## Functional Analysis, Algebra & Geometry

Researchers: Milena Venkova; Colum Watt

### Description of area

Topological & geometric methods in group theory (Colum Watt)

Recent research has centred on finite Coxeter groups and their associated Artin groups. In collaboration, the research has investigated corresponding non-crossing partition lattices, hyperplane arrangements, generalised associahedra and generalised permutahedra. A variety of geometric, topological and combinatorial methods have been used in these investigations. There is considerable interest in solving open conjectures involving non-crossing partitions, in part because they occur in several disparate areas of mathematics such as free probability theory, cluster algebras and Lie algebras. A natural extension of the research is to investigate whether the methods that have been developed can be used to tackle some of these open conjectures.

Inﬁnite-dimensional holomorphy & applications of functional analysis (Milena Venkova)

The primary research interests in the area of functional analysis are associated with infinite-dimensional holomorphy - the study of polynomials and holomorphic mappings on infinite dimensional spaces. Research is also concerned with the applications of functional analysis to classification theory, pattern recognition and optimization.

Of current interest is the problem of approximating holomorphic mappings or polynomials of high degree by polynomials of lower degrees. This problem had been studied using techniques of experimental mathematics, but has many interesting features and challenges even for several variables when the degree of the approximated polynomial is greater than 10. A promising approach is looking at the problem in a Banach space setting, and applying to it results from the theory of linear operators on tensor products.

For more information about any of these areas and to discuss opportunities for research please contact the individuals above.