In the context of 2 + 1â€“dimensional quantum gravity with negative cosmological constant and topology RÃ—T 2, constant matrixâ€“valued connections generate a qâ€“deformed representation of the fundamentalâ€¦ (More)

We describe an approach to the quantisation of (2+1)â€“dimensional gravity with topology IRÃ—T 2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a qâ€“commutationâ€¦ (More)

The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be described by ordered pairs of commuting SL(2, R) matrices modulo simultaneous conjugation by SL(2, R)â€¦ (More)

We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT)â€¦ (More)

The notion of quantummatrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrixâ€¦ (More)

We describe the geometrical ladder of equations for Abelian bundles and gerbes, as well as higher generalisations, in terms of the cohomology of an operator that combines de Rham and ÄŒech cohomology.

We present a general framework for TQFT and related constructions using the language of monoidal categories. We construct a topological category C and an algebraic category D, both monoidal, and aâ€¦ (More)

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are linkâ€¦ (More)

We give an introduction for the non-expert to TQFT (Topological Quantum Field Theory), focussing especially on its role in algebraic topology. We compare the Atiyah axioms for TQFT with the Eilenbergâ€¦ (More)

In the context of quantum gravity for spacetimes of dimension 2 + 1, we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linearâ€¦ (More)