Final Report Summary - MODFLAT (Moduli of flat connections, planar networks and associators)
MODFLAT project lies at the crossroads between several fields of Mathematics: Algebra, Geometry, Topology, and to some extent Analysis. It has interactions with Physics, and in particular with Quantum Field Theory. The main theme of the project was an interaction between three active research directions: the theory of associators in Algebra, moduli spaces in Geometry and planar networks in Combinatorics. Eleven questions ranging from relatively easy to very ambitious ones were stated in the research program.
During the lifetime of the project, we were able to answer most of the easy questions, and we made very significant progress on several difficult directions. The main highlights include a surprizing link between topology of interesections and self-intersections of curves and non-commutative differential calculus, a way to construct families of inequalities by using special functions called potentials, and an approach to an important algebraic problem which uses Feynman graphs. The latter is an example of a fruitful interaction between Mathematics and Physics.
The project opened new research avenues in Algebra, Geometry and Topology and in their relations to Quantum Physics. Several spin-off research projects are currently on the way.
The work on the project was a collective effort by the Pi, several senior collaborators and postdoctoral fellows and an active group of brilliant PhD students.
During the lifetime of the project, we were able to answer most of the easy questions, and we made very significant progress on several difficult directions. The main highlights include a surprizing link between topology of interesections and self-intersections of curves and non-commutative differential calculus, a way to construct families of inequalities by using special functions called potentials, and an approach to an important algebraic problem which uses Feynman graphs. The latter is an example of a fruitful interaction between Mathematics and Physics.
The project opened new research avenues in Algebra, Geometry and Topology and in their relations to Quantum Physics. Several spin-off research projects are currently on the way.
The work on the project was a collective effort by the Pi, several senior collaborators and postdoctoral fellows and an active group of brilliant PhD students.