Analysis, Geometry & Topology

Members of the Analysis, Geometry and Topology cluster in the De Brún Centre undertake research in the following specific areas:

• Analytic topology and its connections with order theory.
• Computational Topology (Graham Ellis)
• Differential Geometry (John Burns, Martin Kerin).
• Fractal geometry and geometric measure theory.
• Functional analysis.
• Geometric rigidity theory (James Cruickshank).
• Group actions on manifolds (Martin Kerin).
• Nonlinear analysis.
• Numerical analysis of PDEs (Niall Madden)

Computational topology 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.

Research in differential geometry within the Centre has a particular focus on homogeneous spaces such as symmetric spaces and generalized flag manifolds, as well as manifolds admitting either positive or non-negative sectional curvature. Many of the problems are related to the representation theory of Lie groups, with both compact Lie groups and finite Coxeter groups 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.

When studying geometric topology or differential geometry, many difficult problems become much more tractable in the presence of group actions on manifolds.  Such symmetry assumptions allow algebraic tools from representation theory, algebraic topology and rational homotopy theory to be used in conjunction with traditional geometric techniques.

Research in numerical analysis focuses on the mathematical foundations of algorithms for solving partial differential equations whose solutions feature boundary and/or interior layers. Recent work includes the analysis of weighted finite element methods, algorithms for adaptive mesh generation, and fast solvers for finite difference and finite element discretizations.

Cluster members

John Burns, James Cruickshank, Graham Ellis, Martin Kerin, Niall Madden, Aisling McCluskey, Donal O'Regan, Ray Ryan, Jerome Sheahan