Yes, but some parts are reasonably concrete.

We give a pedestrian derivation of a formula of Büttiker et. al. (BPT) relating the adiabatically pumped current to the S matrix and its (time) derivatives. We relate the charge in BPT to Berry's phase and the corresponding Brouwer pumping formula to curvature. As applications we derive explicit formulas for the joint probability density of pumping and conductance when the S matrix is uniformly distributed; and derive a new formula that describes hard pumping when the S matrix is periodic in the driving parameters.PACS numbers: 72.10. Bg, Brouwer [3], and Aleiner et. al. [4], building on results of Büttiker, Pretre and Thomas (BPT) [5], pointed out that adiabatic scattering theory leads to a geometric description of charge transport in mesoscopic quantum pumps. Some of these works, and certainly our own work, was motivated by experimental results of Switkes et. al.[6] on such pumps.In this article we examine the formula of BPT [5], which relates adiabatic charge transport to the S matrix and its (time) derivatives, in the special case of singlechannel scattering. We show that the formula admits a simple interpretation in terms of three basic processes at the Fermi energy. Two of these are dissipative and nonquantized. The third integrates to zero for any cyclic variation in the system.Next, we describe the geometric significance of BPT and relate it to Berry's phase [2]. It follows that the pumping formula of Brouwer [3] can be interpreted as curvature and is formally identical to the adiabatic curvature [2]. In spite of the interesting geometry the topological aspects of pumping are trivial. In particular, we prove that all Chern numbers associated to the Brouwer formula are identically zero.We proceed with two applications. First we give an elementary and explicit derivation of the joint probability density for pumping and conductance. This problem was studied in [3]. Brouwer's results go beyond ours as he also calculates the tails of the distributions and we don't. On the other hand, parts of his results are numerical, and they are certainly not elementary. Finally, we calculate, for the first time, the asymptotics of hard pumping for S matrices that depend periodically on two parameters. If the system traverses a circle of radius R in parameter space, with R large, then the amount of charge transported is order √ R, multiplied by a quasi-periodic (oscillatory) function of R leading to ergodic behavior.We shall use units where e = m =h = 1, so the electron charge is −1 and the quantum of conductance is e 2 /h = 1 2π . The mutual Coulombic interaction of the electrons is disregarded.The BPT formula: Consider a scatterer connected to leads that terminate at electron reservoirs. All the reservoirs are initially at the same chemical potential and at zero temperature. The scatterer is described by its (on-shell) S matrix, which, in the case of n channels is an n × n matrix parameterized by the energy and other parameters associated with the adiabatic driving of the system (e.g. gate voltages and magnetic fields).T...

%'e discuss topological invariants

No abstract

We study adiabatic quantum pumps on time scales that are short relative to the cycle of the pump. In this regime the pump is characterized by the matrix of energy shift which we introduce as the dual to Wigner's time delay. The energy shift determines the charge transport, the dissipation, the noise, and the entropy production. We prove a general lower bound on dissipation in a quantum channel and define optimal pumps as those that saturate the bound. We give a geometric characterization of optimal pumps and show that they are noiseless and transport integral charge in a cycle. Finally we discuss an example of an optimal pump related to the Hall effect.

Abstract. We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density.

Abstract. We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve.

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.