Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds

We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev-Nikulin mirror symmetry for K3 surfaces, and an explicit family of threefolds that are not realized as complete intersections in toric varieties.Comment: v2: Section 5 has been completely rewritten to accommodate results removed from Section 5 of arxiv:1501.04019. v3: Final version, to appear in String-Math 2015, forthcoming volume in the Proceedings of Symposia in Pure Mathematics serie

Regional Resilience: Are recessionary shocks persistent or transitory?

The response by regional and national economies to exogenous impulses has a well-established literature in both spatial econometrics and in mainstream econometrics and is of considerable importance given the post-2008 economic crisis, which is characterised by a period of severe global instability resulting from unprecedented economic shocks. Martin et al. (2016) note that in economic geography resilience describes regions’ reactions to, and recovery from, negative economic shocks, based on a concept which has been widely used in the engineering and ecological sciences and which has been increasingly adopted in economic geography. This PhD provides an empirical analysis of resilience at the national, regional, and individual level. Four empirical Chapters are presented which feature econometric analysis in the form of vector error correction (VEC) models, dynamic spatial panel models, and pooled crosssectional models. The national analysis focuses on European counties and the US and analyses the impact of shocks from within the EU and from the US on each country. The second empirical Chapter focuses on US metropolitan statistics areas and analyses the impact of industry structure on the resilience of US metropolitan areas. The third empirical Chapter focuses on the resilience of wages in the US to the global economic crisis. The final empirical Chapter analyses the impact of the crisis on individual’s employment outcomes in select European countries. The results of the analysis clearly indicate that industry structure plays an important role in explaining the resilience of nations, regional, and individuals (who reside within broader regions). The findings suggest that diversity of economic structure and structural change can result in more resilient regions. At the individual level there is significant evidence that education plays a critical role in explaining the resilience of individuals’ wages and employment outcomes

Families of lattice polarized K3 surfaces with monodromy

We extend the notion of lattice polarization for K3 surfaces to families over a (not necessarily simply connected) base, in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fibrewise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces

Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces

We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory.Comment: v2: Significant changes at the request of the referee. Section 3 has been rearranged to accommodate a revised proof of Proposition 3.5 (formerly 3.2). Section 5 has been removed completely, it will instead appear as part of Section 5 in arxiv:1601.0811

Are few-mode fibres a practical solution to the capacity crunch?

In this paper, we compare the nonlinear Shannon capacity of few-mode fibre systems operating with spatial-temporal digital signal processing to the nonlinear Shannon capacity of single-mode fibre systems operating with spectral-temporal digital signal processing. Combining these results with estimates of digital signal processing complexity for each option offers valuable insights to system designers

The costs of inequality: whole-population modelling study of lifetime inpatient hospital costs in the English National Health Service by level of neighbourhood deprivation : Whole-population modelling study of lifetime inpatient hospital costs in the English National Health Service by level of neighbourhood deprivation

BACKGROUND: There are substantial socioeconomic inequalities in both life expectancy and healthcare use in England. In this study, we describe how these two sets of inequalities interact by estimating the social gradient in hospital costs across the life course. METHODS: Hospital episode statistics, population and index of multiple deprivation data were combined at lower-layer super output area level to estimate inpatient hospital costs for 2011/2012 by age, sex and deprivation quintile. Survival curves were estimated for each of the deprivation groups and used to estimate expected annual costs and cumulative lifetime costs. RESULTS: A steep social gradient was observed in overall inpatient hospital admissions, with rates ranging from 31 298/100 000 population in the most affluent fifth of areas to 43 385 in the most deprived fifth. This gradient was steeper for emergency than for elective admissions. The total cost associated with this inequality in 2011/2012 was £4.8 billion. A social gradient was also observed in the modelled lifetime costs where the lower life expectancy was not sufficient to outweigh the higher average costs in the more deprived populations. Lifetime costs for women were 14% greater than for men, due to higher costs in the reproductive years and greater life expectancy. CONCLUSIONS: Socioeconomic inequalities result in increased morbidity and decreased life expectancy. Interventions to reduce inequality and improve health in more deprived neighbourhoods have the potential to save money for health systems not only within years but across peoples' entire lifetimes, despite increased costs due to longer life expectancies

Hodge Numbers from Picard-Fuchs Equations

Given a variation of Hodge structure over $\mathbb{P}^1$ with Hodge numbers $(1,1,\dots,1)$, we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-M\"oller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds