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The Weyl algebra over a field k of characteristic 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z-graded simple rings A of any dimension which have a graded quotient ring of the form K[t, t-1; σ] for a field K. Under some further hypotheses, we classify all such A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdimA = tr. deg(K/k) + 1, we show that K and σ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry. [ABSTRACT FROM AUTHOR]

The article surveys the progress in applying techniques from moduli spaces in mathematics and physics to studying the geometry and topology of three families of macromolecules of interest in biology, including RNAs, proteins and polysaccharides. It is intended for audience such as mathematicians whose expertise does not necessarily extend to either moduli spaces or macromolecules and who may be interested in the nascent application of geometry to problems in biology.

We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving "thin" orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba's Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above. [ABSTRACT FROM AUTHOR]

The article presents a study on intrinsic geometry of disk complex D(M,S). The disk complex has a natural tendency to form a curve complex C(S) which is not a quasi-isometric embedding since some disks may be close to the curve complex but are far in the disk complex. Subsurfaces holes may be formed and these paths in the disk complex must traverse into and out of these holes. The paths formed in the curve complex may skip over a hole by using a vertex to show the boundary of the subsurface.

The article discusses pointwise C2,α estimates at boundary points under specific local conditions on the right hand side and boundary data. The result can be viewed as a boundary Schauder estimate for the Monge-Ampere equation which extends up to the boundary the pointwise interior C2,α estimate Caffarelli (C2). These estimates are reportedly important when dealing with fourth order Monge-Amp`re type equations in geometry.

We study some geometric properties of a compact Kähler manifold (Mn, g) under a certain condition on the bisectional curvature. As an application, we give a new proof for an earlier result which asserts that any boundary class of the Kähler cone of Mn can be represented by a C∞ closed (1,1) form that is parallel and everywhere nonnegative. [ABSTRACT FROM AUTHOR]

The article discusses the uniqueness of enhancement for triangulated categories in mathematics. It looks into the aspects of differential graded (DG) categories, quasi-functors and the quotients of DG categories. It also explores the applications of triangulated categories to commutative and non-commutative geometry, as well as coherent sheaves.

The article examines the problem of a geometric criterion for the breakdown of solutions of the vacuum Einstein equations. It is assumed that a part of space-time is foliated by the level of hypersurface of a time function monotonically increasing towards the future. The space-time region is also assumed to be globally hyperbolic with every causal curve from a point intersecting precisely one point.