Final Report Summary - CAT REPTH (Categorification in Representation Theory)
Executive Summary:
Representation theory is a part of pure mathematics which aims to understand structures arising from symmetries on certain spaces. It has long been known that many two-dimensional geometric phenomena have algebraic explanations via the subject of representation theory. In the last fifteen years it has been conjectured that analogous explanations for problems in three-dimensional geometry should raise from introducing so-called higher symmetries, giving rise to the subject of 2-representation theory, or categorification.
The project has investigated to instances of higher representation theory. The first, building on the fellow's expertise acquired during her Marie-Curie Intra-European Fellowship, investigated, in collaboration with Will Turner (University of Aberdeen), the use of higher representation theoretic methods in understanding all symmetries of the plane. In particular it gave detailed homological information in terms of computing invariants of of these symmetries. The second instance built on the fellow's research experience prior to her Marie-Curie Intra-European Fellowship, and investigated certain prominent algebras in the theory of categorification, the so-called Khovanov-Lauda-Rouquier algebras which have close connections to affine Hecke algebras. In collaboration with Alexander Kleshchev and Joseph Loubert (both University of Oregon) respectively Jérémie Guilhot (then University of East Anglia, now Université de Tours), affine cellularity of Khovanov-Lauda-Rouquier algebras in finite type A respectively of affine Hecke algebras of rank two was proved, thus giving important information about their homological structure. Furthermore the fellow, in joint work with Volodymyr Mazorchuk (University of Uppsala), extended the notion of cellularity to abstract 2-representations of fiat 2-categories, thus initiating a general study of 2-represenations of 2-categories, which had up to date only been studied in examples.
Contact details:
Dr Vanessa Miemietz
School of Mathematics
University of East Anglia
Norwich NR4 7TJ
email: v.miemietz@uea.ac.uk
Website:
http://www.uea.ac.uk/~byr09xgu/