Talks:

Here is a subset of recent and coming talks.

• Segal Infinity Props, Young Topologists Meeting 2024, 8th August 2024, 25 mins, slides
  Abstract

  Props, and their traced variants (wheeled props), are special types of symmetric monoidal categories with two types of strictly associative compositions, the first being categorical, and the second monoidal. They are ubiquitous in mathematics, encoding structures in algebraic topology, deformation theory and knot theory such as Hopf algebras and tangles. However, many interesting mathematical objects, such as Segal’s cobordism categories, don’t admit strictly associative compositions. In this talk, without assuming familiarity with the prior structures, we will introduce a Segal model for homotopy (or infinity) props, which lets us weaken the associativity of the categorical and monoidal compositions up to coherent homotopy. We will then relate this model to other definitions of infinity props, operads, and categories via various nerve theorems. This talk is based on joint work with Philip Hackney and Marcy Robertson.
• Homological Calculus For Operadic Algebras, European Talbot Workshop 2024, 29th July 2024, 60 mins,
  Abstract

  An introduction to the homological calculus of operadic algebras. Covering: twisting morphisms for (co)operadic (co)algebras; their bar constructions; their Koszul resolutions; rectification results; and the benefits of working in the homotopy category of P infinity algebras with infinity morphisms. We shall show that these general constructions applied to the Koszul operads As, Com and Lie, recover the classical constructions for associative, commutative and Lie algebras.
• Graphical and Segal Infinity Props, Topology Seminar, Unversity of Melbourne, 6th May 2024, 60 mins,
  Abstract

  Simplicial sets are of fundamental importance in homotopy theory, as they provide a means to weaken the associativity of categorical composition. For instance, on the category of simplicial sets, the Joyal model structure provides a model of infinity categories, and the Kan-Quillen model structure provides a model of homotopy spaces. A prop is a type of symmetric monoidal category with two types of composition, a categorical composition, and a monoidal composition. In this talk, we will introduce a graphical set model for props, which allows us to weaken both compositions up to coherent homotopy. We will outline a Quillen equivalence to a new definition of a Segal infinity prop, and time permitting, relate these models to existing structures in literature. This talk is based on joint work with Philip Hackney and Marcy Robertson.
• Introduction to Dendroidal Sets, [Homotopy Theory Seminar, Unversity of Melbourne], 10th April 2024, 60 mins,
  Abstract

  Simplicial sets are to categories, as dendroidal sets are to operads. In this talk, we shall recall the first two definitions, unpack the latter two, and formalise this analogy as a commutative square of adjoint pairs.
• Homotopy Probs and Other G-Operadic Structures, AustMS 2023, 5th Dec 2023, 20 mins, slides
  Abstract

  A prop (prob) is a free symmetric (resp. braided) monoidal category generated by a single object. These are useful and ubiquitous structures, for instance encoding bialgebras and having applications in knot theory and topology. Both these classical structures are instances of group-operadic structures, for the symmetric group and braid group respectively. In this talk, we will characterise many G-operadic structures as algebras over quadratic groupoid coloured operads, which admit simple combinatorial descriptions via nestings. We will discuss ongoing work in proving this large family of operads are Koszul, and in using the Koszul machine to form and study homotopy weakened versions of their algebras.
• The Three Little Graces and the Big Bad Basis, Pure Maths Student Seminar, 13th Oct 2023, 60 mins,
  Abstract

  In this talk, we will discuss algebraic operads and a general method for proving they are Koszul. First, we will introduce operads and three key examples, known as the graces. These are the operads whose algebras/representations are associative, commutative, and Lie algebras respectively. After discussing what it means for an operad to be Koszul, we will show that this property is implied by the existence of a conceptually simpler, confluent terminating rewrite system, i.e. a Groebner basis. Finally, we will work through examples showing that the three graces are Koszul, and discuss further applications of this technique to other algebraic structures.
• Relating Diagonals of the Permutahedra, GT Algebraic Combinatorics Day, 4th July 2023, 25 mins, slides
  Abstract

  The study of cellular approximations to the diagonal of polytopes has a long history, mostly owing to applications in homotopy theory. In this talk, we will focus on a well known polytope, the permutahedra. We shall briefly review existing theory and some new enumerative results, before seeking to relate two distinct formulae for cellular operadic diagonals of the permutahedra. Through combinatorial means, we will show that the Saneblidze—Umble diagonal (2004), and Laplante-Anfossi diagonal (2022), are the only such diagonals. Furthermore, we shall relate them via a simple isomorphism. This talk is based on ongoing joint work with Berenice Delcroix-Oger, Matthieu Josuat-Verges, Guillaume Laplante-Anfossi and Vincent Pilaud.
• Homotopy Wheeled Props, Homotopy Theory in Trondheim, 30th June 2023, 25 mins, slides
  Abstract

  A prop is a free symmetric monoidal category generated by a single object, and a wheeled prop is a prop with a trace. They are useful and ubiquitous structures, not only encoding bialgebras (with traces), but also having applications in knot theory and topology. In this talk, without assuming familiarity with these structures, we will present new definitions of (wheeled) props, and characterise them as algebras over Koszul groupoid coloured operads. We will outline how our proof that these operads are Koszul, using an extension of Groebner bases to groupoid coloured operads, circumvents simple obstructions to existing techniques. We will then indicate how the Koszul machine defines a homotopy (wheeled) prop and unpack what exactly this entails. Finally, we will explore homotopy transfer theory applied to these structures, obtaining consequences in formality theory, and re-obtaining a theorem of Mac Lane.
• A Koszul Operad Governing Wheeled Props, AustMS 2022, 6th December 2022, 20 mins, slides
  Abstract

  A prop is a free symmetric monoidal category generated by a single object, meaning that any morphism in a prop is of the form $x^{\otimes m}\to x^{\otimes n}$. A wheeled prop is a prop in which every object has a dual. Props and wheeled props arise naturally in the study of homotopy coherent algebraic structures, deformation theory and knot theory. It is well known that there exist discrete coloured operads (multicategories) which govern props and wheeled props. This arises from the underlying fact that trees can be used to form disconnected graphs possibly with directed cycles. In this talk we'll discuss a groupoid coloured operad governing wheeled props, and prove that this operad is Koszul. One consequence of our construction is that we can give a definition of an infinity wheeled prop, which is to wheeled props as infinity categories are to categories.
• A Koszul Operad Governing Wheeled Props, Categories and Companions Symposium 2022, 23rd September 2022, 20 mins, recording, slides
  Abstract

  A prop is a free symmetric monoidal category generated by a single object, meaning that any morphism in a prop is of the form $x^{\otimes m}\to x^{\otimes n}$. A wheeled prop is a prop in which every object has a dual. Props and wheeled props arise naturally in the study of homotopy coherent algebraic structures, deformation theory and knot theory. It is well known that there exist discrete coloured operads (multicategories) which govern props and wheeled props. This arises from the underlying fact that trees can be used to form disconnected graphs possibly with directed cycles. In this talk we'll discuss a groupoid coloured operad governing wheeled props, and prove that this operad is Koszul. One consequence of our construction is that we can give a definition of an infinity wheeled prop, which is to wheeled props as infinity categories are to categories.
• Quadratic Presentations of Operads Governing Operadic Structures, University of Melbourne Topology Seminar, 9th May 2022, 30 mins,
  Abstract

  There exist coloured operads whose algebras are other operadic structures such as modular operads, wheeled properads and props. In “Massey Products for Graph Homology”, Ben Ward gives a quadratic presentation of a groupoid coloured operad whose algebras are modular operads and shows this operad is Koszul. In this talk we’ll informally discuss what operads governing operadic structures look like, how we can get nice presentations of these operads using ideas of Ward, and some consequences.
• Mini-series on Groebner Bases for Operads, Weekly Student Seminar, 9th-16th February 2022,
  Abstract

  An informal mini-series consisting of two one hour sessions where we unpacked what it means for an operad to have a Groebner basis. We focused in particular on how we can use Groebner bases as a tool to prove operads are Koszul.
• Groebner Bases for Operads Cannot be Generalised to Wheeled Structures, Australian Kittens 2021, 2nd December 2021, 20 mins, slides,
  Abstract

  The theory of Groebner basis for operadic structures provides an algorithmic method of proving that certain structures are Koszul. First introduced for operads by Dotsenko and Khoroshkin, the theory has recently been generalised to coloured operads by Kharitonov and Khoroshkin. In this talk we will discuss the existing theory, some further possible generalisations, and present a simple counter example showing the theory cannot be generalised to wheeled operads, properads and props.