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Algebra & Combinatorics

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

Computation with linear groups, algebraic design theory (Dane Flannery)
Geometric rigidity theory (James Cruickshank)
Linear Algebra (Rachel Quinlan, Niall Madden)
Quantum computing (Michael McGettrick, Mark Howard)
Representation theory (Götz Pfeiffer)
Vertex operator algebras (Michael Tuite)

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. 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.

Research in Numerical Linear Algebra is focused on the design and implementation of algorithms for solving linear systems arising in discretizations of partial differential equations and, especially, the development of preconditioners for boundary layer problems.

Quantum information theory is a framework for understanding the quantum nature of the universe and how to exploit this nature for new information-processing capabilities. Quantum Computation is concerned with constructing algorithms using sequences of unitary operations on quantum states.
Current work includes:
-Algebraic techniques in the development and analysis of quantum algorithms: 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".
-Quantum Error-Correction and Fault-tolerance: Applications of the Stabilizer formalism and the Clifford group to problems like circuit simulation and Magic state distillation.
-Quantum Foundations: The study of Discrete Wigner functions, Mutually unbiased bases, SIC-POVMs, Nonlocality & Contextuality

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.