Last updated: 31 October 2007
|| COBORDISM THEORY || COHERENCE || HOMOTOPY THEORY || HOPF ALGEBRA || TORIC MANIFOLDS ||
This page outlines PhD projects for which I am currently offering supervision; their order may indicate my current level of enthusiasm! The list is not exhaustive, and I am always willing to listen to constructive suggestions and requests on related topics. I very much enjoy working with postgraduate students, so if you fancy the thought of researching into one or more of these areas, send me a message now ....
The projects are unashamedly pure mathematical, and some of my past students now hold academic positions in college and university mathematics departments. Nevertheless, others have found that their higher degrees have been of major importance in getting started in more practical careers; for example in code-busting at GCHQ, or in the rarified financial world of the City. Needless to say, several of the latter now get paid more than I do! Most of my students have also had the opportunity to attend conferences in interesting parts of the world during work on their projects.
For those of you wondering just what PhD study in algebraic topology may actually entail, I suggest you take look at the blog of my current student Craig Laughton; it is very informative!
TORIC TOPOLOGY: Algebraic geometers have studied "torus embeddings" since the 1970s, obtaining many beautiful examples of complex varieties which admit actions by a high dimensional torus. During the 1990s, the underlying ideas spread to algebraic topology, thanks to the work of Mike Davis and Tadeusz Januszkiewicz; the resulting theory of toric manifolds provides a brilliant and exciting mixture of geometry, combinatorics and topology. Much of my work in this area has been joint with Victor Buchstaber and his collaborators, and has also involved my PhD students David Carter and Yusuf Civan. Click here to enjoy a preprint with the flavour! Projects include studying associated loop spaces, generalisations of Bott towers, and calculations with symplectic oriented cohomology theories. During 2001, Taras Panov visited Manchester on a Royal Society/NATO Postdoctoral Fellowship, and returned for a six-month EPSRC Visiting Fellowship in 2003. We are currently organising an international conference in toric topology, to be held in the Alan Turing Building in July 2008. These collaborations have led to associated projects in abstract homotopy theory ...
ABSTRACT HOMOTOPY THEORY: Since Quillen's remarkable pioneering work in the 1960s, attempts to axiomatise homotopy theory as a type of "nonlinear homological algebra" have grown in popularity. The area is now known as model category theory, and has experienced an explosion of interest since the turn of the millennium. One example of the idea is to find a category (or categories) which model standard homotopy theory through the eyes of a particular prime number, in the hope that results will be more straightforward. In the complementary case of the rational numbers, Quillen described several distinct categories of algebraic objects, all of which are equivalent to rational homotopy theory in an appropriate sense. I was led into this area through my collaboration with Taras Panov and Rainer Vogt, and subsequently began working with Dietrich Notbohm. Available projects often involve applications to toric geometry, and include the study of diagram categories, homotopy colimits, and minimal models. Click here to sample the flavour!
COBORDISM THEORY: Cobordism theory is a way of organising and classifying manifolds whose stable tangent bundles admit additional structure. It originally flowered in the hands of Rene Thom and Jack Milnor during the early 1960s, and the study of complex cobordism energised homotopy theory for the next 20 years or so; it is now finding applications in quantum field theory. I have been involved with the development of complex cobordism, framed cobordism, and symplectic cobordism since 1966, and can suggest a variety of projects in the area. These include studying geometrical aspects of the double transfer map, and resolving two conflicting calculations I made in the early 1980s! Underlying the subject is the quest for the topologist's holy grail, namely an understanding of the stable homotopy groups of spheres - otherwise disguised as the framed cobordism ring. The subject is a beautiful amalgam of algebra and geometry, and can accommodate students who wish to lean towards either, but enjoy both. Click here for access to an offprint with the flavour!
HOPF ALGEBRA: Certain types of geometrical and combinatorial object can be subdivided in several different ways; in the right circumstances, this leads to an algebraic coproduct on the abelian group generated by the original objects. These concepts were first developed by Gian-Carlo Rota in the 1970s, and have revolutionised much of combinatorics by providing a rigorous algebraic framework for results which had seemed piecemeal and unstructured. I have been involved in the area since the mid 1980s (often in collaboration with Bill Schmitt), and might still be willing to projects in umbral calculus, quantum group structures, and polynomial invariants of graphs and partially ordered sets. The subject is particularly exciting because applications are constantly emerging in new and unexpected areas. Click here to see a preprint with the flavour! Other projects are relevant to the related areas of Hopf rings and Hopf algebroids, both of which arise naturally in algebraic topology.
COHERENCE: Since the 1960s work of Mike Boardman, Saunders
MacLane, Jim Stasheff, and Rainer Vogt, the study of coherent
commutative diagrams in various algebraic and geometrical categories
has been of major importance. Spurred on by Peter May and his
operads during the 1970s, we have gained an intimate
knowledge of infinite loop spaces and their homology, leading on to
the development of such diverse algebraic notions as Hopf rings and
n-categories. Ever since collaborating with Peter in the
early 1970s, I have been collecting problems on loop spaces with
enthusiasm. The n-categorical aspects of coherence are now
being scrutinised by computer scientists, pure mathematicians, and
theoretical physicists, and suggest remarkable links with geometry and
cobordism theory. There is exciting stuff here for those who like the
idea of interdisciplinary mathematics - but my knowledge is rapidly
falling behind the cutting edge, and I have already stopped thinking
of myself as an expert!
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In order to help you to decide whether or not you should apply for
postgraduate work, you are welcome to come to Manchester at our
expense and talk through the possibilities in person. I will be happy
to arrange for you to meet some of my current students, who will also
show you around. If you wish to proceed with a formal application, you
should contact our Pure Mathematics PG Admissions Tutor Charles Walkden (or
Secretary Karen Stott),
who will guide you through the process. You may prefer to meet us in
the company of other potential postgraduates at our annual Postgraduate
Open Day, which we usually hold in February; you will even get a
free buffet lunch!
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