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.

Ring Out Your Dead : Distribution, form, and function of iron amulets in the late Iron Age grave fields of Lovö

The purpose of this study is to analyze the distribution, forms, and function(s) of iron amulets deposited in the late Iron Age gravefields of Lovö, with the goal of ascertaining how (and so far as possible why) these objects were utilized in rituals carried out during and after burials. Particular emphasis is given to re-interpreting the largest group of iron amulets, the iron amulet rings, in a more relational and practice-focused way than has heretofore been attempted. By framing burial analyses, questions of typology, and evidence of ritualized actions in comparison with what is known of other cult sites in Mälardalen specifically– and theorized about the cognitive landscape(s) of late Iron Age Scandinavia generally– a picture of iron amulets as inscribed objects made to act as catalytic, protective, and mediating agents is brought to light.