1st Leeds And Manchester Event:
Joint University Mathematics Pure Postgraduate Seminar
The 1st Leeds And Manchester Event: Joint University Mathematics Pure Postgraduate Seminar took place in Manchester on Friday the 12th of May 2006. Details of the talks are given below.
All talks were in the School of Mathematics in the MSS building, which is just five minutes walk from Manchester Picadilly train station.
13:30Invariants of supermatrices
Adam Haunch (Manchester)Abstract (click to view)
The aim of this talk is to highlight some of the more unusual aspects of supermatrices. To do this, the properties of the elementary symmetric functions of supermatrices will be studied and a comparison made with the elementary symmetric functions of matrices in GL(n).
14:00Post's Programme and the n-C.E. Hierarchy
Bahareh Afshari (Leeds)Abstract (click to view)
In 1944, Post set out to relate computational structure to its underlying information content. Since then, many computability-theoretic classes have been captured, in the spirit of Post, via their relationships to the lattice of computably enumerable (c.e.) sets. In particular, we have Post's characterisation of the non-computable c.e. Turing degrees as those of the simple, or hypersimple even, sets; Martin's Theorem showing the high c.e. Turing degrees to be those containing maximal sets; and Shoenfield's characterisation of the non-low2 c.e. degrees as those of the atomless c.e. sets (that is, of co-infinite c.e. sets without maximal supersets).
In this talk, I initiate the extension of Post's programme to computability-theoretic classes of the n-c.e. sets.
14:30The normal basis theorem; an introduction to the Galois module structure of field extensions.
Erik Pickett (Manchester)Abstract (click to view)
I will state and prove the normal basis theorem for finite Galois extensions of infinite fields. For a field extension L/K with Galois group G, this theorem sets up an isomorphism between the field L and the group algebra KG. This means we can learn a lot about Galois field extensions by studying the structure of such algebras. In this talk I will try to assume as little Galois theory as possible.
15:30Modules and Mutations
Graham Murphy (Leeds)Abstract (click to view)
I will give a very brief survey of the definitions and basic theorems connected with Auslanderís representation dimension and then introduce the concept of orthogonal modules as defined by Iyama and illustrate a connection between these and the cluster algebras as introduced by Fomin and Zelevinsky.
16:00Counting the infinite; Perron's formula and Tauberian theorems.
Matthew Horsham (Manchester)Abstract (click to view)
We introduce Perronís formula and show how it can be applied to certain counting problems. We explain the major obstacle to this approach and introduce the Tauberian theorems that allow this obstacle to be overcome.
16:30Proving the consistency of arithmetic
Elliott Spoors (Leeds)Abstract (click to view)
This talk shall look at a brief history behind the formalisation of arithmetic at the end of the nineteenth century, and the proof of its consistency by Gentzen nearly 50 years later. The result is presented using a simplified approach due to Schütte. Very little background knowledge of logic or proof theory is assummed.
17:00Post Seminar Meal