The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

A note on the regularity of the free boundaries in the optimal partial transport problem

In this work we study the global regularity of the free boundaries arising in the optimal partial transport problem. Assuming the supports of both the source and the target measure to be convex, we show that the free boundaries of the active regions are globally C ^0,1/2.      

Three equivalent conjectures on the birational geometry of Fano 3-folds

I propose three equivalent conjectures on the birational geometry of Fano 3-folds. Roughly speaking, they suggest that ergodic, or chaotic, behaviour does not occur for Fano 3-folds.     

On groups whose subgroups are closed in the profinite topology

A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

We prove C ¹ regularity of c-convex weak Alexandrov solutions of a Monge–Ampère type equation in dimension two, assuming only a bound from above on the Monge–Ampère measure. The Monge–Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151–183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C ¹ regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge–Ampère equation det D ² u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841–845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Understanding the inelastic electron-tunneling spectra of alkanedithiols on gold

We present results for a simulated inelastic electron-tunneling spectra (IETS) from calculations using the "gDFTB" code. The geometric and electronic structure is obtained from calculations using a local-basis density-functional scheme, and a nonequilibrium Green's function formalism is employed to deal with the transport aspects of the problem. The calculated spectrum of octanedithiol on gold(111) shows good agreement with experimental results and suggests further details in the assignment of such spectra. We show that some low-energy peaks, unassigned in the experimental spectrum, occur in a region where a number of molecular modes are predicted to be active, suggesting that these modes are the cause of the peaks rather than a matrix signal, as previously postulated. The simulations also reveal the qualitative nature of the processes dominating IETS. It is highly sensitive only to the vibrational motions that occur in the regions of the molecule where there is electron density in the low-voltage conduction channel. This result is illustrated with an examination of the predicted variation of IETS with binding site and alkane chain length.     

Hölder Continuity and Injectivity of Optimal Maps

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y). If the source density f ⁺(x) is bounded away from zero and infinity in an open region ${U' \subset \mathbf{R}^n}$ , and the target density f ^?(y) is bounded away from zero and infinity on its support ${\overline{V} \subset \mathbf{R}^n}$ , which is strongly c-convex with respect to U′, and the transportation cost c satisfies the ${\mathbf{A3}_{\rm w}}$ condition of Trudinger and Wang (Ann Sc Norm Super Pisa Cl Sci 5, 8(1):143–174, 2009), we deduce the local Hölder continuity and injectivity of the optimal map inside U′ (so that the associated potential u belongs to ${C^{1,\alpha}_{loc}(U')}$ ). Here the exponent α > 0 depends only on the dimension and the bounds on the densities, but not on c. Our result provides a crucial step in the low/interior regularity setting: in a sequel (Figalli et al., J Eur Math Soc (JEMS), 1131–1166, 2013), we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. Three key tools are introduced in the present paper. Namely, we first find a transformation that under ${\mathbf{A3}_{\rm w}}$ makes c-convex functions level-set convex (as was also obtained independently from us by Liu (Calc Var Partial Diff Eq 34:435–451, 2009)). We then derive new Alexandrov type estimates for the level-set convex c-convex functions, and a topological lemma showing that optimal maps do not mix the interior with the boundary. This topological lemma, which does not require ${\mathbf{A3}_{\rm w}}$ , is needed by Figalli and Loeper (Calc Var Partial Diff Eq 35:537–550, 2009) to conclude the continuity of optimal maps in two dimensions. In higher dimensions, if the densities f ^± are Hölder continuous, our result permits continuous differentiability of the map inside U′ (in fact, ${C^{2,\alpha}_{loc}}$ regularity of the associated potential) to be deduced from the work of Liu et al. (Comm Partial Diff Eq 35(1):165–184, 2010).