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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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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.
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