Morse Spectra, Homology Measures, Spaces of Cycles and Parametric Packing Problems
Morse Spectra, Homology Measures, Spaces of Cycles and Parametric Packing Problems
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莫尔斯谱、同调测度、循环空间和参数填充问题
DOI：
10.2307/j.ctvthhdvv.10
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发表时间：
2019
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影响因子：
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通讯作者：
M. Gromov
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作者：
M. Gromov
空间 X 中（有限或无限多个）粒子的“系综” Ψ = Ψ(X)，例如在欧几里得 3 空间中，通常用集合函数 U ↦ entU(Ψ) = ent(Ψ∣U), U ⊂ X 来表征，该函数将 Ψ 的 U 约简 Ψ∣U 的熵分配给所有有界开子集 U ⊂ X。用物理学家的话说，这个熵是“ E 的状态可以从 U 有效观察”，这 在数理统计力学的背景下，“定义”被翻译成测度/概率论的语言。1但是，如果“有效可观察的状态数”被“运动粒子系综的有效/持久自由度数”取代，会发生什么？我们在本文中提出了“持久自由度”概念的几个数学对应物，并提出了具体问题，其中许多问题受到拉里·古斯（Larry Guth）关于球循环空间上体积能量函数的莫尔斯（共）同调谱的赫尔曼·韦尔（Hermann Weyl）类渐进性的结果和想法的启发。2并且我们经常在本文的不同部分中提出同一想法的不同方面。在我们的论文中几乎找不到任何可以称为“新定理”的东西，但我们重新整理了许多已知的结果，并从特定的角度揭示了它们。本文旨在作为尚未撰写的内容的介绍性章节，其中我们在此介绍的大部分内容摘自我尚未完成的手稿《问题数量》。 1 概念和示例概述。下面我们介绍“参数化打包”的思想以及相关概念，这些概念将在本文的其余部分详细扩展。 A. 设 X 为拓扑空间，例如一个流形，I 是一个可能是有限的可数索引集，特别是如果 X 是紧凑的。非空开放（有时是封闭）子集 Ui ⊂ X, i ∈ I 的 I 元组集合称为 X 的打包或 I 打包（如果这些子集不相交）。用 Ψ(X; I) 表示这些具有某种自然拓扑的堆积的空间，其中，如果 X 是非紧的，则观察到这种拓扑有多个候选者。 1参见：Lanford 的《经典统计力学中的熵和平衡状态》，物理学讲义，第 20 卷，第 1-113 页，1973 年和 Ruelle 的《热力学形式主义：经典平衡统计力学的数学结构》，第 2 版，剑桥数学图书馆 2004 年，其中重点放在（离散）晶格系统上。 “寻找结构，第 1 部分：关于熵”中还建议了玻尔兹曼-香农熵的分类演绎，www.ihes.fr/∼gromov/PDF/structre-serch-entropy-july5-2012.pdf 2与杯幂和 Steenrod 平方相关的极小极大问题，几何和泛函分析，18 (6)， 1917-1987（2009）。
An ”ensemble” Ψ = Ψ(X) of (finitely or infinitely many) particles in a space X, e.g. in the Euclidean 3-space, is customary characterised by the set function U ↦ entU(Ψ) = ent(Ψ∣U), U ⊂ X, that assigns the entropies of the U-reductions Ψ∣U of Ψ, to all bounded open subsets U ⊂ X. In the physicists’ parlance, this entropy is ”the logarithm of the number of the states of E that are effectively observable from U”, This ”definition”, in the context of mathematical statistical mechanics, is translated to the language of the measure/probability theory.1 But what happens if ”effectively observable number of states” is replaced by ”the number of effective/persistent degrees of freedom of ensembles of moving particles”? We suggest in this paper several mathematical counterparts to the idea of ” persistent degrees of freedom ” and formulate specific questions, many of which are inspired by Larry Guth’s results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in balls.2 And often we present variable aspects of the same idea in different sections of this paper. Hardly anything that can be called ”new theorem” can be found in our paper but we reshuffle many known results and expose them from a particular angle. This article is meant as an introductory chapter to something yet to be written with much of what we present here extracted from my yet unfinished manuscript Number of Questions. 1 Overview of Concepts and Examples. We introduce below the idea of ”parametric packing” and of related concepts which are expanded in detail in the rest of the paper. A. Let X be a topological space, e.g. a manifold, and I is a countable index set that may be finite, especially if X is compact. A collection of I-tuples of non-empty open (sometimes closed) subsets Ui ⊂ X, i ∈ I, is called a packing or an I-packing of X if these subsets do not intersect. Denote by Ψ(X; I) the space of these packings with some natural topology, where, observe there are several candidates for such a topology if X is noncompact. 1See: Lanford’s Entropy and equilibrium states in classical statistical mechanics, Lecture Notes in Physics, Volume 20, pp. 1-113, 1973 and Ruelle’s Thermodynamic formalism : the mathematical structures of classical equilibrium statistical mechanics, 2nd Edition, Cambridge Mathematical Library 2004, where the emphasis is laid upon (discrete) lattice systems. Also a categorical rendition of Boltzmann-Shannon entropy is suggested in ”In a Search for a Structure, Part 1: On Entropy”, www.ihes.fr/∼gromov/PDF/structre-serch-entropy-july5-2012.pdf 2Minimax problems related to cup powers and Steenrod squares, Geometric and Functional Analysis, 18 (6), 1917-1987 (2009).