Level structures on the Weierstrass family of cubics

Let W -> A^2 be the universal Weierstrass family of cubic curves over C. For each N >= 2, we construct surfaces parametrizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of A^2. Since W -> A^2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0,0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S^3 with monodromy in SL_2(Z/N).Comment: LaTeX, 12 pages; added section giving a topological interpretation of the result

Geometric properties of curves defined over number fields

The article contains a detailed proof of the famous Belyi theorem on geometry of complex algebraic curves defined over number fields. It also includes the discussion of several constructions and conjectures inspired by Belyi’s result which where brought up by the first author during his colloquium talks at different universities within the period from 1979 to 1984

Modular Invariance and the Odderon

We identify a new symmetry for the equations governing odderon amplitudes, corresponding in the Regge limit of QCD to the exchange of 3 reggeized gluons. The symmetry is a modular invariance with respect to the unique normal subgroup of sl(2,Z) {\,} of index 2. This leads to a natural description of the Hamiltonian and conservation-law operators as acting on the moduli space of elliptic curves with a fixed ``sign'': elliptic curves are identified if they can be transformed into each other by an {\em even} number of Dehn twists.Comment: 9 pages, LaTeX, uses amssym.def for \Bbb 'blackboard math' font

'Nobody really understands' - dementia and the world of family carers.

This thesis provides a comprehensive account of the situation of informal unpaid carers of older people with dementia. Dementia is characterised by a progressive degeneration of intellectual ability, leading to impairment of memory, judgement, and perception, and personality changes. There is no cure, and death typically occurs as a result of pneumonia, strokes, or falls, after a duration of several years. The majority of demented older people are cared for at home often at considerable emotional, financial, social, and physical costs to the carer. This exploratory research highlights the carer’s situation both within the informal network of care as well as in relation to formal care provision. Qualitative interviews were conducted with spouse carers as well as adult children, recruited through carer support groups in Sheffield. The thesis presents the carers’ views and interpretations of their situation and the findings reveal far-reaching misunderstandings and a mis-match between the views of carers and service providers. Overall, an inadequate understanding of the role of carers and of their needs has been identified. Current service provision, which is mainly based on instrumental support such as day care and respite care, has been found to be inappropriate for the majority of carers and their demented relative. The thesis identifies three main areas where reform is needed and suggests improvements for the recognition of dementia, the management of care, and the empowerment of carers

Kummer surfaces associated with Seiberg-Witten curves

By carrying out a rational transformation on the base curve $\mathbb{CP}^1$ of the Seiberg-Witten curve for $\mathcal{N}=2$ supersymmetric pure $\mathrm{SU}(2)$-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure $\mathrm{SU}(2)$-gauge theory to the one for $\mathrm{SU}(2)$-gauge theory with $N_f=2$ massless hypermultiplets extends to define a Nikulin involution on each K3 surface in the family. We show that the desingularization of the quotient of the K3 surface by the involution is isomorphic to a Kummer surface of the Jacobian variety of a curve of genus two. We then derive a relation between the Yukawa coupling associated with the elliptic K3 surface and the Yukawa coupling of pure $\mathrm{SU}(2)$-gauge theory.Comment: 26 pages, LaTe

Homotopy Theory of Strong and Weak Topological Insulators

We use homotopy theory to extend the notion of strong and weak topological insulators to the non-stable regime (low numbers of occupied/empty energy bands). We show that for strong topological insulators in d spatial dimensions to be "truly d-dimensional", i.e. not realizable by stacking lower-dimensional insulators, a more restrictive definition of "strong" is required. However, this does not exclude weak topological insulators from being "truly d-dimensional", which we demonstrate by an example. Additionally, we prove some useful technical results, including the homotopy theoretic derivation of the factorization of invariants over the torus into invariants over spheres in the stable regime, as well as the rigorous justification of replacing $T^d$ by $S^d$ and $T^{d_k}\times S^{d_x}$ by $S^{d_k+d_x}$ as is common in the current literature.Comment: 11 pages, 3 figure

Flip Distance Between Triangulations of a Planar Point Set is APX-Hard

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set $S$ in the Euclidean plane and two triangulations $T_1$ and $T_2$ of $S$, it is an APX-hard problem to minimize the number of edge flips to transform $T_1$ to $T_2$.Comment: A previous version only showed NP-completeness of the corresponding decision problem. The current version is the one of the accepted manuscrip