Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality

This paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint. In this paper we reformulate the inequality of Vafa–Witten [42] in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa–Witten, in the sense of an explicit control of the Vafa–Witten bound under conformal changes of metrics. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated with conformal weights on noncommutative tori, and spectral triples associated with duals of torsion-free discrete cocompact subgroups satisfying the Baum–Connes conjecture.     

Projectively flat Finsler 2-spheres of constant curvature

After recalling the structure equations of Finsler structures on surfaces, I define a notion of "generalized Finsler structure" as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of "generalized path geometry" analogous to that of "generalized Finsler structure". I use these ideas to study the geometry of Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature K and whose geodesic path geometry is projectively flat, i.e., locally equivalent to that of straight lines in the plane. I show that, modulo diffeomorphism, there is a 2-parameter family of projectively flat Finsler structures on the sphere whose Finsler-Gauss curvature K is identically 1.     

A volume comparison theorem for Finsler manifolds

Let be a symmetric Finsler manifold, endowed with the Busemann volume form, and let be its unit disk bundle endowed with the canonical symplectic volume form. It is shown that , where is the volume of the unit disk in . Moreover, equality holds if and only if is Riemannian.

     

Riemann-Finsler Geometry with Applications to Information Geometry

Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce Riemann-Finsler geometry, by which we establish Information Geometry on a much broader base, so that the potential applications of Information Geometry will be beyond statistics.     

On Mascheroni's La geometria del compasso at the beginning of the 19th century

Lorenzo Mascheroni's 1797 work La geometria del compasso, which develops a geometry based solely on compass constructions, is considered by the author as stepping back behind the “demarcation line” of Euclidean geometry. In this work Mascheroni emphasizes the practical aspects of this geometry over a theoretical approach. A century later, in 1899, David Hilbert and his student Michael Feldblum proposed a totally different approach – algebraic and axiomatic – concerning geometric constructions based on various instruments. Taking into account that, at the end of the 18th century, straightedge geometry was also developed, one may ask what happened to the image of instrument-based geometry during the 19th century? By focusing on Mascheroni's book and its reception, this article aims to examine the various views and conceptions of mathematicians with respect to this geometry.     

等周问题的一个初等证明

本文把欧氏平面，半球面和非欧面之中，不含给定边界，含有给定边界和含有边界而且在其上给定端点这样三种等周问题、给以初等、统一的证明。其要点在于把它们的存在性和唯一性简明扼要地归结到下述初等引理，即一个给定凹边边长的四边形的面积以四顶共圆时为其唯一的极大     

To be or to become: how dynamic geometry changes discourse

In this article, we investigate the impact of the introduction of a dynamic geometry environment on mathematical thinking by identifying changes in discourse engendered by its introduction in a high school geometry class. Our focus is on the teacher, and we find significant differences between static and dynamic geometry in terms of the ways in which the teacher talks about geometric objects, makes use of visual artifacts and models geometric reasoning. Even though these changes have major implications for the geometry being studied, they are made only very implicitly in the classroom.     

An unexpected property of quadrilaterals*

These notes discuss several related propositions in geometry that can be explored in a dynamic geometry environment. The propositions involve an unexpected property of quadrilaterals.     

TERNARY OPERATIONS AS PRIMITIVE NOTIONS FOR PLANE GEOMETRY II

We proved in the first part [1] that plane geometry over Pythagorean fields is axiomatizable by quantifier-free axioms in a language with three individual constants, one binary and three ternary operation symbols. In this paper we prove that two of these operation symbols are superfluous.     

The Markov spectra for Fuchsian groups

Applying the Klein model of the hyperbolic plane and identifying the geodesics in with their poles in the projective plane, the author develops a method of determining infinite binary trees in the Markov spectrum for a Fuchsian group. The method is applied to a maximal group commensurable with the modular group and other Fuchsian groups.     

On contact tops and integrable tops

In this paper, we introduce a geometric structure called top, which is a trivialized bundle of plane pencils over a Riemannian 3-manifold, defined as the set of kernels of a circle of l-forms (e.g., of contact and integrable forms) with particular properties with respect to the metric. We classify the manifolds which admit tops and we describe the associated metrics.     

Corrigendum to “Toric codes and finite geometries” [Finite Fields Appl. 45 (2017) 203–216]

This correction adds a necessary hypothesis in Proposition 2.4 of [1]. A counterexample to the previous version was pointed out to the author by Melda Görür and the author thanks her for bringing that to his attention. Theorem 3.2 and Corollary 3.3 are also reformulated so they do not refer to the new, more restricted Proposition 2.4.     

The complexity of plane hyperbolic incidence geometry is ∀∃∀∃

We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ???, but that there is an axiom system, all of whose axioms are ???? sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrable Model of Interacting Elliptic Tops

We suggest a method for constructing a system of interacting elliptic tops. It is integrable and symplectomorphic to the Calogero-Moser model by construction. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 55–64, January, 2006.     

Weighted blowups and mirror symmetry for toric surfaces

This paper explores homological mirror symmetry for weighted blowups of toric varietes. It will be shown that both the A-model and B-model categories have natural semi-orthogonal decompositions. An explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces.     

对话李克正教授：为什么学习代数几何

以与首都师范大学的李克正教授对话的方式,通过其口简要介绍代数几何这门重要学科在我国的早期发展情况,详细回忆李克正教授在国外学习代数几何的过程,以及他回国后的教学与研究成果,并通俗地阐述代数几何方法对于解决重大数论问题所起的关键作用．