Algebra & Combinatorics

Members of the Algebra and Combinatorics cluster in the De Brún Centre undertake research in:   

• Computation with linear groups, algebraic design theory (Dane Flannery) 
  
  
• Geometric rigidity theory (James Cruickshank) 

• Linear algebra over finite fields (Rachel Quinlan) 

• Quantum computing (Michael McGettrick, Mark Howard) 

• Representation theory (Götz Pfeiffer) 

• Vertex operator algebras (Michael Tuite) 

• Combinatorics of posets, Coxeter groups, and hyperplane arrangements (Angela Carnevale, Joshua Maglione, Götz Pfeiffer) 

• Zeta functions of groups, algebras and modules, and Igusa’s local zeta function (Angela Carnevale, Joshua Maglione, Tobias Rossmann) 

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. 

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. 

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. Problems of interest include 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:  Angela Carnevale, James Cruickshank, Dane Flannery, Mark Howard, Kevin Jennings,  Götz Pfeiffer, Rachel Quinlan, Tobias Rossmann, Emil Sköldberg, Michael Tuite. 

Emeritus members: John Burns, Rex Dark, Graham, Ellis, Ted Hurley, John McDermott, Martin Newell, Sean Tobin.