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Personal page of Pierre Clavier

IRIMAS - Département de Mathématiques
6 rue des Frères Lumière
mail: firstname[dot]name[at]uha[dot]fr

NEWS: with Martin Bordemann (UHA), we are offering a PhD position starting in September 2022. The position is sponsored by an Ecole Doctorale. We strongly encourage interested students to apply! The project is about generalisation of MZVs and Drinfeld associator. A description (in french and english) can be found here. Feel free to write me an email if you would like to know more about the project and/or discuss about it.

I currently hold a position of junior lecturer (Maître de Conférences, or MCF for short) Université de Haute Alsace (UHA). I am a member of the algebra groupe of the mathematics department of the IRIMAS.

Research interests

I consider myself a mathematical physicist, and I work on various subjects centered around quantum field theory (QFT) and number theory. I am particularly interested in generalisation of multizeta values and their structures, resurgence applied to non-pertubative QFT and the structures behind Feynman rules. Here are a few words on some topics I am currently working on, or have been recently.

TRAPs and Feynman integrals

With Loïc Foissy and Sylvie Paycha, we have defined TRAPs (TRAces and Permutations) which can be seen as non-unitary wheeled PROPs. We have built the free objects of this category (which are graphs) and established the links between TRAPs and other existing structures. Using the TRAP structure of some analytical spaces allowed us to define a generalised convolution and trace of smooth kernels. The next step of our program is to look if Feynman rules of QFT can be defined using the universal property of graphs in the category of TRAPs.

Resurgence and QFT

Ecalle's resurgence theory allows a very fine analysis of singularities of Borel transform of divergent series. When the Borel transform has some nice analytical properties, resurgence theory gives a resummation procedure that generalises the usual Borel-Laplace resummation method. I try to apply this technics to various divergent series coming from quantum field theory, and to characterise the obtained functions. My main hope is that this procedure would offer a non-pertubative mass generation mecanism for asymptotically free quantum field theories.

Generalisation of MZVs

Multizeta values (MZVs) are numbers that can be written as iterated series or iterated integrals. They enjoy many remarquable algebraic and (conjectural) number theoretic properties. They also admit various generalisations. I am studying one of them, where the series or integrals are iterated following the structure of a rooted forests. These numbers are called branched (or arborified) zeta values (BZVs). Recently, I have been looking at links between BZVs and conical zeta values, and at possible generalisation of BZVs to directed acyclic graphs. This has surprinsing links to combinatorics of graphs and problems from data science.

Locality structures and applications:

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. Recently, with Diego Lopez and Sylvie Paycha, we are working of a generalisation to locality structures of the Milnor-Moore theorem.


I was hired as MCF at UHA in September 2020. Here is a very quick description of my education before this (but after high school!):

Some links