Joachim Kock: Polynomial functors, trees, and opetopes
Polynomial functors are certain functors built from disjoint union, product, and exponentiation. They are a categorification of the notion of polynomial function: you can add and multiply them or substitute them into each other; you can also differentiate them, and there is a Leibniz rule and a chain rule. All these manipulations can be done just in terms of coefficients and exponents. After this introductory material, I will survey some ways in which polynomial functors are intimately linked with trees and operads. Trees arise from free monads on polynomial endofunctors. For such, the disjoint union of all coeffcients is a collection of (decorated) trees, and they correspond to the set of operations of a (coloured) operad, while the exponents correspond to their arities. There is also a canonical way of associating a polynomial functor to a tree, and a characterisation of the polynomial functors that arise in this way; hence the very notion of tree is just a special case of polynomial functor. To finish with some recent research (joint with Joyal, Batanin, and Mascari), I will explain how polynomial functors provide an elegant construction of 'higher trees', the so-called opetopes, which are the basis for the Baez-Dolan approach to higher category theory.
Teimuraz Pirashvili: Topological Hochschild cohomology and categorical rings.
A categorical ring is a category equipped with two monoidal structure, addition and multiplication, such that some coherent conditions hold. We prove that this notion is relevant to get an algebraic model for two-stage ring spectra.