Noncommutative Geometry


Friday, 29 November 2013 

University of Glasgow


2:30 pm - 3:25pm  Jacek Brodzki (Southampton): Noncommutative Geometry, metric spaces and groups  

Coffee break

4:00 pm - 4:55pm  Christian Voigt (Glasgow): Clifford algebras, fermions and categorification

5:00 pm - 5:55pm  Ryszard Nest (Copenhagen): Index and determinant of n-tuples of commuting operators

6:30 pm  Dinner at  "The Bothy"

All talks take place in Lecture Hall 516 in the Mathematics Building of the University of Glasgow. The lecture hall is located on the top floor of the building. You can find travel information and a campus map at the bottom of
this page. The restaurant is located near the Hillhead subway station and within walking distance from the Maths building.
This workshop is a "Celebrating New Appointments" meeting supported by the LMS and the GMJ Trust, in celebration of the appointment of Christian Voigt at the University of Glasgow. If you are interested in attending please contact Christian Voigt for details. There is some limited funding available for external participants.


Jacek Brodzki: "Noncommutative geometry, metric spaces and groups "
Abstract: Noncommutative geometry has established very firm connections between operator algebras and differential geometry. An important outcome of these interactions is a new kind of geometric intuition that can be applied in situations previously considered out of reach of geometry. My talk will focus on applications of the philosophy of non commutative geometry to problems arising from the Baum-Connes programme. In particular, we will discuss relationships between geometric group theory, metric spaces, and operator algebras.

Ryszard Nest: "Index and determinant of n-tuples of commuting operators"
Abstract: Suppose that $ A = (A_1, \dots A_n ) $ is an n-tuple of commuting operators on a Hilbert space and $ f = (f_1, \dots,f_n) $ is an n-tuple of functions holomorphic in a neighbourhood of the (Taylor) spectrum of $ A $. The n-tuple of operators  $ f(A) = (f_1(A_1, \dots, A_n), \dots, f_n(A_1, \dots, A_n) ) $ give rise to a complex $ {\mathcal K}(f(A),H) $, its so called Koszul complex, which is Fredholm whenever $ f^{-1}(0) $ does not intersect the essential spectrum of $ A $. Given that $ f $ satisfies the above condition, we will give a local formula for the index and determinant of $ {\mathcal K}(f(A),H) $. The index formula is a generalisation of the fact that the winding number of a continuous nowhere zero function $ f $ on the unit circle is, in the case when it has a holomorphic extension $ tilde{f} $ to the interior of the disc, equal to the number of zero's of $ tilde{f} $ counted with multiplicity. The explicit local formula for the determinant of $ {\mathcal K}(f(A),H) $ can be seen as an extension of the Tate tame symbol to, in general, singular complex curves.

Christian Voigt: "Clifford algebras, fermions, and categorification"
Abstract: Categorification can be described as the process of adding an additional layer of structure to familiar mathematical concepts. This allows to take into account features which are often swept under the rug, and more importantly, it sometimes leads to completely new insights. In this talk I will outline how to categorify the concept of a Clifford algebra, an algebraic structure which appears in various guises in noncommutative geometry. This turns out to have close links to conformal field theory, more precisely to the theory of free fermions - but for the talk no background knowledge from physics will be assumed.


Rob Archbold (Aberdeen)
Gwendolyn Barnes (Edinburgh)
Jacek Brodzki (Southampton)
Biswarup Das (Leeds)
Matt Daws (Leeds)
Liam Dickson (Glasgow)
Andrew Hawkins (Glasgow)
Rogelio Jante (Edinburgh)
Sergio Inglima (Edinburgh)
Greg Maloney (Newcastle)
Stephen Moore (Cardiff)
Ryszard Nest (Copenhagen)
Anne Thomas (Glasgow)
Aaron Tikuisis (Aberdeen)
Steve Trotter (Leeds)
Christian Voigt (Glasgow)
Simon Wassermann (Glasgow)
Stuart White (Glasgow)
Joachim Zacharias (Glasgow)