Beyond Part III

Young Researchers in Mathematics 2009

16-18 April 2009

Centre for Mathematical Sciences, Cambridge

Algebra and Foundations Session

Talks will be in the Pavillion E committee room, except for Thursday between 4.30 and 6 (MR13) and Friday 4.00-4.45 (MR2).



Homology in Semi-Abelian Categories - Julia Goedecke

I will give an introduction to semi-abelian categories, comparing them with the well-known concept of abelian categories. Semi-abelian categories developed out of the need for a more general framework for homological algebra. While homological algebra is well studied in abelian categories, many of the homological diagram lemmas such as the five lemma or the snake lemma also hold in non-abelian categories such as the category of groups, or Lie algebras. Semi-abelian categories provide a unifying framework for all these different settings, and many others besides. In this context I will discuss different ways of calculating homology, including comonadic homology and Hopf formulae. I will also showcase a long exact homology sequence which differs slightly in appearance to its abelian counterpart. To conclude I will give a hint of how this long exact sequence may be used to define homology without the use of projective objects.

Forcing in set theory and recursion theory - Phil Ellison

Paul Cohen famously proved the independence of the continuum hypothesis using his technique of forcing. Prior to this, Kleene and Post had introduced extension methods for proving results about the Turing degrees, which can be seen as a particular type of forcing argument. In set theory, forcing is used to prove independence results by adjoining sets with suitable "generic" properties to models of ZFC. In recursion theory, sets of natural numbers satisfying weaker forms of genericity turn out to have interesting degree theoretic properties. In this talk, I will briefly introduce the basic concepts of recursion theory, very briefly introduce forcing in set theory (with liberal use of hand waving and very few technical details), sketch Kleene and Post's original construction, and discuss further applications of forcing in both subjects.

Counting, measure, and metrics - Tom Leinster (keynote speaker)

What is the "size" of a thing? The dimension of a vector space, the Euler characteristic of a topological space, the entropy of a probability space and the biodiversity of an ecosystem are all examples of "size". I will explain all of these and more. And although I will not give a general answer to the general question, I will expose some common patterns. I will focus in particular on a new notion of the "size" of a metric space, which springs directly from a categorical line of thinking and appears to encode some very interesting geometrical information.

Conjugation families in Fusion Systems - Betty Fyn-Sydney

We review the theory of conjugation families in a Sylow subgroup of a finite group. We reprove the theorems of Alperin and Goldschmidt on conjugation families and weak conjugation families. We then generalise this theory to the case of saturated fusion systems.

Operads and Induced Adjunctions - James Griffin

In this talk I will describe what an operad is and what it is that they may do. Over the category of chain complexes a morphism between two operads induces an adjunction between the categories of algebras for the respective operads. I'll give some examples of these adjunctions, some very much familiar. There are also familiar functors which do not arise directly in this way as a left or right adjoint, but do fit into a natural double category.

Probabilistic Logics: the new laws of thought? - Roko Mijic

In 1854, George Boole published a seminal paper in the development of Logic called "An investigation on the Laws of Thought", in which (what is now) propositional logic is promoted as the mathematics of human thought - and arguably for almost 150 years the academic community has accepted logic as the de-facto mathematics of thought. First-order predicate calculus is certainly an invaluable tool for working mathematicians.

But in recent years, attempts to use logic to actually build thinking machines have met with failure: logic is too brittle to deal with the uncertainty and noisiness of real world data. Consequently, there is a movement to combine the traditions of Boole and Bayes to make probabilistic logics - laws of thought fit for the real world.

Reverse Mathematics and Ramsey's Theorems (accessible talk) - Julia Erhard

I will give an introduction to the ideas of reverse mathematics and summarise the main results. Reverse mathematics allows us to classify theorems according to their strength. I will place particular emphasis on the classification of Ramsey's Theorem in the context of weak arithmetic. Ramsey's Theorem states that any colouring of the pairs of natural numbers has an infinite monochromatic induced subset. If we colour n-sets (n>2) of natural numbers instead of pairs, the result still holds but is strictly stronger.

The Axiom of Choice - Zachiri McKenzie

The Axiom of Choice (AC to its acquaintances) asserts that for every collection of non-empty disjoint sets there exists a set that picks out exactly one member of each of the sets in the collection. Its non-constructive nature and the fact that it can be shown to be an indispensible to the proofs of important results ranging across almost every field of mathematics has led to AC acquiring the status of the most controversial and studied axiom of set theory. In this talk I will briefly introduce the Axiom of Choice giving some of its equivalent statements and listing some of its applications to other areas of mathematics. I will then introduce Cohen’s forcing method and show how it can be used to produce models in which AC fails. The appeal to symmetry in these models is closely connected to the earlier method developed by Fraenkel, Mostowski and Specker to produce models of Zermelo Fraenkel Set Theory with Urelements (ZFA) that refute AC. If time permits I will examine some models refuting statements about cardinal arithmetic that follow from AC.