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.

Anti-parasitic and anti-viral immune responses in insects

Insects encounter many microorganisms in nature and to survive they have developed counter measures against the invading pathogens. In Drosophila melanogaster research on insect immunity has mainly been focused on infections by bacteria and fungi. We have explored the immune response against natural infections of the parasite Octosporea muscaedomesticae and the Drosophila C virus as compared to natural infections of bacteria and fungi. By using Affymetrix Drosophila GeneChips, we were able to obtain 48 genes uniquely induced after parasitic infection. It was also clearly shown that natural infections led to different results than when injecting the pathogens. In order to search for the ultimate role of the lepidopteran protein hemolin, we used RNA interference (RNAi). We could show that injection of double stranded RNA (dsRNA) of Hemolin in pupae of Hyalophora cecropia led to embryonic malformation and lethality and that there was a sex specific difference. We continued the RNAi investigation of hemolin in another lepidopteran species, Antheraea pernyi, and discovered that hemolin was induced by dsRNA per se. A similar induction of hemolin was seen after infection with baculovirus and we therefore performed in vivo experiments on baculovirus infected pupae. We could show that a low dose of dsHemolin prolonged the period before the A. pernyi pupae showed any symptoms of infection, while a high dose led to a more rapid onset of symptoms. By performing in silico analysis of the hemolin sequence from A. pernyi in comparison with other Hemolin sequences, it was possible to select a number of sites that either by being strongly conserved or variable could be important targets for future studies of hemolin function.