Title: Noncommutative geometry: spaces, bundles and connections

2018.08.03 |

Date | Mon 17 Sep — Fri 21 Sep |

Time | 10:00 — 16:00 |

Location | TBA |

**Abstract**

In the last years, noncommutative geometry has emerged as a very active field of research. Lying at the intersection between operator algebras and differential geometry, noncommutative geometry has shown itself to be a very effective mathematical framework for crucial results in many areas, from spectral theory, index theorems and foliations to field theories and gravity models in physics. In particular, the relation with physics has been clear from the very beginning: indeed, the Heisenberg uncertainty relations clearly call for the use of noncommutative algebras.

This masterclass aims at giving an introduction to several mathematical results of the field. In particular, we plan to explain how the main idea at the basis of noncommutative geometry of translating geometric objects in algebraic terms, fruitfully applies to concepts such as manifolds, vector bundles and connections.

**Program 17-21 september (Mon-Fri)**10:00-10:45 lecture (Roberta Iseppi)

11:00-11:45 lecture (Roberta Iseppi)

12:00-14:00 lunch

14:00-14:45 lecture (Giovanni Landi)

15:00-15:45 lecture (Giovanni Landi)

**Roberta Iseppi, QGM Aarhus University**: *Spectral triples as noncommutative spin manifolds*

The central notion in noncommutative geometry is that of a spectral triple. In this part of the masterclass, we aim to introduce this notion and to explain why a spectral triple could be seen as a noncommutative generalisation of the concept of spin manifold.

Starting with the Gelfand-Naimark theorem, we will explain the correspondence between commutative C* algebras and locally compact Hausdorff spaces. Then, metric and differential aspects will be taken into account, arriving to Connes’ reconstruction theorem, where the equivalence between canonical spectral triples and compact Riemannian spin manifolds has been established.

Finally, the intrinsic relation between spectral triples and gauge theory will be explained, showing how any spectral triple naturally induces a gauge theory. If time allows, the description of the full Standard Model as almost-commutative spectral triple will be quickly illustrated.

**Giovanni Landi, University of Trieste**: *Noncommutative vector and principal bundles *

Starting with the Serre-Swan theorem which establishes a duality between vector bundles and finitely generated projective modules, we view noncommutative vector bundles as projective modules over an algebra.

We give then an introduction to (algebraic) K-theory. This is followed by a theory of connections with respect to the universal calculus and with respect to Connes’ differential forms. Next, we introduce principal bundles as algebra extensions with Hopf algebras as `structure groups’ and the notion of gauge transformations in this context. We construct noncommutative `instantons’ as connections satisfying self-duality equations. Starting with line bundles over noncommutative spaces and using a Pimsner algebra construction we arrive at noncommutative circle bundles. We present several examples of the above constructions. In particular we spell out in details the geometry of the noncommutative torus (irrational rotation algebra) with its spectral geometry, Heisenberg modules and canonical gauge connections.

**List of participants**

Elena Apresyan, A.I. Alikhanyan National Science Laboratory

Adam Magee, SISSA

Sophie Emma Mikkelsen, University of Southern Denmark

Alessandro Rubin, SISSA

Simone Siclari, QGM, Aarhus University

Jim Tao, California Institute of Technology

Daan van de Weem, SISSA