Tag - Non-commutative geometry

Rita Fioresi: Quantum principal bundles on quantum projective varieties

24th June, 2024

In non-commutative geometry, a quantum principal bundle over an affine base is recovered through a deformation of the algebra of its global sections: the property of being a principal bundle is encoded by the notion of Hopf-Galois extension, while the local triviality is expressed by the cleft property. We examine the case of a projective base X in the special case X = G/P, where G is a complex semisimple group and P is a parabolic subgroup. The quantization of G will then be interpreted as the quantum principal bundle on the quantum base space X, obtained via a quantum section.

Nigel Higson: The tangent groupoid in non-commutative geometry

12th September, 2023

This is a 22-lecture course, with each lecture being 90 minutes, given by Nigel Higson.

The tangent groupoid is a geometric construction that can be applied to any smooth manifold. Alain Connes pointed out its relevance to the Atiyah-Singer index theorem, and ever since he did so the tangent groupoid has appeared regularly in noncommutative geometry, often in ways related to index theory but usually illuminating other issues at the same time. Good examples of this are the elegant and simple ways of understanding pseudodifferential operators that have been developed recently by Claire Debord and Georges Skandalis, and by Erik van Erp and Bob Yuncken. I shall start with pseudodifferential operators, then introduce the tangent groupoid through them, and go on to examine applications in representation theory, hypoelliptic partial differential equations and elsewhere.

Francesco Genovese: Deforming t-structures

21st June, 2023

A guiding principle of non-commutative algebraic geometry is that geometric objects (i.e. rings and schemes) are replaced by categories of modules/sheaves thereof. In order to keep track of the homological information, we actually take derived categories of such modules/sheaves. From this point of view, we are now interested in understanding typical geometric concepts directly in this categorical framework. A key example is given by deformations. In this talk, I will report on joint work with W. Lowen and M. Van den Bergh, where we attempt to define and study deformations categorically, in the framework of (enhanced) triangulated categories with a t-structure. This will also shed light on Hochschild cohomology.

Maxime Fairon: Around Van den Bergh’s double brackets

10th April, 2023

The notion of a double Poisson bracket on an associative algebra was introduced by M. Van den Bergh in order to induce a (usual) Poisson bracket on the representation spaces of this algebra. I will start by reviewing the basics of this theory and its relation to other interesting operations, such as Leibniz brackets and H0-Poisson structures. I will then explain some recent results and generalisations related to double Poisson brackets.

Camille Male: Freeness over the diagonal and the global fluctuations of Wigner matrices

18th August, 2021

We characterize the limiting second order distributions of independent complex Wigner and deterministic matrices using Voiculescu’s notions of freeness over the diagonal. For unitary invariant random matrices, Mingo and Speicher’s notion of second-order freeness gives a universal rule to compute the global fluctuation. But this one is in general not valid for non-Gaussian Wigner matrices, since the fluctuations are not universal, depending in particular on the moment of order 4 of the matrices. Yet, it is possible to adapt Mingo-Speicher’s formulation and reformulate this notion for operator-valued random variables in a second-order probability space, and prove a universal rule for more general Wigner matrices (for which the marginal second-order distributions are not universal).

Lisa Glaser: A picture of a spectral triple

17th August, 2021

A compact manifold can be described through a spectral triple, consisting of a Hilbert space H, an algebra of functions A and a Dirac operator D. But what if we are given a spectral triple? Then the situation is more complicated, it is not clear how to reconstruct geometry from a spectral triple, in particular one with a non-commutative algebra A, or a finite Hilbert space H. But these are questions one would like to ask if trying to use spectral triples to possibly quantize gravity. In this talk I will show how we recover metric information from a truncation of a spectral triple to make an image, and show some odd shrinking spectral triples.

Roberta Iseppi: The BV-BRST cohomology for U(n)-gauge theories induced by finite spectral triples

17th August, 2021

The Batalin–Vilkovisky (BV) formalism provides a cohomological approach for the study of gauge symmetries: given a gauge theory, by introducing extra (non-existing) fields, we can associate to it two cohomology complexes, the BV and the BRST complex. The relevance of these complexes lies in the fact that their cohomology groups capture interesting physical information on the initial theory. In this talk we describe how both these complexes can be seen as Hochschild complexes of a graded algebra B over a bimodule M. By focusing on U(n)-gauge theories induced by a finite spectral triple on Mn(ℂ), we explain how the pair (B,M) is naturally encoded, respectively, in the BV spectral triple associated to the theory for the BV complex and in its gauge - fixed version for the BRST one. This result further reinforces the idea that the BV construction naturally inserts in the framework provided by non-commutative geometry.

Bertrand Eynard: (Mixed) topological recursion and the two-matrix model

17th August, 2021

In this series of lectures we will introduce the 2-matrix model and the issue of mixed traces, then we shall give the answers as formulas. Some formulas will be proved during the lectures, but the main goal is to explain how to use the formulas for practical computations. We shall largely follow the Chapter 8 of the book Counting surfaces, B. Eynard, Birkhäuser 2016.

Roland Speicher: Free probability theory

16th August, 2021

Usual free probability theory was introduced by Voiculescu in the context of operator algebras. It turned out that there exists also a relation to random matrices, namely it describes the leading order of expectation values of the trace for multi-matrix models. Higher order versions of free probability were later introduced by Collins, Mingo, Sniady, Speicher in order to capture in the same way the leading order of correlations of several traces. A prominent role in free probability theory is played by “free cumulants” and “moment-cumulant formulas”, and the underlying combinatorial objects are “non-crossing partitions” and, for the higher order versions, “partitioned permutations”. I will give in my talks an introduction to free probability theory, with special emphasis on the higher order versions, and an eye towards possible relations to topological recursion. In particular, it seems that the problem of symplectic invariance in topological recursion has, at least in the planar sector, something to do with the transition between moments and free cumulants.