DEPARTMENT OF MATHEMATICS

Projects in Topology, Geometry and Combinatorics

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Projects with Peter Eccles, Ron Ledgard, Nige Ray, Igor Rivin, Grant Walker, and Reg Wood.

The following list gives outlines of some representative MPhil and PhD projects for which we offer supervision. The list is not exhaustive, and we are always willing to listen to constructive requests and suggestions on related topics! We all enjoy working with postgraduate students, so if you fancy the thought of researching into one or more of these areas, send us a message now. If you feel you need further information before making an application, we are likely to invite you to Manchester at our expense and talk through the possibilities with you in person; we will arrange for you to meet some of our current students, who will also show you around.

Projects with Peter Eccles

SELF-INTERSECTIONS OF IMMERSIONS: Examples of immersions are given by the figure eight in the plane (an immersion of the circle with one double point), by the usual picture of the Klein bottle in three-space (which has a circle of double points), and by Boy's surface, a model of the projective plane in three-space with an immersed circle of double points and a single triple point. Although my initial research concerned the homotopy of infinite loop spaces (see the projects on coherence), I was amazed to discover applications to the study of manifolds which arise as the self-intersections of immersions; these led me to the solution of problems which had been outstanding since the pioneering work of Hassler Whitney in the 1940s. My viewpoint combines many fundamental aspects of algebraic and differential topology, and offers projects which involve aspects of those in cobordism theory and the Steenrod Algebra. I expect them to yield further striking geometrical results, and to provide fascinating connections with some classical problems in homotopy theory. Click here to see a paper illustrating the latter.

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Projects with Ron Ledgard

CODES FROM ALGEBRAIC VARIETIES: In the 1980s the Russian mathematician V Goppa discovered a remarkable connection between algebraic curves over finite fields and the theory of error-correcting linear codes. Modular curves, which are quotients of the upper half-plane by certain groups of 2 x 2 integer matrices, are of particular importance, and there is a construction of polynomial complexity for the codes which arise. Unfortunately, the algorithms which establish this fact are impractical to implement, and it is an interesting and worthwhile project to find effective alternatives. Higher dimensional varieties also give rise to codes, many of whose features are ripe for investigation - for example, the generalisations of Hamming weight as defined by Wei in 1991 (in connection with cryptography). In the case of Grassmann varieties, a valuable project is to determine all such higher weights; this builds on joint work which I am currently pursuing.

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Projects with Nige Ray

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 Renee 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 supervise work on several problems which remain unsolved. Underlying these 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 to find an offprint with the flavour!

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. I collaborated with Peter in the early 1970s, and have been collecting unresolved problems on loop spaces ever since! More recently I have worked with colleagues and students on Hopf rings, which serve to organise an amazing amount of algebraic structure under the slogan of "rings in the category of coalgebras". I am currently learning about $n$-categorical aspects of coherence; these are coming under scrutiny by theoretical physicists, and provide remarkable links with geometry and cobordism theory. There is exciting stuff here!

HOPF ALGEBRAS: 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, and can supervise work on 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!

TORIC MANIFOLDS: Algebraic geometers have studied "torus embeddings" since the 1960s, obtaining many beautiful examples of complex varieties which admit actions by a high dimensional torus. During the 1990s, the underlying ideas have 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. I am currently developing several projects in this area, which include investigating simple convex polytopes from an unusual viewpoint - but I have no preprints quite ready to go online yet. Much of this work is joint with Victor Buchstaber and his students in Moscow, and involves my current PhD student Yusuf Civan, from Turkey.

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Projects with Igor Rivin

Igor's projects will soon appear here ...

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Projects with Grant Walker and Reg Wood

STEENROD ALGEBRA: Much of our work is collaborative, and centres around the Steenrod Algebra of stable operations in mod 2 cohomology and its action on rings of polynomials. This algebra was discovered by the topologist Norman Steenrod in 1947, but we are now developing applications in areas which are purely algebraic, and even in iteration theory and computation. Its study has led to many unsolved problems, some of which pertain to the structure of the Steenrod algebra itself, and others to the module structure of the polynomial rings; variants include substituting 2 by an odd prime, and replacing the entire Steenrod Algebra by the divided differential operator algebra, which is defined over the integers and consists of formal differential operators with polynomial coefficients. Such problems may be approached by methods of topology, algebra and combinatorics, and provide a source of projects for students with interests in any of those areas; for more details, click on Reg Wood's Problems in the Steenrod Algebra.

REPRESENTATIONS: The action of the Steenrod Algebra commutes with that of matrices on polynomials by linear substitution, bringing the representation theory of the general linear groups over a finite field into play. Likewise, the action of the divided differential operator algebra commutes with permutations of the variables, and so involves the representation theory of the symmetric groups. As well as providing purely algebraic projects, these interactions lead in turn to a highly developed area of combinatorics, centred around the theory of symmetric functions. For the computationally minded, there is scope for the use of algebraic packages, of which we are currently finding MAPLE to be a particularly appropriate tool.

QUANTUM ALGEBRA: For the more theoretically motivated, we suggest the blue skies project of extending as many of the above concepts as possible to analogues for quantum groups and algebras.

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Page last modified: January 27, 1999