@manuascript {31458, title = {(Co)-Higgs Bundles on Non-K{\"a}hler Elliptic Surfaces}, year = {Submitted}, abstract = {In this paper, we study Higgs and co-Higgs bundles on non-K{\"a}hler elliptic surfaces. We show, in particular, that non-trivial stable Higgs bundles only exist when the base of the elliptic fibration has genus at least two and use this existence result to give explicit topological conditions ensuring the smoothness of moduli spaces of stable rank-2 sheaves on such surfaces. We also show that non-trivial stable co-Higgs bundles only exist when the base of the elliptic fibration has genus 0, in which case the non-K{\"a}hler elliptic surface is a Hopf surface. We then given a complete description of non-trivial co-Higgs bundles in the rank 2 case; these non-trivial rank-2 co-Higgs bundles are examples of non-trivial holomorphic Poisson structures on\ ℙ1-bundles over Hopf surfaces.}, url = {https://arxiv.org/abs/2210.09839}, author = {Eric Boulter and Ruxandra Moraru} } @unpublished {30117, title = {Moduli Spaces of Stably Irreducible Sheaves on Kodaira Surfaces}, year = {Submitted}, abstract = {Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic symplectic manifolds by deformation type. This paper studies a natural Lagrangian fibration on these moduli spaces to determine that they are not K{\"a}hler or simply connected, ruling out most of the known deformation types of holomorphic symplectic manifolds.}, url = {https://arxiv.org/abs/2112.00058}, author = {Eric Boulter} } @mastersthesis {32138, title = {Vector Bundles on Non-K{\"a}hler Elliptic Surfaces}, year = {2023}, type = {PhD thesis}, abstract = {

This thesis studies two problems relating to moduli spaces of vector bundles on non-K{\"a}hler elliptic surfaces. The first project involves the holomorphic symplectic structure on smooth and compact moduli spaces of sheaves on Kodaira surfaces. We show that these moduli spaces are neither K{\"a}hler nor simply connected. Comparing \ to other known examples of compact holomorphic symplectic manifolds, this shows that if the moduli spaces are deformation equivalent to a known example, then they are Douady spaces of points on a Kodaira surface.\ 

The second problem deals with the interplay between singularities of moduli spaces of rank-2 vector bundles and existence of stable Vafa--Witten bundles on non-K{\"a}hler elliptic surfaces. By constructing a Vafa--Witten bundle in each filtrable Chern class of the elliptic surface when the base has genus $g\geq 2$, we show that such a moduli space is smooth as a ringed space if and only if every bundle in the moduli space is irreducible.

}, author = {Eric Boulter} } @mastersthesis {31473, title = {Born Metrics on Compact Complex Surfaces}, year = {2018}, type = {Masters research paper}, author = {Eric Boulter} } @article {15902, title = {The N-body problem in spaces with uniformly varying curvature}, journal = {Journal of Mathematical Physics}, volume = {58}, year = {2017}, abstract = {

We generalize the curved\ N-body problem to spheres and hyperbolic spheres whose curvature\ κ\ varies in time. Unlike in the particular case when the curvature is constant, the\ equations of motion\ are non-autonomous. We first briefly consider the analog of the Kepler problem and then investigate homographic orbits for any number of bodies, proving the existence of several such classes of solutions on spheres. Allowing the curvature to vary in time offers some insight into the effect of an expanding universe in the context of the curved\ N-body problem, when\ κ\ satisfies\ Hubble{\textquoteright}s\ law. The study of these\ equations\ also opens the possibility of finding new connections between classical mechanics and general relativity.

}, url = {https://doi.org/10.1063/1.4983681}, author = {Eric Boulter and Florin Diacu and Shuqiang Zhu} }