Pure Postgraduate Seminars
The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgrads to present mathematical ideas. The merger of both UMIST and the Victoria University of Manchester had now occurred and the newly formed School of Mathematics was now larger than ever. The academic year 2004/2005 seminars were organised by Matthew Horsham.
The seminars were held fortnightly on Fridays at 4.00pm. In the Autumn semester the seminars were held in the now non-existant Newman Building. In the Spring semester, the seminars were held in the MSS Building.
Autumn Semester 2004
8 OctAn Introduction to Bordism
22 Oct 2004The Weyl Group
5 Nov 2004Framed Cobordism and Freudenthal Theorems
19 Nov 2004Locally Finitely Presented Categories
26 Nov 2004The Imaginary Category of a Coherent Theory
Ravi RajaniAbstract (click to view)
The aim of this talk is to generalise some functorial ideas in the model theory of modules to a more general setting. Of particular interest is the nonabelian equivalent of Ivo Herzog's category "eq+". This is the category of finitely presented functors in (fp(RMod), Ab) which can be described in terms of positive primitive formulas. We show that for any locally finitely presented category C of models of a first order theory, the category of finitely presented functors in (fp C, Set) has a description in terms of positive existential formulas. Moreover, as in the abelian situation, a definable subcategory D⊆C (that is a subcategory with a coherent axiomatisation) gives rise to a category of sheaves in (fp C, Set) which is itself a locally finitely presented category. The category of finitely presented sheaves corresponding to D will also have a nice description in terms of positive exisitential formulas. This will enable us to define the category Deq of imaginaries corresponding to D. We will define interpretations between definable subcategories as appropriate functors between the corresponding imaginary categories.
Spring Semester 2005
25 Feb 2005Very Basic Introduction to Homotopy
Jerry HopkinsonAbstract (click to view)
Beginning from intuitive ideas of deforming curves in a space, we build up to the homotopy groups of spheres. The ideas, definitions and simple results will be presented, with plenty of examples. Topics: Homotopic maps; homotopic spaces; homotopy groups; homotopy groups of spheres.
18 Mar 2005Polynomial Maps Between Spheres
Jerry HopkinsonAbstract (click to view)
Amongst all the maps from any sphere to itself, there are some that are polynomials. The question is when such polynomial maps exist. The question is explained; these maps are related to representatives of the elements of a homotopy group of the sphere. Polynomial maps are constructed in the cases where they are known to exist. Examples of why polynomial maps fail are shown in some other cases.
29 Apr 2005Putting Numbers in Boxes, a Standard Tableau Problem
Marianne JohnsonAbstract (click to view)
A partition n can be represented as a Young diagram (which is essentially a collection of boxes arranged with a weakly decreasing number of boxes in each row). A standard tableau is a filling of a Young diagram with the numbers 1-n such that the numbers increase as you read across each row and down each column.
Klyachko's remarkable theorem (1974) on the module structure of the n'th Lie representation of Sn can be restated solely in terms of the objects just described. I leave all the hardcore algebra and representation theory at the door and suggest a strategy for proving Klyachko's theorem just using properties of standard tableaux.
6 May 2005An Introduction to Supermanifolds
Adam HaunchAbstract (click to view)
The super-sphere of dimension (2|2) has zero curvature. The aim of this talk is to explain the above statement and show how it is possible for a non-trivial supermanifold to have zero curvature. In the process of doing so, the concepts of odd and even variables, the Berezinian and the supertrace will be introduced.
20 May 2005The Size of Self Similar Sets
Thomas JordanAbstract (click to view)
Self-similar sets are sets which are invariant under a finite (usually) collection of similarites. They often have fractal structure. It is usual to assume that the images of the similarities are essentially disjoint. However I will include sets with arbitrary overlap. We will look at various notions of size for these sets in Rn including Lebesgue measure, Hausdorff dimension and address the question of whether they contain open sets (in Rn).
List of 2008/2009 seminars (Ali Everett)
List of 2007/2008 seminars (Jacob George)
List of 2006/2007 seminars (Stephen Clegg)
List of 2005/2006 seminars (Marianne Johnson)
List of 2004/2005 seminars (Matt Horsham)
List of 2003 seminars (Matthew Craven/Sara Santos)
List of 2002 seminars (Sarah Perkins/Sara Santos)