Projects in Previous Years

1st Semester 2013/14: Duality Theory

Luca Spada
If you are interested in this project, please contact the instructor email.
Dualities are an important general theme that has manifestations in almost every area of mathematics. A vast majority of them can be phrased in the language of category theory. Generally speaking, dualities translate concepts, theorems or mathematical structures into (apparently) different concepts, theorems or structures. In this sense they formalise the practice of "changing perspective". In mathematics, many dualities, e.g., Pontryagin duality or Gelfand duality, had a prolific impact in their realms. In particular in logic they usually provide concrete semantics and tools to study abstract calculi, e.g., Stone duality or Priestly duality.

In a recent paper we have put forward a general framework that affords a unification of known result and establishes connections between the theory of duality, the classical Nullstellensatz in algebraic geometry and Birkhoff's subdirect representation theorem. The aim of this project is to have a closer look to this approach and see how it emanates through concrete examples.

The new approach mentioned above will be presented in a round of three introductory classes. Discussion meetings will follow, the participants will be asked to apply the abstract framework to specific examples and find new applications of the framework. This will lead to either some reports or small research papers.
Some mathematical maturity of the participants will be assumed. Familiarity with some category theory and universal algebra would be helpful but is not strictly necessary.
This is a 6 EC project. Students will be asked to demonstrate understanding of the course materials through a written report and an oral presentation. Participation in the group discussions will also contribute to the assessment.