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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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

We review the current state of knowledge of phase separation and phase equilibria in porous materials. Our emphasis is on fundamental studies of simple fluids (composed of small, neutral molecules) and well-characterized materials. While theoretical and molecular simulation studies are stressed, we also survey experimental investigations that are fundamental in nature. Following a brief survey of the most useful theoretical and simulation methods, we describe the nature of gas‐liquid (capillary condensation), layering, liquid‐liquid and freezing/melting transitions. In each case studies for simple pore geometries, and also more complex ones where available, are discussed. While a reasonably good understanding is available for phase equilibria of pure adsorbates in simple pore geometries, there is a need to extend the models to more complex pore geometries that include effects of chemical and geometrical heterogeneity and connectivity. In addition, with the exception of liquid‐liquid equilibria, little work has been done so far on phase separation for mixtures in porous media.

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Phase separation in confined systems

We present a review of experimental, theoretical, and molecular simulation studies of confinement effects on freezing and melting We consider both simple and more complex adsorbates that are confined in various environments (slit or cylindrical pores and also disordered porous materials) The most commonly used molecular simulation, theoretical and experimental methods are first presented We also provide a brief description of the most widely used porous materials The current state of knowledge on the effects of confinement on structure and freezing temperature, and the appearance of new surface-driven and confinement-driven phases are then discussed We also address how confinement affects the glass transition

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Effects of confinement on freezing and melting.

A review of experimental work on freezing and melting in confinement is presented. A range of systems, from metal oxide gels to porous glasses to novel nanoporous materials, is discussed. Features such as melting-point depression, hysteresis between freezing and melting, modifications to bulk solid structure and solid-solid transitions are reviewed for substances such as helium, organic fluids, water and metals. Recent work with well characterized assemblies of cylindrical pores like MCM-41 and graphitic microfibres with slit pores has suggested that the macroscopic picture of melting and freezing breaks down in pores of molecular dimensions. Applications of the surface force apparatus to the study of freezing and melting phenomena in confinement are discussed in some detail. This instrument is unique in allowing the study of conditions in a single pore, without the complications of pore blockage and connectivity effects. The results have confirmed the classical picture of melting-point depression in larger pores, and allowed the direct observation of capillary condensation of solid from vapour. Other results include the measurement of solvation forces across apparently fluid films below the bulk melting point and a solid-like response to shear of films above the bulk melting point. These somewhat contradictory findings highlight the difficulty of using bulk concepts to define the phase state of a substance confined to nanoscale pores.

https://iopscience.iop.org/article/10.1088/0953-8984/13/11/201/pdf

Confinement effects on freezing and melting

Physicochemical characterization of colloidal drug delivery systems such as reverse micelles, vesicles, liquid crystals and nanoparticles for topical administration.

Magnetic resonance and calorimetric studies of 4'-n-pentyl-4-cyanobiphenyl confined to Vycor glass, a random network of pores with average diameter ∼70 A, are reported. Under such severely constrained conditions, the nematic-to-isotropic transition is replaced by a continuous evolution of orientational order in the pores. Our findings are explained by a Landau type of model where the liquid crystal is confined to independent pore segments, suggesting that the glassylike orientational order is to a large extent locally determined

Randomly constrained orientational order in porous glass.

Lyotropic chromonic liquid crystals (LCLCs) are formed by molecules with ionic groups at the periphery that associate into stacks through noncovalent self-assembly while in water. The very existence of the nematic (N) phase in the typical LCLC, the dye Sunset Yellow (SSY) is a puzzle, as the correlation length associated with the stacking, as measured in the X-ray experiments, is too short to explain the orientational order by the Onsager model. We propose that the aggregates can be more complex than simple rods and contain “stacking faults” such as junctions with a shift of neighboring molecules, 3-fold junctions, etc. We study how ionic additives, such as salts of different valency and pH-altering agents, alter the N phase of SSY purified by recrystallization. The additives induce two general trends: (a) stabilization of the N phase, caused by the mono and divalent salts (such as NaCl), and evidenced by the increase of the N-to-I transition temperature and the correlation length; (b) suppression of the ...

Self-Assembly of Lyotropic Chromonic Liquid Crystal Sunset Yellow and Effects of Ionic Additives

The microscopic properties of the inner surface of the cavities of Anopore and Nuclepore membranes are investigated with scanning electron microscopy (SEM), nitrogen adsorption isotherms, and deuterium nuclear magnetic resonance (2H‐NMR). Useful information about the cavity orientation and morphology, and internal surface area is obtained. Analysis of SEM photographs yields estimates of the surface area and the porosity of these membranes which complements the adsorption results. The orientation of liquid crystals permeated in the cavities of Anopore and Nuclepore membranes is probed with the 2H‐NMR technique. It is found that the orientation of the liquid crystal molecules is governed by the confining volume and surface conditions. The 2H‐NMR spectral line shape of the confined liquid crystal also provides information on the substrate morphology and roughness that is consistent with SEM and adsorption experiments.

Characterization of the cylindrical cavities of Anopore and Nuclepore membranes

Using an ac calorimetry technique we measured the specific heat at the nematic-isotropic, smectic-A--nematic, and smectic-A--isotropic phase transitions, for the alkylcyanobiphenyl liquid crystal nCB series confined to the 0.2-\ensuremath{\mu}m-diam pores of Anopore membranes. Studies were also performed as a function of nematic director alignment within the pores. Novel effects regarding specific-heat amplitudes, transition-temperature shifts, and broadening and rounding of phase transitions are found. They are strongly dependent on the director orientation and on the order of the phase transition.

Calorimetric study of phase transitions in confined liquid crystals

$^{2}\mathrm{H}$ NMR of a liquid crystal confined to submicron cylindrical cavities, in a 17 K interval above the nematic-isotropic transition, reveals a weakly orientationally ordered molecular layer at the cavity wall governed by local interactions. The order (${\mathit{S}}_{0}$=0.02) is temperature independent so that a partial wetting is realized with no prewetting transition expected. A molecular exchange rate between the surface layer and the bulk is measured to be \ensuremath{\sim}${10}^{3}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$. The coherence length measuring the order penetrating further into the cavity is described by \ensuremath{\alpha}=0.4 and ${\ensuremath{\xi}}_{0}$=0.6 nm.

Daniele Finotello

Papers

Randomly constrained orientational order in porous glass.

Self-Assembly of Lyotropic Chromonic Liquid Crystal Sunset Yellow and Effects of Ionic Additives

Characterization of the cylindrical cavities of Anopore and Nuclepore membranes

Calorimetric study of phase transitions in confined liquid crystals

Ordering and self-diffusion in the first molecular layer at a liquid-crystal-polymer interface.