Lawvere Theories and monads are two ways of handling algebraic theories. They are related but subtly different; one way in which they differ is that models for a given Lawvere Theory can automatically be taken in many different base categories, whereas monads have a fixed base category.

Distributive laws give a way of combining two algebraic structures expressed as monads, so one might naturally ask whether something analogous can be done for Lawvere Theories; in fact Bénabou asked me this in December 2009 after I had given a talk on distributive laws for monads. In this talk I will give a way of doing this, using a reformulation of Lawvere Theories that I learnt from Martin Hyland and Tom Leinster, which is of interest in its own right. I will also discuss an illuminatingly wrong way of doing it.

I will begin by briefly covering the basics of Lawvere Theories, their relationship with monads, and the notion of distributive laws between monads.

In this talk, I will present some results from my paper (joint with John Power) recently accepted at CALCO 2011. First part of the talk will provide a gentle introduction to Logic programming and some basic proof-search algorithms associated with it. The second part of the talk will be devoted to the category theoretic (coalgebraic) semantics for logic programming; and the role Lawvere Theory plays in the semantics.

When one mixes algebra with homotopy, higher operations appear naturally. The combinatorics of such higher structures become quickly complicated and they can hardly be handled by hand. Hence there is a need of a mathematical object, which models these algebraic operations. It is the notion of an operad. Homological algebra on the level of operads allows us to define categories of algebras with good homotopy properties.

The purpose of this talk is to give a gentle survey of the methods and results of the operad theory (bar-cobar adjunction, Koszul duality theory, homotopy transfer theorem). It also aims to show the relationship with other fields (rewriting systems, Gröbner bases, syzygies, deformation theory). [I will not assume that the audience is familiar with the notion of an operad.]