Formal theory of modular operads and related structures

Modular operads were introduced in the study of moduli spaces. Their combinatorics, which are governed by graphs with cycles, are very intricate in comparison with ordinary operads, which are governed by trees.

My PhD solved a `problem of loops' in the formal theory that had hindered the development of modular operads for some years. The problem arises since (finite) graphs with cycles contain paths of infinite length. And this means that the rules for simpler structures like trees cannot be directly modified to graphs.

Graphical combinatorics and a distributive law for modular operads, Advances in Mathematics, 2021.

This work presents a detailed analysis of the combinatorics of modular operads. These are operad-like structures that admit a contraction operation as well as an operadic multiplication. Their combinatorics are governed by graphs that admit cycles, and are known for their complexity. In 2011, Joyal and Kock introduced a powerful graphical formalism for modular operads. This paper extends that work. A monad for modular operads is constructed and a corresponding nerve theorem is proved, using Weber's abstract nerve theory, in the terms originally stated by Joyal and Kock. This is achieved using a distributive law that sheds new light on the combinatorics of modular operads.

arXiv preprint

Brauer diagrams, modular operads and a nerve theorem for circuit algebras (preprint)

Circuit algebras, used in the study of finite-type knot invariants, are a symmetric analogue of Jones's planar algebras. They are very closely related to circuit operads, which are a variation of modular operads admitting an extra monoidal product. This paper gives a description of circuit algebras in terms categories of Brauer diagrams. An abstract nerve theorem for circuit operads -- and hence circuit algebras -- is proved using an iterated distributive law, and an existing nerve theorem for modular operads.
  • I am currently completing a similar construction for compact closed categories.
  • Together with Marcy Robertson, we are looking at models of higher circuit algebras<\i>, and applications in low-dimensional topology and algebra.
  • Together with Ross Street, we are looking Tannaka duality, stable representation theory, and modular operads.
  • (More details to come.)

    Posted on:
    October 1, 2021
    2 minute read, 348 words
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