From Cracked Polytopes to Fano Threefolds

We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes - polytopes whose intersection with a complete fan forms a set of unimodular polytopes - using Laurent inversion; a method developed jointly with Coates-Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen-Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as 'pieces' of a cracked polytope.Comment: New introduction and section on the connection with the Gross-Siebert program. 46 page

Timing analysis of low-energy gamma ray emission from galactic compact objects using the Gamma Ray Observatory

The principal goal of our phase 1 investigation was the development of techniques and data analysis tools for pulsar searches and timing. After the launch of the Compton Observatory, we received from the Burst and Transient Source Experiment (BATSE) team one day of discriminator large area (DISCLA) data for use in the development and testing of data analysis techniques. Using this first day of data for testing and optimizing our timing tools we detected four x-ray binary pulsars, Vela X-1, Cen X-3, 4U 0115+63, and GX 301-2. Subsequently, we received four more days of data, allowing us to test our timing tools with data from a variety of days. In summary, using the tools we developed based on the first day of data that we received, we have detected 8 pulsars in 5 days of data, or roughly one quarter of the approximately 30 known x-ray binary pulsars. In addition to the pulsars listed above, we detected GX 1+4, 4U 1626-67, OAO 1657-415, and Her X-1. Many of the data analysis tools that we developed have been ported to MSFC and are being used for the analysis of BATSE data. This appendix describes some of the timing tools and presents preliminary pulse period and phase profile results

Minimality and mutation-equivalence of polygons

We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine the mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface; or (ii) a surface with only singularities of type 1/3(1,1).Comment: 29 page

Applications of mirror symmetry to the classification of Fano varieties

In this dissertation we discuss two new constructions of Fano varieties, each directly inspired by ideas in Mirror Symmetry. The first recasts the Fanosearch program (Coates--Corti--Kasprzyk et al.) for surfaces in terms of a construction related to the SYZ conjecture. In particular we construct Q-Gorenstein smoothings of toric varieties via an application of the Gross-Siebert algorithm to certain affine manifolds. We recover the theory of combinatorial mutation, which plays a central role in the Fanosearch program, from these affine manifolds. Combining this construction and the work of Gross--Hacking--Keel on log Calabi--Yau surfaces we produce a cluster structure on the mirror to a log del Pezzo surface proposed by Coates--Corti--et al. We exploit the cluster structure, and the connection to toric degenerations, to prove two classification results for Fano polygons. This cluster variety is equipped with a superpotential defined on each chart by a so-called maximally mutable Laurent polynomial. We study an enumerative interpretation of this superpotential in terms of tropical disc counting in the example of the projective plane (with a general choice of boundary divisor). In the second part we develop a new construction of Fano toric complete intersections in higher dimensions. We first consider the problem of finding torus charts on the Hori--Vafa/Givental model, adapting the approach taken by Przyjalkowski. We exploit this to identify 527 new families of four-dimensional Fano manifolds. We then develop an inverse algorithm, Laurent Inversion, which decorates a Fano polytope P with additional information used to construct a candidate ambient space for a complete intersection model of the toric variety defined by P. Moving in the linear system defining this complete intersection allows us to construct new models of known Fano manifolds, and also to construct new examples of Fano manifolds from conjectured mirror Laurent polynomials. We use this algorithm to produce families simultaneously realising certain collections of 'commuting' mutations, extending the connection between polytope mutation and deformations of toric varieties.Open Acces