Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

Allegory, It Happens: A Multi-Perspective Case Study of The Lord of the Rings

Allegory is not obsolete as Samuel Coleridge and Johann Wolfgang von Goethe have claimed. It is alive and well and has transformed from a restrictive concept to a concept that is flexible and can form to meet the needs of the author or reader. The most efficient way to evidence this is by making a case study of it with a suitable work that will allow us to perceive its plasticity. This essay uses J.R.R. Tolkien’s The Lord of the Rings as a multi-perspective case study of the concept of allegory; the size and complexity of the narrative make it a suitable choice. My aim is to illustrate the plasticity of allegory as a concept and illuminate some of the possibilities and pitfalls of allegory and allegoresis. As to whether The Lord of the Rings can be treated as an allegory, it will be examined from three different perspectives: as a purely writerly process, a middle ground of writer and reader and as a purely readerly process. The Lord of the Rings will then be compared to a series of concepts of allegorical theory such as Plato’s classical “The Ring of Gyges”, William Langland’s classic The Vision of William Concerning Piers the Plowman and contemporary allegories of racism and homoeroticism to demonstrate just how adaptable this concept is. The position of this essay is that the concept of allegory has changed over time since its conception and become more malleable. This poses certain dangers as allegory has become an all-round tool for anyone to do anything that has few limitations and has lost its early rigid form and now favours an almost anything goes approach.