with A. Alexander, A. Carey, G. Levitina.

Eva-Maria Hekkelman: The Sound of Sequins

Abstract: The principal symbol of a (pseudo)differential operator typically arises as a function on a cotangent sphere bundle: a space which can be imagined as sequins stitched to a geometric space. I will discuss how to think of sequins in NCG. Combined with the paradigm of spectral truncations by Connes–van Suijlekom, this provides a noncommutative version of a theorem relating geodesic flow to quantum ergodicity. If time permits, I will also discuss non-compact manifolds, based on Melrose’s scattering pseudodifferential operators. Based on joint work with Galina Levitina, Edward McDonald, Fedor Sukochev and Dmitriy Zanin.