Department of Mathematics and Computer Studieshttps://dspace.mic.ul.ie/handle/10395/342021-09-26T23:44:03Z2021-09-26T23:44:03ZThe stability space of the derived category of holomorphic triples and further investigationshttps://dspace.mic.ul.ie/handle/10395/29782021-04-15T02:00:36Z2021-04-14T00:00:00ZThe stability space of the derived category of holomorphic triples and further investigations
In this thesis we give a complete description of the Bridgeland stability space for the bounded derived category of holomorphic triples over a smooth projective curve of genus one as a connected, four dimensional complex manifold.
We will then prove a number of helpful facts that characterise the bounded derived category of holomorphic triples and will subsequently generalise some of the results on the stability space of the bounded derived category of holomorphic triples to that of holomorphic chains.
2021-04-14T00:00:00ZConstructing isostatic frameworks for the l1 and l infinity planehttps://dspace.mic.ul.ie/handle/10395/29642021-03-26T03:00:34Z2020-06-12T00:00:00ZConstructing isostatic frameworks for the l1 and l infinity plane
We use a new coloured multi-graph constructive method to prove that if the edge-set of a graph G = (V,E) has a partition into two spanning trees T1 and T2 then there is a map p : V → R2, p(v) = (p(v)1,p(v)2), such that |p(u)i −p(v)i| > |p(u)3−i−p(v)3−i| for every edge uv in Ti (i = 1,2). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the `1 or `∞-plane. We also show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces.
2020-06-12T00:00:00ZGraph rigidity for unitarily invariant matrix normshttps://dspace.mic.ul.ie/handle/10395/29632021-03-26T03:00:30Z2020-11-15T00:00:00ZGraph rigidity for unitarily invariant matrix norms
A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of -sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2 x 2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.
2020-11-15T00:00:00ZSymbol functions for symmetric frameworkshttps://dspace.mic.ul.ie/handle/10395/29622021-03-26T03:00:29Z2021-05-15T00:00:00ZSymbol functions for symmetric frameworks
We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.
2021-05-15T00:00:00Z