Hopf algebroids and operads

One of the first steps in most mathematical theories is to exhibit the full structure of the objects under consideration. For example, the set of integers alone is not the structure one is after in number theory, there are the operations of addition and multiplication, and only the whole package gives rise to truly deep questions and applications.

In this research project, we will study such algebraic structures that are present on the cohomology of certain mathematical objects. Recall that there is for example the (say integral) cohomology of a topological space. This is an invariant that encodes essential information about a given space and can be used e.g. to prove rigorously that a sphere can not be deformed into a torus. Similarly there is the cohomology of a group, an algebra and most other objects in algebra, topology or geometry, and these invariants have found a wide range of applications not only within the area that has defined them. For example, the behaviour of certain field theories in physics is understood via topological charges associated to fields, and these are nothing but elements of the cohomology of the space-time on which the theory lives.

Now, what type of structure do these cohomologies have? In pretty much every example they have a natural addition, and in many cases they also have a multiplication, so they become a ring like the integers. However, often there is more, namely a so-called Gerstenhaber bracket which is a third operation that is compatible with the two others in a prescribed manner. The precise axioms look more mysterious than that of an addition and a multiplication, and this might prompt the question whether this structure is really so natural and fascinating. Fortunately, there are enough results such as for example Kontsevich's famous formality theorem that demonstrate how relevant this structure is, and how far-reaching applications can emerge from a better understanding of its properties; see the main part of the proposal for further details.

The concrete research that will be carried out in this project will further clarify for which type of cohomology theories there is such a third operation, and what the properties of the resulting algebraic structure tell us about the original object whose cohomology we are talking about.

An important aspect of the project will be the language and setting in which the questions will be studied. There are roughly speaking two main approaches to all this, one called operads and one called derived categories, and we will investigate in how far results already obtained in one of them have analogues in the other. The principal investigator has been working in one of the two settings so far, thus an important objective is to learn also the other language, and to stimulate interaction and communication between the two communities.

Dually to cohomology there is an invariant called homology - for instance, the cohomology of a finite group with coefficients in a complex representation is the subspace of the representation on which the group acts trivially, whereas the homology is the (largest) quotient space on which it does so. On homology, potential additional algebraic structures are an action of the cohomology ring, or a certain differential that gives rise to a second notion of homology called cyclic homology. In particularly nice cases, cohomology and homology turn out to be isomorphic, and the isomorphism relates the Gerstenhaber bracket and the cyclic differential in what is called a Batalin-Vilkovisky algebra. To understand when this happens and what is the role of these algebraic structures that first were introduced in a completely different context, namely quantum field theory, is a long-term objective of this project.

There are areas of mathematics that despite being of very pure nature can easily claim to have impact on everyone's health and wealth, as the applications of elliptic curves in cryptography show, or the relevance of low-dimensional topology in the understanding of the dynamics of our DNA.

The research itself that I propose here is surely not anticipated to have such direct societal or economic impact in the nearer future. Its impact will be mostly academic, it is meant to make a relevant contribution to an area of mathematics that is rapidly developing in itself and through its links to other fields.

First of all, the results obtained will answer some open questions and raise new ones, thus opening the way to further research afterwards. Secondly, it will more generally attempt to stimulate and enhance the communication between two larger communities of pure mathematics which I am convinced will be beneficial for both, as I feel that the language and techniques used within the two can efficiently be employed to solve problems of the respectively other one. As a good example, I will speak in the main part of the proposal about the paper "A-infinity algebras for ring theorists" by Lu, Palmieri, Wu and Zhang, which is exactly in the spirit of this proposal, demonstrating the impact that methodology from the operadic world can have on purely ring-theoretic questions. In particular, I myself want to properly learn the theory of operads in its full breadth, and part of the project will be a reading of the long awaited monograph that Loday and Vallette have just finished writing. Working group seminars here in Glasgow and discussions with my friends and colleagues elsewhere will then help making this knowledge accessible and known to a wider audience of algebraists. I request some travel money that will facilitate this exchange, and plan to prepare a separate application to organise a workshop on the topic of the project towards its end.

However, I think the most immediate and manifest impact of this project will be personalised in the RA. I have decided to reduce what I request for myself to the minimum so that I can generate through this project a postdoc position. Ideally this will be filled with an applicant bringing in some expertise on operads who wants to learn some more algebraic topics so that there will be a true transfer of knowledge from the RA to our group and back. In any case this position will help some young scientist to fill a gap year, learn some new theory and build up new scientific connections.

Either this will become a crucial stepping stone in a longer research career, or it will confront him or her with new people and thoughts that will help shaping a job applicant who will afterwards be well positioned on the job market outwith academia.

Pure mathematics in general and the interface of algebra and geometry in particular offers like almost no other area in sciences and arts a simultaneously deep and broad training in abstract analytical and structural thinking, and it is only logical that employers are looking exactly for applicants who have worked seriously on projects like the proposed one.