Topological materials have become the focus of intense research in recent years, since they exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This review provides a pedagogical introduction into the field of topological quantum matter with an emphasis on classification schemes. Both fully gapped and gapless topological materials and their classification in terms of nonspatial symmetries, such as time reversal, as well as spatial symmetries, such as reflection, are considered. Furthermore, the classification of gapless modes localized on topological defects is surveyed. The classification of these systems is discussed by use of homotopy groups, Clifford algebras, K theory, and nonlinear sigma models describing the Anderson (de) localization at the surface or inside a defect of the material. Theoretical model systems and their topological invariants are reviewed together with recent experimental results in order to provide a unified and comprehensive perspective of the field. While the bulk of this article is concerned with the topological properties of noninteracting or mean-field Hamiltonians, a brief overview of recent results and open questions concerning the topological classifications of interacting systems is also provided.

Galaxy masses play a fundamental role in our understanding of structure formation models. This review addresses the variety and reliability of mass estimators that pertain to stars, gas, and dark matter. The different sections on masses from stellar populations, dynamical masses of gas-rich and gas-poor galaxies, with some attention paid to our Milky Way, and masses from weak and strong lensing methods all provide review material on galaxy masses in a self-consistent manner.

This article presents an overview of current understanding of the interaction of low-energy positrons with molecules with emphasis on resonances, positron attachment, and annihilation. Measurements of annihilation rates resolved as a function of positron energy reveal the presence of vibrational Feshbach resonances (VFRs) for many polyatomic molecules. These resonances lead to strong enhancement of the annihilation rates. They also provide evidence that positrons bind to many molecular species. A quantitative theory of VFR-mediated attachment to small molecules is presented. It is tested successfully for selected molecules (e.g., methyl halides and methanol) where all modes couple to the positron continuum. Combination and overtone resonances are observed and their role is elucidated. Molecules that do not bind positrons and hence do not exhibit such resonances are discussed. In larger molecules, annihilation rates from VFR far exceed those explicable on the basis of single-mode resonances. These enhancements increase rapidly with the number of vibrational degrees of freedom, approximately as the fourth power of the number of atoms in the molecule. While the details are as yet unclear, intramolecular vibrational energy redistribution (IVR) to states that do not couple directly to the positron continuum appears to be responsible for these enhanced annihilation rates. In connection with IVR, experimental evidence indicates that inelastic positron escape channels are relatively rare. Downshifts of the VFR from the vibrational mode energies, obtained by measuring annihilate rates as a function of incident positron energy, have provided binding energies for 30 species. Their dependence upon molecular parameters and their relationship to positron-atom and positron-molecule binding-energy calculations are discussed. Feshbach resonances and positron binding to molecules are compared with the analogous electron-molecule (negative-ion) cases. The relationship of VFR-mediated annihilation to other phenomena such as Doppler broadening of the gamma-ray annihilation spectra, annihilation of thermalized positrons in gases, and annihilation-induced fragmentation of molecules is discussed. Possible areas for future theoretical and experimental investigation are also discussed.

Understanding the predictions of general relativity for the dynamical interactions of two black holes has been a long-standing unsolved problem in theoretical physics. Black-hole mergers are monumental astrophysical events, releasing tremendous amounts of energy in the form of gravitational radiation, and are key sources for both ground-and space-based gravitational-wave detectors. The black-hole merger dynamics and the resulting gravitational wave forms can only be calculated through numerical simulations of Einstein's equations of general relativity. For many years, numerical relativists attempting to model these mergers encountered a host of problems, causing their codes to crash after just a fraction of a binary orbit could be simulated. Recently, however, a series of dramatic advances in numerical relativity has allowed stable, robust black-hole merger simulations. This remarkable progress in the rapidly maturing field of numerical relativity and the new understanding of black-hole binary dynamics that is emerging is chronicled. Important applications of these fundamental physics results to astrophysics, to gravitational-wave astronomy, and in other areas are also discussed.

This paper reviews recent progress in understanding phenomena such as crumpling, in which elastic membranes or sheets subject to structureless forces develop sharply curved structure over a small fraction of their surface. In the limit of zero thickness, these structures become singular. After reviewing several related phenomena, the paper recalls the physical elements that give rise to the singular behavior: elasticity and the nearly inextensible behavior of thin sheets. This singular behavior has counterparts in higher dimensions. Then the paper discusses the most basic of these singularities, the vertex. The paper recounts mathematical progress in describing the d-cone, a simple realization of a vertex. After discussing the size of the core that governs departure from singularity, the paper concludes that fundamental understanding is lacking. It points out further mysterious behavior at the region where a d-cone is supported. Next comes a discussion of an emergent singularity that appears when two or more vertices are present: the stretching ridge. The paper offers several explanations of the scale of this singularity, ranging from qualitative scaling arguments to a formal asymptotic analysis. It discusses recent experiments and theories about the interaction of ridges and vertices and reviews evidence that these ridges dominate the mechanics of crumpled sheets.

Super-resolution, extraordinary transmission, total absorption, and localization of electromagnetic waves are currently attracting growing attention. These phenomena are related to different physical systems and are usually studied within the context of different, sometimes rather sophisticated, approaches. Remarkably, all these seemingly unrelated phenomena owe their origin to the same underlying physical mechanism, namely, wave interaction with an open resonator. Here we show that it is possible to describe all of these effects in a unified way, mapping each system onto a simple resonator model. Such description provides a thorough understanding of the phenomena, explains all the main features of their complex behavior, and enables one to control the system via the resonator parameters: eigenfrequencies, Q factors, and coupling coefficients.

The properties of quasi-two-dimensional semiconductor quantum dots are reviewed. Experimental techniques for measuring the electronic shell structure and the effect of magnetic fields are briefly described. The electronic structure is analyzed in terms of simple single-particle models, density-functional theory, and "exact" diagonalization methods. The spontaneous magnetization due to Hund's rule, spin-density wave states, and electron localization are addressed. As a function of the magnetic field, the electronic structure goes through several phases with qualitatively different properties. The formation of the so-called maximum-density droplet and its edge reconstruction is discussed, and the regime of strong magnetic fields in finite dot is examined. In addition, quasi-one-dimensional rings, deformed dots, and dot molecules are considered.

Attempts to estimate the influence of global cosmological expansion on local systems are reviewed. Here "local" is taken to mean that the sizes of the considered systems are much smaller than cosmologically relevant scales. For example, such influences can affect orbital motions as well as configurations of compact objects, like black holes. Also discussed are how measurements based on the exchange of electromagnetic signals of distances, velocities, etc. of moving objects are influenced. As an application, orders of magnitude of such effects are compared with the scale set by the apparently anomalous acceleration of the Pioneer 10 and 11 spacecrafts, which is 10(-9) m/s(2). There is no reason to believe that the latter is of cosmological origin. However, the general problem of gaining a qualitative and quantitative understanding of how the cosmological dynamics influences local systems remains challenging, with only partial clues being so far provided by exact solutions to the field equations of general relativity.

The progress in solving problems involving nonrelativistic fast ion (atom)-atom collisions with two actively participating electrons is reviewed. Such processes involve, e.g., (i) scattering between a bare nucleus (projectile) P of charge Z(P) and a heliumlike atomic system consisting of two electrons e(1) and e(2) initially bound to the target nucleus T of charge Z(T), i.e., the Z(P)-(Z(T);e(1),e(2))(i) collisions; (ii) scattering between two hydrogenlike atoms (Z(P),e(1))(i1) and (Z(T),e(2))(i2), etc. A proper description of these collisional processes requires solutions of four-body problems with four active particles including two nuclei and two electrons. Among various one- as well as two-electron transitions which can occur in such collisions, special attention will be paid to double-electron capture, simultaneous transfer and ionization, simultaneous transfer and excitation, single-electron detachment and single-electron capture. Working within the four-body framework of scattering theory and imposing the proper Coulomb boundary conditions on the entrance and exit channels, the leading quantum-mechanical theories are analyzed. Both static and dynamic interelectron correlations are thoroughly examined. The correct links between scattering states and perturbation potentials are strongly emphasized. Selection of the present illustrations is dictated by the importance of interdisciplinary applications of energetic ion-atom collisions, ranging from thermonuclear fusion to medical accelerators for hadron radiotherapy.

This Colloquium addresses the issue of the shape of hadrons and, in particular, that of the proton. The concept of shape in the microcosm is critically examined. Special attention is devoted to properly define the meaning of shape for bound-state systems of near massless quarks. The ideas that lead to the expectation of nonsphericity in the shape of hadrons, the calculations that predict it, and the experimental information obtained from recent high-precision measurements are examined. Particular emphasis is given to the study of the electromagnetic transition between the nucleon and its first excited state, the Delta(1232) resonance. The experimental evidence is critically examined and compared with lattice calculations, as well as with effective-field theories and phenomenological models.

Remarkable and varied pattern-forming phenomena occur in fluids and in phase transformations. The authors describe and compare some of these phenomena, offer reflections on their similarities and differences, and consider possibilities for the future development of this field. [S0034-61(99)04702-9].

In view of future plans for accurate measurements of the theoretically clean branching ratios Br(K+->pi(+)nu(nu) over bar) and Br(K-L ->pi(0)nu(nu) over bar), which should occur in the next decade, the relevant formulas for quantities of interest are collected and their theoretical and parametric uncertainties are analyzed. In addition to the angle beta in the unitarity triangle (UT), the angle gamma can also be determined from these decays with respectable precision and in this context the importance of the recent NNLO QCD calculation on the charm contribution to K+->pi(+)nu(nu) over bar and of the improved estimate on the long-distance contribution by means of chiral perturbation theory are presented. In addition to known expressions, several new ones that should allow transparent tests of the standard model (SM) and of its extensions are presented. While the review is centered around the SM, models with minimal flavor violation and scenarios with new complex phases in decay amplitudes and meson mixing are also discussed. A review of existing results within specific extensions of the SM, in particular the littlest Higgs model with T-parity, Z(') models, the MSSM, and a model with one universal extra dimension are given. A new "golden" relation between B and K systems is derived that involves (beta,gamma) and Br(K-L ->pi(0)nu(nu) over bar), and the virtues of (R-t,beta), (R-b,gamma), (beta,gamma), and ((eta) over bar,gamma) strategies for the UT in the context of K ->pi nu(nu) over bar decays with the goal of testing the SM and its extensions are investigated.

In this paper the authors discuss recent advances and trends in laser spectroscopy and quantum optics. It is obvious that both are fields that experienced a tremendous development in the last twenty years. Therefore the survey must be incomplete, and only a few highlights are touched on. [S0034-6861(99)04802-3].

This review discusses the heavy-electron compounds and related materials from the perspective of their electrodynamic response. The investigation of the electrodynamic response by means of optical methods, extending over a very broad spectral range, should reveal, in principle, the complete excitation spectrum. This study incorporates several important sources of information on the intrinsic properties of the investigated materials. In particular, attention will be focused on the optical properties of prototype heavy-electron systems and on Kondo systems with low-temperature non-Fermi-liquid behavior or insulating characteristics. In the discussion, the electrodynamic response will be related to other relevant results arrived at by various experimental methods and to the theoretical state of the art. [S0034-6861(99)00303-7].