The mathematical structure of renormalizable quantum field theories

A quantum field theory is specified by the one-particle irreducible (1PI) Green functions of the theory.
For a renormalizable theory, a finite number of amplitudes needs to be fixed by experiment
to determine the theory completely.

The mathematical structure of the corresponding set of 1PI Green functions
reveals itself upon the study of the underlying Hopf algebra structure
of a perturbative expansion in the coupling. Before we describe the main results following this approach, we point out some introductory material.

A good source reviewing the combinatorial approach to renormalization is Dominique Manchon's review of my work. Having digested that, you might consider my paper with Christoph Bergbauer in IRMA Lect.Math.Theor.Phys.10:133-164,2006. This clarifies the structure of equations of motion originating from Hochschild cohomology or Lie algebra cohomology, for that matter.
This has consequences for quantum field theory beyond perturbation theory. Look at Karen Yeats thesis for an introduction.
Apart from this mainly combinatorical aspects, there are hard analytic questions in quantum field theory, which connect it to beautiful problems in algebraic geometry.
On the interface of these two aspects lives my recent work with Spencer Bloch on a connection between limiting mixed Hodge structures and renormalization Commun.Num.Theor.Phys.2:637-718,2008. There, you will find also the necessary details on graph polynomials and the core Hopf algebra.
An account of the connection between periods and values of Feynman integrals from a mathematicians perspective, which is actually quite illuminating to a physicist, is given by Francis Brown in his paper in Commun.Math.Phys.287 (2009) DOI 10.1007/s00220-009-0740-5.

So highlights of this approach then include:

i) We have learned recently how to make progress with quantum field theory beyond perturbation theory
in such an approach, see beta function QED and references there, a collaboration with Karen Yeats, Guillaume van Baalen and David Uminsky.

ii) Slavnov--Taylor identities find a natural explanation in this approach, see for example Anatomy of a gauge theory and Walter van Suijlekom's paper. From here, nice connections to the core Hopf algebra, to unitarity of the S-matrix and recursive relations between multi-leg tree-level ampltudes emerge. My very recent paper with Walter is a first step in this direction.

iii) As mentioned above, the renormalization process is captured as a limiting mixed Hodge structure in collaboration with Spencer Bloch. Much work remains to connect this to

iv) Amplitudes in beta-functions and anomalous dimensions typically deliver multiple zeta values as found in collaboration with David Broadhurst,
and hence should be of motivic origin, as indicated by Spencer and Hélène Esnault in particular.

v) quantum gravity has some very interesting features with regard to these Hopf algebra structures, see this remark and a short recent review. It suggests recursive relations between n-point and (n-1)-point amplitudes which need better understanding, and which should have bearing on gauge theories as well. The above-quoted paper is a first result illuminating this connection.

vi) a field theoretic prescription of a Brownian gas can lead to a Poissonian ground state relating to similar algebraic structures, a collaboration with Andrea Velenich, Claudio Chamon and Letitia Cugliandolo.

vii) Minimal subtraction schemes in the context of dimensional regularization connect the perturbative expansion to a Birkhoff decomposition, as worked out in collaboration with Alain Connes in several papers, NCG and renormalization, RHI, RHII. See the work of Connes and Matilde Marcolli for mathematical consequences.

viii) Matt Szczesny and Kobi Kremnizer developed a connection to Ringel-Hall algebras.

ix) Kurusch Ebrahimi-Fard and collaborators Dominique Manchon and Frederic Patras clarified largely the Rota--Baxter aspects of all this.

x) Loic Foissy investigates the Hochschild cohomological classification of Dyson--Schwinger equations, a connection established together with

xi) Christoph Bergbauer, who continues to work on relations between Epstein-Glaser renormalization, Hopf algebras and Fulton-MacPherson compactifications.

xii) Other work which relates to this approach includes aspects of the calcul moulien, of the works of Li Guo and of Sylvie Paycha, and also the results in collaboration with Igor Mencattini demand further clarification.