Mathematical Models of Abstract Systems: Knowing abstract geometric forms

Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models

Building a case for a Planck-scale-deformed boost action: the Planck-scale particle-localization limit

"Doubly-special relativity" (DSR), the idea of a Planck-scale Minkowski limit that is still a relativistic theory, but with both the Planck scale and the speed-of-light scale as nontrivial relativistic invariants, was proposed (gr-qc/0012051) as a physics intuition for several scenarios which may arise in the study of the quantum-gravity problem, but most DSR studies focused exclusively on the search of formalisms for the description of a specific example of such a Minkowski limit. A novel contribution to the DSR physics intuition came from a recent paper by Smolin (hep-th/0501091) suggesting that the emergence of the Planck scale as a second nontrivial relativistic invariant might be inevitable in quantum gravity, relying only on some rather robust expectations concerning the semiclassical approximation of quantum gravity. I here attempt to strengthen Smolin's argument by observing that an analysis of some independently-proposed Planck-scale particle-localization limits, such as the "Generalized Uncertainty Principle" often attributed to string theory in the literature, also suggests that the emergence of a DSR Minkowski limit might be inevitable. I discuss a possible link between this observation and recent results on logarithmic corrections to the entropy-area black-hole formula, and I observe that both the analysis here reported and Smolin's analysis appear to suggest that the examples of DSR Minkowski limits for which a formalism has been sought in the literature might not be sufficiently general. I also stress that, as we now contemplate the hypothesis of a DSR Minkowski limit, there is an additional challenge for those in the quantum-gravity community attributing to the Planck length the role of "fundamental length scale".Comment: 12 pages, LaTe

Enumerative aspects of the Gross-Siebert program

We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.Comment: A version of these notes will appear as a chapter in an upcoming Fields Institute volume. 81 page

Facets and Levels of Mathematical Abstraction

International audienceMathematical abstraction is the process of considering and manipulating operations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content connected to particular applications. There is no one single way of performing mathematical abstraction. The term "abstraction" does not name a unique procedure but a general process, which goes many ways that are mostly simultaneous and intertwined ; in particular, the process does not amount only to logical subsumption. I will consider comparatively how philosophers consider abstraction and how mathematicians perform it, with the aim to bring to light the fundamental thinking processes at play, and to illustrate by significant examples how much intricate and multi-leveled may be the combination of typical mathematical techniques which include axiomatic method, invarianceprinciples, equivalence relations and functional correspondences.L'abstraction mathématique consiste en la considération et la manipulation d'opérations, règles et concepts indépendamment du contenu dont les nantissent des applications particulières et du rapport qu'ils peuvent avoir avec les phénomènes et les circonstances du monde réel. L'abstraction mathématique emprunte diverses voies. Le terme " abstraction " ne désigne pasune procédure unique, mais un processus général où s'entrecroisent divers procédés employés successivement ou simultanément. En particulier, l'abstraction mathématique ne se réduit pas à la subsomption logique. Je vais étudier comparativement en quels termes les philosophes expliquent l'abstraction et par quels moyens les mathématiciens la mettent en oeuvre. Je voudrais parlà mettre en lumière les principaux processus de pensée en jeu et illustrer par des exemples divers niveaux d'intrication de techniques mathématiques récurrentes, qui incluent notamment la méthode axiomatique, les principes d'invariance, les relations d'équivalence et les correspondances fonctionnelles

Quantum Gravity as Topological Quantum Field Theory

The physics of quantum gravity is discussed within the framework of topological quantum field theory. Some of the principles are illustrated with examples taken from theories in which space-time is three dimensional.Comment: 23 pages, amstex, JMP special issue (deadline permitting). (Text not changed

Navigation of Spacetime Ships in Unified Gravitational and Electromagnetic Waves

On the basis of a "local" principle of equivalence of general relativity, we consider a navigation in a kind of "4D-ocean" involving measurements of conformally invariant physical properties only. Then, applying the Pfaff theory for PDE to a particular conformally equivariant system of differential equations, we show the dependency of any kind of function describing "spacetime waves", with respect to 20 parametrizing functions. These latter, appearing in a linear differential Spencer sequence and determining gauge fields of deformations relatively to "ship-metrics" or to "flat spacetime ocean metrics", may be ascribed to unified electromagnetic and gravitational waves. The present model is based neither on a classical gauge theory of gravitation or a gravitation theory with torsion, nor on any Kaluza-Klein or Weyl type unifications, but rather on a post-Newtonian approach of gravitation in a four dimensional conformal Cosserat spacetime.Comment: 28 pages. Relative to the second version some changes in the mathematical results have been corrected without consequences in the physical model. The conformally flatness of the substratum spacetime which is an assumption used throughout in the mathematical developements from chapter 2, has been well precised in the first chapter. Clearer explanations at the very end of chapter 3 about accelerating frames are given. New references are indicated and some of them correcte