Dyson-Schwinger Equations in Modern Mathematics & Physics
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Mario Pitschmann (Vienna University of Technology)

Craig D. Roberts (Argonne National Laboratory)

Wolfgang Lucha (Austrian Academy of Sciences)








Dyson-Schwinger equations (DSEs) are ubiquitous in physics. In local relativistic quantum gauge field theories this tower of equations serves, at the simplest level, as a generating tool for perturbation theory. In condensed matter physics, the gap equation is playing an important role in elucidating the properties of graphene and in particle physics, the DSEs are being used to explore strong-interaction alternatives to string-theory as a basis for extending the Standard Model. 

In parallel there has been progress in mathematics with DSEs. This began roughly fifteen years ago with the realisation that the process of renormalisation is naturally expressed via a Hopf algebra structure, which allows for a comprehensive description of the algebraic and combinatorial structures underpinning renormalisation. This enables a mathematically sound approach to the problem of computing the beta-function. The Hopf algebra description permits mathematicians to comprehend and explore basic ideas of renormalisation, driving new applications of those ideas in the context of pure and applied mathematics.

The purpose of the workshop DSEMP2014 is to bring together experts in: the Hopf algebra structure of Dyson-Schwinger equations and renormalisation; perturbative QCD; the application of DSEs to phenomena in hadro-nuclear and -particle physics; and strong interaction physics beyond the Standard Model. Our immediate goals are to open a dialogue between mathematicians and physicists, so that each may come to appreciate the challenges and needs of the other, and to build fruitful collaborations between mathematicians and physicists working on nonperturbative phenomena.