This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

Band bending in semiconductors: chemical and physical consequences at surfaces and interfaces.

Oxygen vacancies in transition metal and rare earth oxides: Current state of understanding and remaining challenges

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Studies in Surface Science and Catalysis

http://w0.rz-berlin.mpg.de/hjfdb/pdf/260e.pdf

Metal deposits on well-ordered oxide films

In this work, the geometric and electronic structure of N species in N-doped carbon nanotubes (NCNTs) is derived by X-ray photoemission (XPS) and absorption spectroscopy (NEXAFS) of the N 1s core excitation. Substitutional N species in pyridine-like configuration and another form of N with higher thermal stability are found in NCNTs. The structural configuration of the high thermally stable N species, in the literature often referred to as graphitic N, is assessed in this work by a combined theoretical and experimental study as a 3-fold substitutional N species in an NCNT basic structural unit (BSU). Furthermore, the nature of the interaction of those N species with a Pd metal center immobilized onto NCNTs is of σ-type donation from the filled π-orbital of the N atom to the empty d-orbital of the Pd atom and a π back-donation from the filled Pd atomic d-orbital to the π* antibonding orbital of the N atom. We have found that the interaction of pyridine N with Pd is characterized by a charge transfer typica...

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Nature of the N-Pd interaction in nitrogen-doped carbon nanotube catalysts

We report an analysis of the interaction of the closed shell, lone pair ligands CO, NH3 and PF3 with a Cu surface. The surface is represented by one and by five atom clusters. The interaction is analyzed with the constrained space orbital variation, CSOV, method which leads to a division of the interaction into several steps and permits an unambiguous assignment for the energetic importance of each step. The importance of the σ repulsion between the ligand lone pair and the metal is demonstrated and the significance of the dipole moment of the ligand is shown. The charge rearrangements are divided into intra‐unit polarization and inter‐unit donation between the units. Formation of covalent chemical bonds is not particularly important for NH3; the bonding can be viewed as electrostatic and arising because of the large NH3 dipole moment. For CO and PF3, covalent bonds are formed by metal to ligand π donation; for CO, the relative importance of dπ and valence 4pπ for this bonding is examined.

On the nature of the bonding of lone pair ligands to a transition metal

Size dependence of surface cluster models: CO adsorbed on Cu(100)

Density-functional theory (DFT) studies are performed to examine geometric and electronic properties of orthorhombic bulk ${\mathrm{V}}_{2}{\mathrm{O}}_{5}$ as well as of its (010) oriented surface. Electronic states, total energies, as well as atom forces (used to obtain equilibrium geometries) are computed with the ab initio full-potential linear augmented plane wave method. The ${\mathrm{V}}_{2}{\mathrm{O}}_{5}(010)$ surface is modeled by periodic single layers in a repeated slab geometry, which is justified by the weak electronic interlayer coupling found in the bulk calculations. The electronic structure of the ${\mathrm{V}}_{2}{\mathrm{O}}_{5}(010)$ single-layer slabs, represented by their valence densities of states (DOS) and its atom contributions, is compared with results of bulk ${\mathrm{V}}_{2}{\mathrm{O}}_{5}$ and with previous results obtained by DFT surface cluster studies. The comparison yields good qualitative agreement between the different approaches, which confirms the local nature of interatomic binding in ${\mathrm{V}}_{2}{\mathrm{O}}_{5}.$ Further, the computed valence DOS is used to interpret recent experimental results from photoemission on ${\mathrm{V}}_{2}{\mathrm{O}}_{5}(010),$ which suggests that differently coordinated oxygen sites at the surface can be identified in the spectrum. Thus, ${\mathrm{V}}_{2}{\mathrm{O}}_{5}(010)$ photoemission spectra may be used to monitor the participation of oxygen ions in respective surface reactions.

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Geometric and electronic structure of vanadium pentoxide: A density functional bulk and surface study

The bonding and the temperature-driven metalation of 2H-tetraphenylporphyrin (2H-TPP) on the Cu(111) surface under ultrahigh vacuum conditions were investigated by a combination of x-ray photoelectron spectroscopy (XPS) and near-edge x-ray absorption fine structure (NEXAFS) spectroscopy with density functional theory calculations. Thin films were prepared by organic molecular beam epitaxy and subsequent annealing. Our systematic study provides an understanding of the changes of the spectroscopic signature during adsorption and metalation. Specifically, we achieved a detailed peak assignment of the 2H-TPP multilayer data of the C1s and the N1s region. After annealing to 420 K both XPS and NEXAFS show the signatures of a metalloporphyrin, which indicates self-metalation at the porphyrin-substrate interface, resulting in Cu-TPP. Furthermore, for 2H-TPP monolayer samples we show how the strong influence of the copper surface is reflected in the spectroscopic signatures. Adsorption results in a strongly deformed macrocycle and a quenching of the first NEXAFS resonance in the nitrogen edge suggesting electron transfer into the LUMO. For Cu-TPP the spectroscopic data indicate a reduced interaction of first-layer molecules with the substrate as demonstrated by the relaxed macrocycle geometry.

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Klaus Hermann

Papers

Nature of the N-Pd interaction in nitrogen-doped carbon nanotube catalysts

On the nature of the bonding of lone pair ligands to a transition metal

Size dependence of surface cluster models: CO adsorbed on Cu(100)

Geometric and electronic structure of vanadium pentoxide: A density functional bulk and surface study

Self-metalation of 2H-tetraphenylporphyrin on Cu(111): An x-ray spectroscopy study