Research in Algebra and Number Theory in Edinburgh focuses on the following topics:
- Noncommutative Ring Theory (Lenagan,
Smoktunowicz)
- Representation Theory (Gordon)
- Number Theory (Smyth)
- Commutative Algebra (O'Carroll)
together with several research fellows and post-docs, and PhD students.
There are close links in our work with Geometry and Mathematical PhysicsIn the last few years, Agata Smoktunowicz has won a European Mathematical Society Prize, the Whitehead Prize of the London Mathematical Society, the Whittaker Prize of the Edinburgh Mathematical Society and the Waclaw Sierpinski Prize of the Polish Academy of Sciences, as well as giving an invited lecture at the 2006 International Congress of Mathematicians. Iain Gordon was awarded the Berwick Prize of the London Mathematical Society, is currently an EPSRC Leadership Fellow and is an invited speaker at the 2010 International Congress of Mathematicians.
As well as running a weekly algebra seminar, and several informal working seminars, we are heavily involved in international conferences and giving lectures throughout the world. Recent destinations include Grenoble, Vancouver, Palo Alto, Montreal, Shanghai.
Below we describe some of our research interests in greater detail.
Gwyn Bellamy is a research student working with Iain Gordon.
Iain Gordon studies representation theory and its applications to algebraic geometry and combinatorics. The work has Lie theoretic strand (algebraic groups, Lie algebras, reflection groups) and a noncommutative algebra angle (differential operators, noncommutative algebraic geometry, PI algebras) and it involves a variety of methods from geometry. Possible PhD projects: (1) Representations of algebras and resolutions of singularities; (2) Equivalences of categories of algebras, with special reference to symplectic reflection algebras; (3) Noncommutative algebraic geometry.
Rollo Jenkins is a research student working with Iain Gordon.
Tom Lenagan works on noncommutative algebra, in particular on representation theory of noetherian rings and quantum algebras, and on growth properties of finitely-generated algebras over fields, especially on the notion of the Gelfand-Kirillov dimension of an algebra. Possible PhD projects: 1.Structure theory of noetherian rings and modules; 2.Quantum algebras; 3.Growth of algebras.
Liam O'Carroll works on research problems in commutative algebra which have a geometrical flavour, such as reductions of ideals in local rings, the smoothness of blowings-up, deformations and maximal Cohen-Macaulay modules, and Hilbert Series. Possible PhD projects: 1.Joint and complete reductions of ideals; 2.Smoothness of Blow-ups; 3.Resolutions for deformations of maximal Cohen-Macaulay modules; 4.Thin Hilbert Series.
Agata Smoktunowicz Agata works on problems in noncommutative ring theory, especially problems concerning nil rings, the Jacobson radical, prime and primitive rings, and combinatorial problems in ring theory including growth of algebras (Gelfand-Kirillov dimension).
Chris Spencer is a research student working with Iain Gordon.
Chris Smyth has interests in algebraic number theory, with subsidiary interests in algorithmic aspects of algebraic curves, and in combinatorial aspects of network design. In number theory, he is interested in particular in the study of algebraic integers which are confined in some way (e.g. their conjugates could be restricted to a certain region of the plane, or constrained to satisfy some identity). Possible PhD projects: 1.Cyclotomic points on curves, with application to factoring polynomials; 2.The Euclidean algorithm and Bezout's theorem; 3.Pisot numbers and trees.
Graeme Taylor is a research student working with Chris Smyth.
Michal Ziembowski is a research student working with Agata Smoktunowicz.