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The group conducts research in two main domains.
Computational Group Theory (CGT)
Our purpose is to establish a new area of CGT: computing with matrix groups over infinite fields. We are developing special techniques for the efficient handling of linear groups over infinite fields by computer, and are using those techniques to answer a number of vital computational questions. A primary objective is the development of software that enables practical computing in this class of groups.
Algebraic Design Theory
A separate area of our research combines computational algebra and combinatorial design theory, and deals with problems arising in a new meta-theory covering all known types of pairwise combinatorial designs. These problems include deriving computational approaches to solving the 'norm equation' over group rings; determining the structure of an ideal in a certain infinite associative ring given a presentation of that ring; and calculating the automorphism group of a design and its regular subgroups. The ideal question is closely related to the problem of proving the famous Hadamard Conjecture. (details)
See the project web page for more details.
The group is interested in designing and implementing algorithms for computing homological properties of classifying spaces of algebraic objects such as groups, Lie algebras, crossed modules and homotopy n-types. Details can be found in the web pages for the Homological Algebra Programming software package and in the preprints page of the principal investigator in this area.
The group also participates in the European Science Foundation ACAT network on applied and computational algebraic topology. The revolutionary growth of experimental data in the sciences and the availability of unprecedented computing power pose many challenges to contemporary mathematics. The ACAT network aims to combine efforts of researchers from thirteen European countries to develop mathematical tools for the following broad research themes:
Coxeter groups form a fascinating class of groups of symmetries which appear frequently in nature, and which play an important role in many different branches of Mathematics.
This research project is concerned with the design, implementation and application of computational tools for the structural investigation of finite Coxeter groups and related algebraic objects, such as Iwahori-Hecke algebras, Solomon's descent algebras, hyperplane arrangements and Orlik-Solomon algebras. For further details see the preprints page of the principal investigator in this area.
The mapping class group of a topological surface is a central object of study in low-dimensional topology and geometric group theory, two closely interrelated areas of mathematics that have seen remarkable developments in recent years.
Mapping class groups appear in other areas of mathematics, such as algebraic geometry, dynamics, hyperbolic geometry, etc. Notably, the geometry of the mapping class group plays a central role in the proof of Thurston's Ending Lamination Conjecture, recently given by Brock-Canary-Minsky.
Our current research in this area centers around rigidity phenomena for mapping class groups, and the various spaces on which they act. For further details see the preprints page of the principal investigator in this area.
Formal Languages in Group Theory
Elements of finitely generated groups are naturally given as words (strings) over the generators and their inverses and it is an intersting question whether, and if so how, algebraic properties of the underlying group are related to language theoretic properties of certain sets of words. Some nice results have been obtained in recent years but many open problems remain. For more details see the preprints page of the principal investigator in this area.
Asymptotic Group Theory
This area of Group Theory studies infinte groups on a large scale by associating a counting function to them and then relating the growth rate of this function to the algebraic properties of the groups. Famous examples are Gromov's classification of groups of polynomial word growth and Lubotzky, Mann and Segals work on groups with polynomial subgroup growth. At the time of writing geodesic growth and representation growth are hot topics. For more details see the preprints page of the principal investigator in this area.
The theory of Vertex Operator Algebras (VOAs) is a relatively new branch of mathematics with connections to many areas of mathematics including Lie and Kac-Moody algebras, analytic number theory, Riemann surfaces and theoretical physics.
The VOA group is primarily working on two areas.
VOAs on Riemann surfaces
This project is concerned with understanding VOA partition functions and correlation functions on Riemann surfaces of genus two and higher. This involves applying new combinatorial techniques to VOAs and developing appropriate geometrical descriptions of sewn Riemann surfaces.
This project is concerned with exploring VOAs with exceptional symmetry groups (such as the Monster finite group) from a unified point of view based on properties of the Virasoro algebra.