Mark Powell's home page

Contact

Mail:       Mark Powell
            University of Edinburgh School of Mathematics
            Room 5405 JCMB
            King's Buildings
            Edinburgh EH9 3JZ


email:      M.A.C.Powell 'at' sms.ed.ac.uk

Research interests

I am a graduate student of Professor Andrew Ranicki, and I am working on topology, specifically on the classical knot concordance group.

Some general nonsense

I am trying to understand the filtration of the classical concordance group which was defined by Tim Cochran, Kent Orr and Peter Teichner using representations into solvable groups, and so-called higher order Alexander modules and Blanchfield forms. It is beautifully geometrically motivated and explained using Whitney Towers and Gropes. This filtration encompasses all previously known concordance invariants in the first few stages. It is of infinite length, and at each stage it has been shown that the step up (going in integral steps) is non-trivial and indeed of infinite rank; i.e. the quotient has elements represented by non-slice knots. It is my hope to be able to better understand the Cochran-Orr-Teichner obstruction theory by looking at the chain complex of the universal cover of the knot exterior.

Some of my work

Please email me in particular if you have comments or suggestions on, find errors in, or have questions about the work below.

My First Year Report.

My First Year Talk.

An expanded version which I gave during my semester (October - December 2008) at Universität Münster, Germany Münster Talk.

Some notes on constructing duality chain equivalences for some simple manifolds, in the sense that the word problem is easy for their fundamental groups.

The exterior of the trefoil as an example of my first year project.

A short note on the homology of a branched covering space over a knot (now contained in 2nd year report).

A note on the basic operation which removes intersections of immersed surfaces in 4-manifolds.

Some lecture notes based on a course on Slice Knots given by Peter Teichner which Julia Collins and I have typed up. The first part of these notes can be found here, but I have amalgamated both our work into a single file.

The slides of a talk which I gave in Geometry Club which surveys the contents of these lecture notes.

My Second Year Report, which contains the latest on an attempt to define an algebraic group which contains the same information as the 1.5 level of the COT filtration, and my attempts to calculate Twisted Blanchfield forms.

Slides of a talk on Twisted Blanchfield forms which I gave at Indiana University Bloomington in September 2009.

A description of my research to first order: in one slide.

Previous Education

2001 - 2004 Lady Margaret Hall, Oxford, BA Hons Oxon Mathematical Sciences

2005 - 2006 King's College, Cambridge Part III Mathematical Tripos (CASM)

Conferences Attended

Khovanov Homology Conference, University of Liverpool, December 2006.

British Topology Meeting, University of Sheffield, September 2007.

Graduate Homotopy Conference, University of Cambridge, December 2007. I gave a talk here defining the concordance groups and explaining how it is filtered using gropes.

Andrew Ranicki's 60th Birthday Conference: Surgery and Manifold Theory, Muenster, June 2008.

The mathematics of knots: theory and applications, Heidelberg, December 2008.

Other interests

Sailing: my favourite results sheet. I like Lasers, Musto Skiffs, Team Racing, and Small Keelboats like Etchells.

Also windsurfing, cycling, football, and much else besides.

Links

The Graduate School of the University of Edinburgh.

Graduate Students Geometry Club.

SMSTC, the Scottish Mathematical Sciences Training centre, which runs lectures for 1st year graduates via video conferencing across the Scottish universities; I attended (2007-2008) the courses in Algebra, Pure Analysis and Geometry and Topology.

Julia Collins, my 'mathematical sister'.

Arf invariant: an article which I had some part in writing.

Pictures of Laser sailing