Derived autoequivalences of generalised Kummer varieties

In this article, we construct new derived autoequivalences of generalised Kummer varieties. Together with Huybrechts-Thomas twists around $\mathbb{P}^n$-objects, these are the only known examples of such symmetries.Comment: Final version. To appear in Math. Res. Let

A note on spherical functors

We provide a new and very short proof of the fact that a spherical functor between certain triangulated categories induces an autoequivalence.Comment: Final version. Accepted for publication in Bull. Lond. Math. So

Birational Geometry of Singular Moduli Spaces of O'Grady Type

Following Bayer and Macr\`{i}, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macr\`{i} map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland semistable objects on $X$. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of $X$ and the N\'{e}ron-Severi lattice of $M$ which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of $M$ is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.Comment: Final versio

The physiological significance of insemination in programming pregnancy outcome

The cellular and molecular environment of the uterus during the pre - and peri - implantation period of early pregnancy is critical for implantation success and optimal fetal and placental development. Perturbations to this environment not only have consequences for the success of pregnancy and neonatal health and viability, but can also drive adverse health outcomes in the offspring after birth, particularly the development of metabolic disorders such as obesity, hypertension and insulin resistance. The influence of seminal plasma on the cytokine and immune uterine environment has been previously well characterised in mice, however the effects of disruption in uterine seminal plasma exposure for pregnancy outcome have not been investigated. The studies described in this thesis employed the use of surgical seminal vesicle ablation in males and embryo transfer experiments to investigate the physiological significance of uterine seminal plasma exposure on programming fetal and neonatal outcomes, and growth and metabolic status in adult offspring. We demonstrate that in the absence of seminal plasma, oocyte fertilisation and embryo implantation are reduced, showing that seminal plasma acts primarily to facilitate fertilisation, possibly by promoting sperm transport and survival in the reproductive tract. In addition we show that pregnancies initiated in the absence of seminal plasma give rise to offspring which display accelerated growth after birth and increased adiposity in adulthood, compared to those developed in a tract exposed to seminal plasma at the time of conception. Offspring conceived in the absence of seminal plasma also displayed alterations in serum leptin and adiponectin content, similar to those known to be associated with obesity in the mouse. Using embryo transfer experiments, we showed that some, but not all aspects of the perturbed postnatal development are recapitulated when embryos fertilised in the presence of what semen are transferred to a recipient tract which has not been exposed to seminal plasma. More severe perturbations were seen in 2 - cell transfer than in blastocyst transfer experiment. Additionally, there was a significant effect of the embryo transfer procedure, irrespective of seminal plasma exposure, on fetal and postnatal development that confounded interpretation of these experiments. In addition, we investigated the potential mechanisms by which the influence of seminal plasma is exerted. Mediators of pre - implantation embryo development, implantation and the modulation of the maternal immune response to pregnancy were all assessed for regulation by seminal plasma using QRT - PCR. It was demonstrated that seminal plasma exposure induces the up - regulation of key embryotrophic factors, LIF, GM - CSF and IL - 6, in the oviduct following insemination. Factors important in tissue remodelling required for implantation and angiogenesis, MMP - 2, MMP - 3 and VEGF - C, were also shown to be increased at the time of implantation after seminal plasma exposure. Additionally the generation of T - regulatory cells in uterine tissues, demonstrated by the up - regulation of the transcription factor FOXp3 was shown to be dependent on semen exposure. The influence of seminal plasma on embryonic development, implantation and modulation of the maternal immune response to pregnancy may therefore be mechanisms which contribute to the adverse outcomes seen in pregnancies initiated in the absence of seminal plasma. Together these experiments show a role for seminal plasma signalling at the time of insemination in influencing the pre - implantation embryo to program later fetal and neonatal development, thereby impacting on the metabolic health of offspring. We conclude that seminal plasma is not simply a transport medium for sperm, but acts also as a key regulator of a female tract environment providing optimal support for the developing embryo.Thesis (Ph.D.)--School of Paediatrics and Reproductive Health, 2006

Frobenius and spherical codomains and neighbourhoods

Given an exact functor between triangulated categories which admits both adjoints and whose cotwist is either zero or an autoequivalence, we show how to associate a unique full triangulated subcategory of the codomain on which the functor becomes either Frobenius or spherical, respectively. We illustrate our construction with examples coming from projective bundles and smooth blowups. This work generalises results about spherical subcategories obtained by Martin Kalck, David Ploog and the first author.Comment: 40 pages, same content as published versio

A note on spherical functors

We provide a new and very short proof of the fact that a spherical functor between certain triangulated categories induces an auto‐equivalence

Moduli of Bridgeland-Stable objects

In this thesis we investigate wall-crossing phenomena in the stability manifold of an irreducible principally polarized abelian surface for objects with the same invariants as (twists of) ideal sheaves of points. In particular, we construct a sequence of fine moduli spaces which are related by Mukai flops and observe that the stability of these objects is completely determined by the configuration of points. Finally, we use Fourier-Mukai theory to show that these moduli are projective

Rank One Bridgeland Stable Moduli Spaces on A Principally Polarized Abelian Surface

We compute moduli spaces of Bridgeland stable objects on an irreducible principally polarized complex abelian surface corresponding to twisted ideal sheaves. We use Fourier-Mukai techniques to extend the ideas of Arcara and Bertram to express wall-crossings as Mukai flops and show that the moduli spaces are projective.Comment: 17 pages, 3 figure