Operads: Hopf algebras and coloured Koszul duality

Abstract

Operads are tools designed to study not mathematical objects themselves, but operations on these. A simplified example: instead of integers, one studies multiplication. Multiplication is a map that takes two integers as input and gives one new integer as output (2*3 = 6 says that the inputs 2 and 3 give 6 as output). The origin of operads is to look at operations with an arbitrary number of inputs and one output (2*3*4*5=120 describes an operation with 4 inputs that with inputs 2, 3, 4, and 5 gives output 120). One should think of an operad as a system of such maps together with composition rules (in our example (2*3)*(4*5) = 2*3*4*5 is a composition rule). This point of view seems rather straightforward, but has led to important developments in several branches of mathematics. Operads enable handling complicated combinatorics where many operations appear. Imagine a surface together with a rule called multiplication, that assigns to every two points on the surface a third point. One might ask what are the possible ways to change the multiplication a little bit. The study of such changes is a branch of Deformation Theory. Deformation theory can be extended to a purely algebraic context, where the surface in the example above is replaced by an algebraic object. In fact, for systems of operations described by an operad one can often develop a deformation theory. The interplay between operads and Deformation Theory has contributed to a large extent to the understanding of one of the major results in mathematics in the previous decennium: Kontsevich’ theorem on the Quantisation Deformation of Poisson Manifolds. This result can be interpreted as a theorem about the correspondence between Classical and Quantum Mechanics. More precisely: it is an instance of Bohr’s Correspondence Principle, which says that to every system in Classical Mechanics there exists a system in Quantum Mechanics that looks the same if one observes it at sufficiently large scale. My research consists in part in generalising existing theory on deformations on the basis of operads, and finding new applications and relations to classical results. Furthermore, the author is currently studying not deformations of structures described by operads, but deformations of operads themselves. From a mathematical point of view this is a natural thing to do. Moreover, this approach reveals a subtle interplay of deformation results at different levels. Apart from the above, I am studying applications of operads in Combinatorics. These applications are related to so called Hopf Algebras. Hopf algebras have applications in Perturbative Quantum Field Theory. The essential problem is that computations of physical quantities from Feynmann integrals often give infinity as a result. Hopf Algebras suggest a solution to this problem. My work shows that the Hopf Algebras used in renormalisation often stem from operads. This approach simplifies the combinatorics involved, and shows relations to other Hopf Algebras.