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Personal page of Pierre Clavier
IRIMAS - Département de Mathématiques
6 rue des Frères Lumière
68 093 MULHOUSE
mail: firstname[dot]name[at]uha[dot]fr
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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. I defended my habilitation to direct research (HDR) titled "TRAPs, Generalisations of MZVs, Locality and Resurgence for Quantum Field Theories"
the 26th of May 2025.
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.
Branching and cobranching for species
With Yannic Vargas and Sylvie Paycha, we are attempting to generalise the classical branching map of rooted forests to species.
Many properties of the branching map still hold in this very general setup, and in particular the universal properties of rooted
forests, which now has such a property in a category of species. Graftings (and co-graftings) provide a way to built balanced pairs of
up-down operators, which in turns give candidates for the creation-annihilation operators of quantum field theories. Furthermore, these
operators have interesting links to operads.
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.
Education
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!):
- In the year 2019-2020 I held a research position at Potsdam University and the Technische Universität of Berlin.
- From 2015 to 2019 I was Mitarbeiter at Potsdam University in the Institut für Mathematik.
- From 2012 to 2015, I did my PhD in the University Paris VI, in the mathematical physics group of the LPTHE, under the supervision of Marc Bellon.
- I received a magistère of Fundamental Physics (speciality theoretical physics) from the University Paris XI, in 2012. I had my lectures at the Imperial College of London and at the ENS
Ulm in Paris.
- I was awarded a bachelor of Fundamental Physics from the University Paris XI (Orsay) in 2010. Before this I did two years of "classe prépa" and one year at Orsay University.
Some links