In the wilderness of noncommutative topology

The research area of the thesis is noncommutative topology theory. Topology is the study of continuous functions of a space, in a broad sense. An important theorem shows that a unital commutative C*-algebra is the algebra of the continuous function of a compact space, where the property commutativity is like 6x2 is the same as 2x6. So, noncommutative C*-algebras can be interpreted as noncommutative topology. Several attempts have been made to sort the C*-algebras out and to classify them, and Guillaume’s work adds to this. Two properties of the C*-algebras have been studied: residually finite dimensionality (RFD) and stability (in the sense of matricial semiprojectivity). If the C*-algebra has one of these properties, it is easier to handle.

The C*algebra is RFD if it has many finite dimensional representations, or said otherwise, if representations into matrix spaces separate points. One particular example of stability for a C*-algebra would be that for any two objects of this C*-algebra that almost satisfies a condition, for example that they commute, there exist two other objects, close to the first one that actually satisfy the condition. For this property Guillaume has looked at unital graph C*-algebras and C*-algebras of some groups.

A clear characterisation of property

Here, it could be concluded that the unital graph C*-algebra had the RFD property if and only if there is no cycle with an entry, which gave a clear characterisation of this broad property. To prove this, Guillaume relied on the idea of a decomposition of a graph, since for some graphs the C*-algebras can be written as an amalgamated free product. In a second article, this result has been generalised, with some extra conditions, to all graphs. This has been done with the help of a completely different method, using groupoids.

For the stability property Guillaume has given a characterisation for a unital graph C*-algebra to be stable, which is that a particular subpgraph has to be finite. It has been done using the idea of decomposition into amalgamated free product from the first article. He has also proved, in a third article, that the soft deformation of the C*-algebra of the group P2 is not stable. P2 is one of the groups of the isometries of the plane, also called the crystallographic groups.

Thoughts of time flows ultimately led to a PhD

Guillaume has always been interested in mathematics. After taking an exam in financial mathematics, he worked as an actuary for three years. But he continued to think about more abstract things – as the theoretical explanation of time flows, where a noncommutative geometry setup can be used as a modelling tool. At first, he tried to understand the ideas on his own but then decided to do a master course.

– When I started my master’s studies, I used extremely heavy formulas which were a nightmare for me. One formula could cover an entire A4 page. Over time I have shifted from noncommutative geometry to noncommutative topology. The backbone of the two areas is the same, but topology is much more elegant to my taste.

Guillaume applied for several PhD positions and started doing his PhD in Warszaw. When his supervisor Tatiana Shulman after some months got a position in Gothenburg and moved, Guillaume moved as well. He did not know much about the Swedish PhD system when he arrived, but the general impression is very positive. In France the PhD time is three years and to Guillaume the idea seems to be that these years should be tough, almost painful, to sort out those who survive. The Swedish system, which includes taking courses and teaching duty and covers five years, he finds overall more careful and better for the personal life.

– My advice is to think about which path to choose from the very beginning: if you are like me and prefer to concentrate, do as much as possible of the courses and teaching part in the beginning and do not worry about the research, you will have time for that. But others might like to spread courses, teaching and research more evenly over time.

Next step for Guillaume will probably be as a teacher in upper secondary school in France. These positions are in France acquired through exams, where the first part took place in February and the second will be in the summer. After that, Guillaume will know where the next move will take him.