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.

Catalytic Processes Mediated by Metal−Organic Frameworks : Reactivity and Mechanistic Studies

The present thesis describes the development of heterogeneous catalytic methodologies using metal−organic frameworks (MOFs) as porous matrices for supporting transition metal catalysts. A wide spectrum of chemical reactions is covered. Following the introductory section (Chapter 1), the results are divided between one descriptive part (Chapter 2) and four experimental parts (Chapters 3–6). Chapter 2 provides a detailed account of MOFs and their role in heterogeneous catalysis. Specific synthesis methods and characterization techniques that may be unfamiliar to organic chemists are illustrated based on examples from this work. Pd-catalyzed heterogeneous C−C coupling and C−H functionalization reactions are studied in Chapter 3, with focus on their practical utility. A vast functional group tolerance is reported, allowing access to substrates of relevance for the pharmaceutical industry. Issues concerning the recyclability of MOF-supported catalysts, leaching and operation under continuous flow are discussed in detail. The following chapter explores puzzling questions regarding the nature of the catalytically active species and the pathways of deactivation for Pd@MOF catalysts. These questions are addressed through detailed mechanistic investigations which include in situ XRD and XAS data acquisition. For this purpose a custom reaction cell is also described in Chapter 4. The scope of Pd@MOF-catalyzed reactions is expanded in Chapter 5. A strategy for boosting the thermal and chemical robustness of MOF crystals is presented. Pd@MOF catalysts are coated with a protecting SiO2 layer, which improves their mechanical properties without impeding diffusion. The resulting nanocomposite is better suited to withstand the harsh conditions of aerobic oxidation reactions. In this chapter, the influence of the nanoparticles’ geometry over the catalyst’s selectivity is also investigated. While Chapters 3–5 dealt with Pd-catalyzed processes, Chapter 6 introduces hybrid materials based on first-row transition metals. Their reactivity is explored towards light-driven water splitting. The heterogenization process leads to stabilized active sites, facilitating the spectroscopic probing of intermediates in the catalytic cycle.