Algebra and Geometry

Members of the Algebra and Geometry cluster in the De Brún Centre undertake research in the following specific areas.

  • Differential geometry (John Burns)
  • Geometric rigidity theory (James Cruickshank)
  • Computational homology (Graham Ellis)
  • Computation with linear groups, algebraic design theory (Dane Flannery)
  • Quantum computing (Michael McGettrick)
  • Representation theory (Götz Pfeiffer)
  • Linear algebra (Rachel Quinlan)
  • Vertex operator algebras (Michael Tuite)

Research in differential geometry within the Centre has a particular focus on homogeneous spaces such as symmetric spaces and generalized flag manifolds. Many of the problems are related to the representation theory of Lie groups, with finite Coxeter groups also playing an important role.

Geometric Rigidity Theory studies bar and joint frameworks and various other related structures. Recent work is on rigidity theory of surface graphs and other related families of graphs such as block and hole graphs. This involves ideas from algebraic geometry, graph theory and low-dimensional topology.

Computational homology uses algorithms and techniques from symbolic computation to describe cohomological properties of topological spaces. Current work focuses on the cohomology of classifying spaces of a range of finite and infinite groups as well as on the low dimensional cohomological properties of spaces arising in the emerging area of applied computational topology.

Computation with linear groups has entered a new phase with the development and implementation of effective algorithms for computing with finitely generated groups over infinite domains. Problems solved include testing finiteness, recognizing finite groups, obtaining a computational analogue of the Tits alternative, and computing ranks. Current work focuses on computing with arithmetic subgroups of linear algebraic groups, or, more generally, Zariski dense groups.

Algebraic Design Theory treats pairwise combinatorial designs (such as Hadamard matrices and their generalizations) from the perspective of abstract algebra. It draws on techniques from group theory, matrix algebra and cohomology, to explore the structure of automorphism groups in the twin contexts of constructing and classifying designs.

Quantum information theory (also known as quantum computation) applies algebraic techniques in the development and analysis of quantum algorithms. In particular, efficient quantum algorithms are constructed and analyzed using quantum walks where the asymptotic properties of quantum Markov chains (using unitary matrices) are studied. In a different direction, game theory techniques are extended to the quantum domain where iterated quantum games played on a network (graph) are studied, to determine the properties of "quantum agents".

Linear algebra has many interactions with group theory, combinatorics and the theory of finite fields. Of particular interest are linear and affine matrix spaces in which rank behaves in a controlled way, and completion problems for partial matrices and related objects over fields. Well-known problems in the area include the "Netflix problem", which (roughly) seeks to complete a sparsely filled array to a matrix of low rank; and the minimum rank problem for graphs, which asks for the minimum rank of a matrix with a specified (possibly symmetric) pattern of zero and non-zero entries.

Vertex operator algebras is a relatively new mathematical theory based on ideas originally arising in string theory and conformal field theory in theoretical physics. Vertex operator algebras explore deep relationships between algebra, complex geometry, group theory, Riemann surfaces, number theory and combinatorics. For example, they provide the setting for understanding `Monstrous Moonshine', which relates modular forms to the Monster finite simple group. Recent research has concentrated on the relationship of vertex operator algebras to (i) higher genus Riemann surfaces and (ii) Jacobi forms.

Cluster members

John Burns, James Cruickshank, Graham Ellis, Dane Flannery, Kevin Jennings, Niall Madden, Michael McGettrick, Götz Pfeiffer, Rachel Quinlan, Emil Sköldberg, Michael Tuite,

Emeritus members

Rex Dark, Ted Hurley, John McDermott, Martin Newell, Sean Tobin,