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In the early 1980's Margulis startled the world by showing the existence of proper affine actions of free groups on 3-space, answering a provocative and suggestive question Milnor posed in 1977. In this paper we discuss the historical background motivating this question, recent progress on this subject, and future directions inspired by this discovery., Comment: 74 pages, 8 figures. Solicited for: Dynamics, Geometry, Number Theory: the Impact of Margulis on Modern Mathematics

Let $\Sigma$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\Gamma$ of $\Sigma$ acts on the SU(3)-character variety of $\Sigma$. We show that the action is ergodic with respect to the natural symplectic measure on the character variety., Comment: 18 pages, 1 figure. Revised version to appear in the journal of "Ergodic Theory and Dynamical Systems"

The magnetic-field-tuned quantum superconductor-insulator transitions of disordered amorphous indium oxide films are a paradigm in the study of quantum phase transitions, and exhibit power-law scaling behavior. For superconducting indium oxide films with low disorder, such as the ones reported on here, the high-field state appears to be a quantum-corrected metal. Resistance data across the superconductor-metal transition in these films are shown here to obey an activated scaling form appropriate to a quantum phase transition controlled by an infinite randomness fixed point in the universality class of the random transverse-field Ising model. Collapse of the field-dependent resistance vs. temperature data is obtained using an activated scaling form appropriate to this universality class, using values determined through a modified form of power-law scaling analysis. This exotic behavior of films exhibiting a superconductor-metal transition is caused by the dissipative dynamics of superconducting rare regions immersed in a metallic matrix, as predicted by a recent renormalization group theory. The smeared crossing points of isotherms observed are due to corrections to scaling which are expected near an infinite randomness critical point, where the inverse disorder strength acts as an irrelevant scaling variable., Comment: 8 pages, 6 figures

The behavior of a disordered amorphous thin film of superconducting Indium Oxide has been studied as a function of temperature and magnetic field applied perpendicular to its plane. A superconductor-insulator transition has been observed, though the isotherms do not cross at a single point. The curves of resistance vs. temperature on the putative superconducting side of this transition, where the resistance decreases with decreasing temperature, obey two-dimensional Mott variable-range hopping of vortices over wide ranges of temperature and resistance. To estimate the parameters of hopping, the film is modeled as a granular system and the hopping of vortices is treated in a manner analogous to hopping of charges. The reason the long range interaction between vortices over the range of magnetic fields investigated does not lead to a stronger variation of resistance with temperature than that of two-dimensional Mott variable-range hopping remains unresolved., Comment: 7 pages, 7 figures

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the derormation space and of each stratum of the Teichmueller unit sphere bundle over the Riemann moduli space. We prove ergodicity for the analogous lift of the Weil-Petersson geodesic local. flow., Comment: 18 pages, no figures, presented at the Oxford conference honoring Nigel Hitchin's 70th birthday (9 September 2016) and to appear in the companion volume published by Oxford University Press

The complexity of the healthcare ecosystem and the trans-disciplinary convergence which is essential for its function, makes it difficult to address healthcare as one domain. Data curation and analysis of the information may boost our health related knowledge. Increasing connectivity and improving infrastructure may help, among other things, to uncover facts and observations which may influence the future of global health.

A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to $M$ is a noncompact complete hyperbolic surface $\Sigma$. We study double extensions of $\pi_1 (M) \cong \pi_1 (\Sigma)$ when $\Sigma$ is homeomorphic to a projective plane minus two discs. We classify proper actions of this double extension on Minkowski space and show that there exist proper actions that do not admit crooked fundamental domains., Comment: 17 pages, 5 figures

Finding more efficient ways of exploring the doping phase diagrams of high temperature superconductors as well as probing the fundamental properties of these materials are essential ingredients for driving the discovery of new materials. We use a doping technique involving gating with ionic liquids to systematically and continuously tune the T$_{c}$ of superconducting La$_{2}$CuO$_{4+x}$ thin films. We probe both the transport properties and the penetration depth of these samples and find that Homes scaling $\lambda^{-2}\propto\sigma T_{c}$ is obeyed, consistent with these materials being in the dirty limit. This result is independent of the precise mechanism for the gating process as all of the parameters of the scaling relationship are determined by direct measurements on the films.

We derive an identity for Margulis invariants of affine deformations of a complete orientable one-ended hyperbolic sur- face following the identities of McShane, Mirzakhani and Tan- Wong-Zhang. As a corollary, a deformation of the surface which infinitesimally lengthens all interior simple closed curves must in- finitesimally lengthen the boundary., Comment: resubmitted after error revising another submission

High mobility and dilute two-dimensional electron systems exhibit metallic behavior down to the lowest experimental temperatures. In studies of ionic liquid gated insulating silicon, we have observed transitions to a metallic state in low mobility samples at much higher areal carrier densities than found for samples of high mobility. We have also observed a mobility peak in metallic samples as the carrier density was increased beyond $10^{13} \text{cm}^{-2}$.

In order to clarify the physics of the gating of solids by ionic liquids (ILs) we have gated lightly doped $p$-Si, which is so well studied that it can be called the "hydrogen atom of solid state physics" and can be used as a test bed for ionic liquids. We explore the case where the concentration of induced holes at the Si surface is below $10^{12}\text{cm}^{-2}$, hundreds of times smaller than record values. We find that in this case an excess negative ion binds a hole on the interface between the IL and Si becoming a surface acceptor. We study the surface conductance of holes hopping between such nearest neighbor acceptors. Analyzing the acceptor concentration dependence of this conductivity, we find that the localization length of a hole is in reasonable agreement with our direct variational calculation of its binding energy. The observed hopping conductivity resembles that of well studied $\text{Na}^{+}$ implanted Si MOSFETs.

A Margulis spacetime is a complete flat Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface S homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when S is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of S. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces. This extends our previous work on affine deformations of three-holed sphere and two-holed cross surfaces., Comment: 47 pages, 5 figures

Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is the interior of the quotient of Einstein space by an involution fixing an Einstein plane. The purpose of this note is to show that the crooked planes defined in anti-de Sitter space recently by Danciger-Gu\'eritaud-Kassel lift to restrictions of crooked surfaces in Einstein space which are adapted under the involution of Einstein space defining anti-de Sitter space., Comment: 37 pages, no figures. Revised version, based on comments of two referees. Accepted for publication in Geometriae Dedicata

We define deformations of $RP^2$-manifolds., Comment: 7 pages, 7 figures; unpublished note (with pictures) from 12/18/2004

The evolution with carrier concentration of the electrical properties of amorphous indium oxide (InO) thin films has been studied using electronic double layer transistor configurations. Carrier variations of up to 7 X 10^(14) carriers/cm^2 were achieved using an ionic liquid as a gate dielectric. The superconductor-insulator transition was traversed and the magnitude and position of the large magnetoresistance peak found in the insulating regime were modified. The systematic variation of the magnetoresistance peak with charge concentration was found to be qualitatively consistent with a simulation based on a model involving granularity.

Using an electronic double layer transistor we have systematically studied the superconductor to insulator transition in La$_{2}$CuO$_{4+\delta}$ thin films growth by ozone assisted molecular beam epitaxy. We have confirmed the high crystalline quality of the cuprate films and have demonstrated the suitability of the electronic double layer technique to continuously vary the charge density in a system that is otherwise characterized by the presence of miscibility gaps. The transport and magneto-transport results highlight the role of electron-electron interactions in the mechanism of the transition due to the proximity of the Mott insulating state.

We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature -1. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry., Comment: Not a research article in the usual sense but rather a list of open questions. 19 pages

We survey developments arising from Milnor's 1958 paper, "On the existence of a connection with curvature zero" and his 1977 paper, "On fundamental groups of complete affinely flat manifolds"., Comment: 29 pages, 7 figures, presented at "Frontiers in Complex Dynamics" in celebration of John Milnor's 80th birthday (Banff International Research Station, 24 February 2011)

We successfully tuned an underdoped ultrathin YBa2Cu3O7-x film into the overdoped regime by means of electrostatic doping using an ionic liquid as a dielectric material. This process proved to be reversible. Transport measurements showed a series of anomalous features compared to chemically doped bulk samples and a different two-step doping mechanism for electrostatic doping was revealed. The normal resistance increased with carrier concentration on the overdoped side and the high temperature (180 K) Hall number peaked at a doping level of p$\sim$0.15. These anomalous behaviors suggest that there is an electronic phase transition in the Fermi surface around the optimal doping level., Comment: 4 pages,3 figures

A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic to a two-holed cross-surface, that is, the complement of two disjoint discs in the real projective plane. We show that every such manifold is homeomorphic to a solid handlebody of genus two, and admits a fundamental polyhedron bounded by crooked planes. Furthermore, the deformation space is a bundle of convex quadrilateral cones over the space of marked hyperbolic structures. The sides of each quadrilateral cone are defined by invariants of the two boundary components and the two orientation-reversing simple curves. The two-holed cross-surface, together with the three-holed sphere, are the only topologies for which the deformation space of complete affine structures is finite-sided., Comment: 29 pages, 8 figures. Accepted for publication in the Journal of Topology

The electrical transport properties of ultrathin YBCO films have been modified using an electric double layer transistor configuration employing an ionic liquid. The films were grown on SrTiO3 substrates using high pressure oxygen sputtering. A clear evolution from superconductor to insulator was observed in nominally 7 unit cell thick films. Using a finite size scaling analysis, curves of resistance versus temperature, R(T), over the temperature range from 6K to 22K were found to collapse onto a single scaling function, which suggests the the presence of a quantum critical point. However the scaling failed at the lowest temperatures suggesting the presence of an additional phase between the superconducting and insulating regimes., Comment: 4 pages, 6 figures

Let M be a Margulis spacetime whose associated complete hyperbolic surface S has compact convex core. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on M and recurrent geodesics on S. In contrast, no timelike geodesic recurs in either forward or backwards time., Comment: 12 pages, no figures, to appear in the issue of "Ergodic Theory and Dynamical Systems" dedicated to the memory of Daniel Rudolph

A surprisingly strong variation of resistance with perpendicular magnetic field, and a peak in the resistance vs. field, R(B) has been found in insulating films of a sequence of homogeneous, quench-condensed films of amorphous Bi undergoing a thickness-tuned superconductor-insulator transition. Isotherms of magnetoresistance, rather than resistance, vs. field were found to cross at a well-defined magnetic field higher than the field corresponding to the peak in R(B). For all values of B, R(T) was found to obey an Arrhenius form. At the crossover magnetic field the prefactor became equal to the quantum resistance of electron pairs, h/4e^2, and the activation energy returned to its zero field value. These observations suggest that the crossover is the signature of a quantum phase transition between two distinct insulating ground states, tuned by magnetic field., Comment: 4 pages, 4 figures

Electric double layer transistor configurations have been employed to electrostatically dope single crystals of insulating SrTiO_{3}. Here we report on the results of such doping over broad ranges of temperature and carrier concentration employing an ionic liquid as the gate dielectric. The surprising results are, with increasing carrier concentration, an apparent carrier-density dependent conductor-insulator transition, a regime of anomalous Hall effect, suggesting magnetic ordering, and finally the appearance of superconductivity. The possible appearance of magnetic order near the boundary between the insulating and superconducting regimes is reminiscent of effects associated with quantum critical behavior in some complex compounds.

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic surface which is smooth at infinity and whose ideal locus is a generic tritangent plane arises as a relative SL(2)-character variety of the 4-holed sphere. Every such affine cubic for which all the periodic automorphisms of the tritangent plane extend to automorphisms of the cubic arises as a relative SL(2)-character variety of a 1-holed torus., Comment: 10 pages, no figures, various corrections and a shorter proof; some literature references added; manuscript now submitted for publication

In his 1990 doctoral thesis, Todd Drumm showed that proper affine deformations of free Fuchsian groups could be constructed as Schottky groups using a new family of hypersurfaces called "crooked planes." The existence of proper affine deformations of Fuchsian Schottky groups was demonstrated by Margulis in the early 1980's, answering a question raised by Milnor in 1977. This paper expounds Drumm's result, at least in the case of Fuchsian Schottky groups (that is, when the group contains no parabolic elements)., Comment: This paper appeared in the Contemporary Mathematics volume, "Crystallographic groups and their generalizations", the proceedings of the conference in Kortrijk, Belgium in 1999 and represents an expanded version of the lecture presented by the second-named author. The paper contains 18 figures.

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology S and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of G/H into S. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on S, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of S. In this talk I will survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups., Comment: Invited address for Geometry Section for 2010 International Congress of Mathematicians

Let \Sigma be a compact orientable surface with genus g and n boundary components B = (B_1,..., B_n). Let c = (c_1,...,c_n) in [-2,2]^n. Then the mapping class group MCG of \Sigma acts on the relative SU(2)-character variety X_c := Hom_C(\pi, SU(2))/SU(2), comprising conjugacy classes of representations \rho with tr(\rho(B_i)) = c_i. This action preserves a symplectic structure on the smooth part of X_c, and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J be the subgroup of MCG generated by Dehn twists along null homologous simple loops in \Sigma. Then the action of J on X_c is ergodic for almost all c., Comment: This new version includes a ribbon graph for clarity and corrects some typographic errors and a few misstatements between Remark 3.1 and Lemma 3.2 and between Remark 3.7 and Lemma 3.8 in the previous version

A systematic study has been carried out on the previously reported "magnetic-field-induced superconductivity" of Zn nanowires. By varying parameters such as magnetic field orientation and wire length, the results provide evidence that the phenomenon is a nonequilibrium effect associated with the boundary electrodes. They also suggest there are two length scales involved, the superconducting coherence length and quasiparticle relaxation length. As wire lengths approach either of these length scales, the effect weakens. We demonstrate that it is appropriate to consider the effect to be a stabilization of superconductivity, that has been suppressed by an applied current., Comment: (Updated Version) 9 pages, 8 figures

Magnetic field and electrostatically tuned superconductor-insulator (SI) transitions of ultrathin metal films with levels of disorder that place them near the disorder-tuned SI transition appear to be direct, continuous quantum phase transitions. When films with lower levels of disorder are subjected to a perpendicular magnetic field, instead of a direct transition, a mixed superconductor-nonsuperconductor regime emerges at the lowest temperatures. The zero temperature limit of the resistance is either insulating or superconducting, depending upon the value of the field, suggesting that the behavior in this limit is governed by percolation physics. At high fields and low temperatures, in the nominally insulating regime, the resistance rather than the conductance is found to be a logarithmic function of temperature corresponding to predicitons for the weak localization of bosons., Comment: 4 pages, 4 figures

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Sigma, the deformation space identifies with two opposite octants in R^3. Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4,Z)., Comment: 30 pages, 7 figures

This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesimally lengthens every closed geodesic. The proof interprets the derivative of the geodesic length function as the Margulis invariant (signed marked Lorentzian length spectrum) of the corresponding affine deformation. The aforementioned results imply that the affine deformation is proper, and hence by Margulis's Opposite Sign Lemma, every closed geodesic infinitesimmaly lengthens., Comment: 10 pages, no figures, to appear in the Proceedings of the 2008 Ahlfors-Bers Colloquium

Four-terminal resistance measurements have been carried out on Zn nanowires formed using electron-beam lithography. When driven resistive by current, these wires re-enter the superconducting state upon application of small magnetic fields. The data are qualitatively different from those of previous experiments on superconducting nanowires, which revealed either negative magnetoresistance near T_{c} or magnetic field enhanced critical currents. We suggest that our observations are associated with the damping of phase slip processes by the enhancement of dissipation by the quasiparticle conductance channel resulting from the application of a magnetic field.

Let S be a compact orientable surface with genus g and n boundary components d_1,...,d_n. Let b = (b_1, ..., b_n) where b_n lies in [-2,2]. Then the mapping class group of S acts on the relative SU(2)-character variety X comprising conjugacy classes of representations f of the fundamental group F of S, where trace f(d_i) = b_i. This action preserves a symplectic structure on the open dense smooth submanifold of X. corresponding to irreducible representations. This subset has full measure and is connected. In this note we use the symplectic geometry of this space to give a new proof that this action is ergodic., Comment: to appear in the Festschrift honoring Robert Zimmer's 60th birthday

The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which may be an arbitary element of C^3. This paper gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a representation of the index-two supergroup which is a free product of three groups of order two, is described in detail, and related to hyperbolic geometry. When n > 2, the classification of conjugacy-classes of n-tuples in SL(2,C) is more complicated. We describe it in detail when n= 3. The deformation spaces of hyperbolic structures on some simple surfaces S whose fundamental group is free of rank two or three are computed in trace coordinates. (We only consider the two orientable surfaces whose fundamental group has rank 3.), Comment: To appear in "Handbook of Teichmueller theory, volume II". The paper substantially expands, and extends, math.GM/0402103 An exposition of results of Fricke

This paper concerns the relationship between locally homogeneous geometric structures on topological surfaces and the moduli of polystable Higgs bundles on Riemann surfaces, due to Hitchin and Simpson. In particular we discuss the uniformization of Riemann surfaces by hyperbolic geometry from this viewpoint, and survey more recent developments in this theory., Comment: 29 pages, To appear in "The Many Facets of Geometry: a Tribute to Nigel Hitchin," Bourgignon, Garcia-Prada & Salamon, eds, Oxford Univ. Press

Exchange bias has been studied in a series of La2/3Ca1/3MnO3 / La1/3Ca2/3MnO3 bilayers grown on (001) SrTiO3 substrates by ozone-assisted molecular beam epitaxy. The high crystalline quality of the samples and interfaces has been verified using high-resolution X-ray diffractometry and Z-contrast scanning transmission electron microscopy with electron energy loss spectroscopy. The dependence of exchange bias on the thickness of the antiferromagnetic layer has been investigated. A critical value for the onset of the hysteresis loop shift has been determined. An antiferromagnetic anisotropy constant has been obtained by fitting the results to the generalized Meiklejohn-Bean model., Comment: 7 pages. 8 figures

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension $2 + 1$, in which there is a rich interplay with symplectic geometry., Comment: 56 pages, 6 figures

These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata., Comment: 26 pages

Lead clusters deposited on Si(111) substrates have been studied at low temperatures using scanning tunneling microscopy and spectroscopy. The current-voltage characteristics exhibit current peaks that are irregularly spaced and varied in height. The statistics of the distribution of peak heights and spacings are in agreement with random matrix theory for several clusters. The distributions have also been studied as a function of cluster shape., Comment: 10 pages, 9 figures, to appear in Phys. Rev. B

Measurements of transport properties of amorphous insulating indium oxide thin films have been interpreted as evidence of the presence of superconducting islands on the insulating side of a disorder-tuned superconductor-insulator transition. Although the films are not granular, the behavior is similar to that observed in granular films. The results support theoretical models in which the destruction of superconductivity by disorder produces spatially inhomogenous pairing with a spectral gap., Comment: Revised title and content/argument. Totals: 4 pages, 3 figures

The parallel magnetic field tuned two-dimensional superconductor-insulator transition has been investigated in ultrathin films of amorphous Bi. The resistance is found to be independent of temperature on both sides of the transition below approximately 120 mK. Several observations suggest that this regime is not intrinsically "metallic" but results from the failure of the films' electrons to cool. The onset of this temperature-independent regime can be moved to higher temperatures by either increasing the measuring current or the level of electromagnetic noise. Temperature scaling is successful above 120 mK. Electric field scaling can be mapped onto temperature scaling by relating the electric fields to elevated electron temperatures. These results cast doubt on the existence of an intrinsic metallic regime and on the independent determination of the correlation length and dynamical critical exponents obtained by combining the results of electric field and temperature scaling., Comment: 4 pages, 4 figures

The 2D superconductor-insulator transition in disordered ultrathin amorphous bismuth films has been tuned both by electrostatic electron doping using the electric field effect and by the application of parallel magnetic fields. Electrostatic doping was carried out in both zero and nonzero magnetic fields, and magnetic tuning was conducted at multiple strengths of electrostatically induced superconductivity. The transitions were analyzed using finite size scaling with critical exponent products nu*z = 0.65-0.7. The parallel critical magnetic field increased with electron transfer as (dn_c-dn)^0.33, where dn is the electron transfer and dn_c is its critical value, and the critical resistance decreased linearly with dn. However at lower temperatures, in the insulating regime, the resistance became larger than expected from extrapolation of its temperature dependence at higher temperatures, and scaling failed. These observations imply that although the electrostatic- and parallel magnetic field- tuned superconductor-insulator transitions would appear to belong to the same universality class and to be delineated by a robust phase boundary that can be crossed either by tuning electron density or magnetic field, in the case of the field-tuned transition at the lowest temperatures, some different type of physical behavior turns on in the insulating regime., Comment: About 11 pages, with 14 figures. To be submitted to Phys Rev B

Deformation spaces Hom($\pi$,G)/G of representations of the fundamental group $\pi$ of a surface $\Sigma$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_\Sigma$, preserving a Poisson structure. When $G$ is compact, the actions are ergodic. In contrast if $G$ is noncompact semisimple, the associated deformation space contains open subsets containing the Fricke-Teichm\"uller space upon which $Mod_\Sigma$ acts properly. Properness of the $Mod_\Sigma$-action relates to (possibly singular) locally homogeneous geometric structures on $\Sigma$. We summarize known results and state open questions about these actions., Comment: 32 pages, no figures

When $G$ is a connected compact Lie group, and $\pi$ is a closed surface group, then $Hom(\pi,G)$ contains an open dense $Out(\pi)$-invariant subset which is a smooth symplectic manifold. This symplectic structure is $Out(\pi)$-invariant and therefore defines an invariant measure $\mu$, which has finite volume. The corresponding unitary representation of $Out(\pi)$ on $L^2(Hom(\pi,G)/G,\mu)$ contains no finite-dimensional subrepresentations besides the constants. This note gives a short proof that when $G=SU(n)$, the representation $L^2(Hom(\pi,G)/G,\mu)$ contains many other invariant subspaces., Comment: 6 pages, no figures

For n>2, the action of the outer automorphism group of the rank n free group F_n on the SU(2)-character variety Hom(F_n,SU(2))/SU(2)$ is ergodic with respect to the Lebesgue measure class., Comment: 12 pages, 4 figures The proof of Lemma 3.1 was rewritten, following advice of David Fisher

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_\rho$ on Teichm\"uller space $\Teich$ which is a qualitative invariant of the holonomy representation $\rho$ of $\pi_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that the energy function $E_\rho$ is proper for any convex cocompact representation of the fundamental group. More generally, if $\rho$ is a discrete embedding onto a normal subgroup of a convex cocompact group $\Gamma$, then $E_\rho$ defines a proper function on the quotient $\Teich/Q$ where $Q$ is the subgroup of the mapping class group defined by $\Gamma/\rho(\pi_1(S))$. When the image of $\rho$ contains parabolic elements, then $E_\rho$ is not proper. Using the recent solution of Marden's Tameness Conjecture, we show that if $\rho$ is a discrete embedding into $\SLtC$, then $E_\rho$ is proper if and only if $\rho$ is quasi-Fuchsian. These results are used to prove that the mapping class group acts properly on the subset of convex cocompact representations., Comment: 21 pages; a few corrections were made based on a referee's report

Low temperature scanning tunneling microscope images and spectroscopic data have been obtained on subnanometer size Pb clusters fabricated using the technique of buffer layer assisted growth. Discrete energy levels were resolved in current-voltage characteristics as current peaks rather than current steps. Distributions of peak voltage spacings and peak current heights were consistent with Wigner-Dyson and Porter-Thomas distributions respectively, suggesting the relevance of random matrix theory to the description of the electronic eigenstates of the clusters. The observation of peaks rather than steps in the current-voltage characteristics is attributed to a resonant tunneling process involving the discrete energy levels of the cluster, the tip, and the states at the interface between the cluster and the substrate surface., Comment: 4 pages, 4 figures

Superconductivity has been induced in insulating ultra-thin films of amorphous bismuth using the electric field effect. The screening of electron-electron interaction was found to increase with electron concentration in a manner correlated with the tendency towards superconductivity. This does not preclude an increase in the density of states being important in the development of superconductivity. The superconductor-insulator transition appears to belong to the universality class of the three dimensional XY model., Comment: Four pages, three figures. Revised slightly to reflect referees' comments

Ultrathin La0.8Ca0.2MnO3 films have been measured in a field-effect geometry. The electric field due to the gate produces a large ambipolar decrease in resistance at low temperatures. This is attributed to the development of a pseudogap in the density of states and the couple of localized charge to strain. The gate effect and mangetoresistance are interpreted in a consistent framework. The implications for the low temperature behavior of a manganite film in the two dimensional limit are discussed., Comment: 4 pages, 3 figures