“ALGEBRAIC AND COMBINATORIAL ASPECTS OF QUANTUM FIELD THEORIES”
A lot of algebraic or combinatorial notions are used in Quantum Fields theory:
1) Feynman graphs, which represent possible interactions between several
elementary particles. They are organized by several combinatorial operations,
such as insertion, extraction, or contraction.
2) These combinatorial operations allow to define a Hopf algebra structure on
Feynman graphs. With this, Feynman rules are seen as characters, and the
renormalization process can entirely be seen as a Birkhoff decomposition on the
character groups of these Hopf algebras. The Dyson-Schwinger equations of the
studied Quantum Field Theory are also defined on these Hopf algebras, with the
help of insertion operators.
3) These insertion operators defined on Feynman graphs are 1-cocycles for the
Cartier-Quillen cohomology, and with the help of a universal property, this
allows to replace Feynman graphs by rooted trees.
We would like to explain how these structures help to describe and classify all
"physically meaningful" Dyson-Schwinger equations, that is to say systems such
that the unique solution generates a Hopf sub algebra of the Hopf algebra of
Feynman graphs of the theory, and how this is related to the Faà di Bruno group
of composition of formal series.
Doron Zeilberger
“SIEVE METHODS IN NUMBER THEORY AND COMBINATORICS”
Our Patron Saint, Gian-Carlo Rota, initiated a marriage of combinatorics and
number theory in his seminal 1964 article "On the Foundation of Combinatorial
Theory I: The theory of Moebius functions".
Recently, the world of mathematics was astounded when an unknown lecturer at the
University of New Hampshire, Yitang ("Tom") Zhang, proved that there are
infinitely many prime-pairs that are less than seventy million apart, getting
ever-so-close to proving the twin prime conjecture. He in turn, stood on the
shoulders of the 2005 breakthrough of Goldston- Pintz-Yildirim, who in turn,
stood on the shoulders of Bombieri, Friedlander and Iwaniec,
who were greatly inspired by Atle Selberg, who was inspired by Viggo Brun, who
was inspired by Jean Merlin, who stood on the shoulders of Eratosthenes, who
invented the very first sieve (also known as " Principle of
Inclusion-Exclusion", also known as "Moebius inversion").
Now is the time to heed the advise of our beloved Gian-Carlo Rota and try to
strengthen the ties between combinatorics and number theory. Perhaps we,
combinatorialists, can help number theorists go even further?. Conversely, I am
sure that we can learn a lot from their ingenious methods and try and apply them
to enumeration!