– Online Seminar
Numerical semi-groups and applications
Abdallah ASSI (University of Angers)
I am a mathematical physicist, working on various subjects centered around quantum field theory (QFT) and number theory. More precisely a short list of my current research interests is:
Locality plays a crucial role in QFT. In particular, in perturbative QFT, it is implemented through the requirement that renormalisation map has to be an algebra morphism (for the concatenation product of Feynman graphs) and is realised by a Birkhoff-Hopf factorisation. We have defined locality structures as symmetric partial structures in order to implement the concept of locality in mathematics. This allow us to build multivariate renormalisation schemes. We were then able to show that minimal subtractions preserve locality.
The Hopf algebra of rooted forests plays an essential role in understanding the combinatorics of renormalisation, as had been made clear through the work of Connes, Kreimer and Foissy among others. This could be seen as a consequence of the universal property of this object, namely that it is the initial object in a category of operated Hopf algebras. I am interested into extending this universal property into the locality structures setting, and also to more rigorously construct objects such as branched zeta values and renormalisation schemes.
Branched Zeta Values (BZVs) generalise Multi Zeta Values in the same way that rooted forests generalise words. However the algebraic relations obeyed by BZVs are not, by far, as well known as these of MZVs. I am therefore studying them, in particular by making an extensive use of the universal property of trees. Using combinatorial techniques (e.g. Rota-Baxter algebras) I have been able to describe the relations obeyed by these objects, and in particular that they are algebra morphisms for generalisations of the shuffle and stuffle products.
In the usual approaches to renormalisation, one regularisation parameter is used. However, there is no mathematical nor physical reason to use such regularisation schemes. I am working on regularisation schemes where one regularisation variable is introduced for each possible singularity. This uses multivariate complex analysis and locality structure
With my former PhD advisor Dr. Marc Bellon I am working on non-perturbative aspects of QFTs through Schwinger-Dyson equation. We study these equations in the light of Ecalle's resurgence theory to compute non-perturbative contributions to Green functions. With this approach we have been able to propose a non-perturbative mechanism for mass generation in QFT. We aim to apply it to QCD to compute masses of hadrons.
Here is a very quick description of my education after high school:
A more complete description of my experience can be found on my cv.
Here is a link to all my preprints on arXiv:
Furthermore, my PhD thesis can be found there:
Analytical and geometric approaches of non-perturbative quantum field theories: arxiv.org/abs/1511.09190
Institut für Mathematik
Campus Golm, Haus 9