Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

Materials science and the study of the electronic properties of solids are a major field of interest in both physics and engineering. The starting point for all such calculations is single-electron, or non-interacting, band structure calculations, and in the limit of strong on-site confinement this can be reduced to graph-like tight-binding models. In this context, both mathematicians and physicists have developed largely independent methods for solving these models. In this paper we will combine and present results from both fields. In particular, we will discuss a class of lattices which can be realized as line graphs of other lattices, both in Euclidean and hyperbolic space. These lattices display highly unusual features including flat bands and localized eigenstates of compact support. We will use the methods of both fields to show how these properties arise and systems for classifying the phenomenology of these lattices, as well as criteria for maximizing the gaps. Furthermore, we will present a particular hardware implementation using superconducting coplanar waveguide resonators that can realize a wide variety of these lattices in both non-interacting and interacting form

Log Fano varieties over function fields of curves

Consider a smooth log Fano variety over the function field of a curve. Suppose that the boundary has positive normal bundle. Choose an integral model over the curve. Then integral points are Zariski dense, after removing an explicit finite set of points on the base curve.Comment: 18 page

Hyperbolic Lattices in Circuit Quantum Electrodynamics

After close to two decades of research and development, superconducting circuits have emerged as a rich platform for both quantum computation and quantum simulation. Lattices of superconducting coplanar waveguide (CPW) resonators have been shown to produce artificial materials for microwave photons, where weak interactions can be introduced either via non-linear resonator materials or strong interactions via qubit-resonator coupling. Here, we highlight the previously-overlooked property that these lattice sites are deformable and allow the realization of tight-binding lattices which are unattainable, even in conventional solid-state systems. In particular, we show that networks of CPW resonators can create a new class of materials which constitute regular lattices in an effective hyperbolic space with constant negative curvature. We present numerical simulations of a series of hyperbolic analogs of the kagome lattice which show unusual densities of states with a spectrally-isolated degenerate flat band. We also present a proof-of-principle experimental realization of one of these lattices. This paper represents the first step towards on-chip quantum simulation of materials science and interacting particles in curved space

On Sasaki-Einstein manifolds in dimension five

We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.Comment: 18 pages, Exposition was expanded and a reference adde

The Role of Surgical Expertise and Surgical Access in Retroperitoneal Sarcoma Resection - A Retrospective Study.

Background Retroperitoneal sarcoma (RPS) is a rare disease often requiring multi-visceral and wide margin resections for which a resection in a sarcoma center is advised. Midline incision seems to be the access of choice. However, up to now there is no evidence for the best surgical access. This study aimed to analyze the oncological outcome according to the surgical expertise and also the incision used for the resection. Methods All patients treated for RPS between 2007 and 2018 at the Department of Visceral Surgery and Medicine of the University Hospital Bern and receiving a RPS resection in curative intent were included. Patient- and treatment specific factors as well as local recurrence-free, disease-free and overall survival were analyzed in correlation to the hospital type where the resection occurred. Results Thirty-five patients were treated for RPS at our center. The majority received their primary RPS resection at a sarcoma center (SC = 23) the rest of the resection were performed in a non-sarcoma center (non-SC = 12). Median tumor size was 24 cm. Resections were performed via a midline laparotomy (ML = 31) or flank incision (FI = 4). All patients with a primary FI (n = 4) were operated in a non-SC (p = 0.003). No patient operated at a non-SC received a multivisceral resection (p = 0.004). Incomplete resection (R2) was observed more often when resection was done in a non-SC (p = 0.013). Resection at a non-SC was significantly associated with worse recurrence-free survival and disease-free survival after R0/1 resection (2 vs 17 months; Log Rank p-value = 0.02 respectively 2 vs 15 months; Log Rank p-value < 0.001). Conclusions Resection at a non-SC is associated with more incomplete resection and worse outcome in RPS surgery. Inadequate access, such as FI, may prevent complete resection and multivisceral resection if indicated and demonstrates the importance of surgical expertise in the outcome of RPS resection

Degree formula for connective K-theory

We apply the degree formula for connective $K$-theory to study rational contractions of algebraic varieties. Examples include rationally connected varieties and complete intersections.Comment: 14 page

The cone of curves of Fano varieties of coindex four

We classify the cones of curves of Fano varieties of dimension greater or equal than five and (pseudo)index dim X -3, describing the number and type of their extremal rays.Comment: 27 pages; changed the numbering of Theorems, Definitions, Propositions, etc. in accordance with the published version to avoid incorrect reference

Fibrations on four-folds with trivial canonical bundles

Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and prove that there is no such fibration in the sixth class. We classify all such fibrations whose generic fibre is the Jacobian of a genus two curve.Comment: 28 page

Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.Comment: 72 pages, 6 figures. A shortened version of this paper has appeared in Publications math\'ematiques de l'IH\'ES. The final publication is available at http://www.springerlink.co