In various mathematical contexts it is possible to find a single object which, when subjected to a countable process, yields approximations to the whole universe under study. Such an object is termed "universal" and, contrary to expectations, such objects often turn out to be generic rather than exceptional. This talk will focus on this phenomenon in respect of the Taylor series of a holomorphic function, and how the partial sums behave outside the domain of the function. It will discuss how potential theory reveals much about the boundary behaviour of such functions, and their relationship with conformal mappings.

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Ordered commutative monoids are mathematical structures that are simple to define, yet display a very diverse phenomenology. I will introduce these structures and explain how they formalize situations in which one deals with resource objects and how they can be combined with each other or converted into each other, such as the molecules in a chemical reaction like $2H_2 + O_2 \rightarrow 2H_2O$. Some standard theorems of functional analysis yield results on ordered commutative monoids, which in turn have applications to Shannon's theory of communication and the ordered commutative monoid of graphs.