Collège de France

About: Collège de France is a education organization based out in Paris, France. It is known for research contribution in the topics: Population & Dopamine. The organization has 6541 authors who have published 11983 publications receiving 648742 citations. The organization is also known as: College de France.

Noncommutative Finite-Dimensional Manifolds. I. Spherical Manifolds and Related Examples

Abstract: We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations \(\) of the standard 3-sphere S3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space ℝ4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed ℝ4u only depend on two parameters and are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. It follows that different \(\) can span the same \(\). This equivalence generates a foliation of the parameter space Σ. This foliation admits singular leaves reduced to a point. These critical points are either isolated or fall in two 1-parameter families \(\). Up to the simple operation of taking the fixed algebra by an involution, these two families are identical and we concentrate here on C+. For \(\) the above isomorphism with the Sklyanin algebra breaks down and the corresponding algebras are special cases of θ-deformations, a notion which we generalize in any dimension and various contexts, and study in some detail. Here, and this point is crucial, the dimension is not an artifact, i.e. the dimension of the classical model, but is the Hochschild dimension of the corresponding algebra which remains constant during the deformation. Besides the standard noncommutative tori, examples of θ-deformations include the recently defined noncommutative 4-sphere \(\) as well as m-dimensional generalizations, noncommutative versions of spaces \(\) and quantum groups which are deformations of various classical groups. We develop general tools such as the twisting of the Clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative \(\)-dimensional spherical manifolds \(\). A key result is the differential self-duality properties of these projections which generalize the self-duality of the round instanton.

The Dynamics of Inequality

Abstract: The past forty years have seen a rapid rise in top income inequality in the United States While there is a large number of existing theories of the Pareto tail of the long-run income distributions, almost none of these address the fast rise in top inequality observed in the data We show that standard theories, which build on a random growth mechanism, generate transition dynamics that are too slow relative to those observed in the data We then suggest two parsimonious deviations from the canonical model that can explain such changes: “scale dependence” that may arise from changes in skill prices, and “type dependence,” that is, the presence of some “high-growth types” These deviations are consistent with theories in which the increase in top income inequality is driven by the rise of “superstar” entrepreneurs or managers

Unified picture of anionic redox in Li/Na-ion batteries.

Abstract: Anionic redox in Li-rich and Na-rich transition metal oxides (A-rich-TMOs) has emerged as a new paradigm to increase the energy density of rechargeable batteries. Ever since, numerous electrodes delivering extra anionic capacity beyond the theoretical cationic capacity have been reported. Unfortunately, most often the anionic capacity achieved in charge is partly irreversible in discharge. A unified picture of anionic redox in A-rich-TMOs is designed here to identify the electronic origin of this irreversibility and to propose new directions to improve the cycling performance of the electrodes. The electron localization function is introduced as a holistic tool to unambiguously locate the oxygen lone pairs in the structure and follow their participation in the redox activity of A-rich-TMOs. The charge-transfer gap of transition metal oxides is proposed as the pertinent observable to quantify the amount of extra capacity achievable in charge and its reversibility in discharge, irrespective of the material chemical composition. From this generalized approach, we conclude that the reversibility of the anionic capacity is limited to a critical number of O holes per oxygen, hO ≤ 1/3.