Geometry, Topology and Combinatorics of large random simplicial complexes

大型随机单纯复形的几何、拓扑和组合

基本信息

批准号：
1936239
负责人：
金额：
--
依托单位：
Queen Mary University of London
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-1936239
关键词：
Geometry Topology Combinatorics large random

项目摘要

This project belongs to the field of stochastic topology which studies propertiesof large random spaces and predicts their geometric, topological and combinatorialproperties. The relationship between stochastic and classical topologyis similar to the relationship between statistical and classical mechanics.The predictions of stochastic topology become increasingly accurate when therandom space becomes "large" (in certain sense), i.e. when the methods ofclassical topology become inadequate. The tools of stochastic topology may be used formodelling large complex systems in various practical applications. The methodsand results of stochastic topology might also be useful in pure mathematics fornon-constructive existence proofs in topology.First models of random simplicial complexes and smooth compact manifoldsappeared around 2006 and are the object of intensive current research. We maymention random surfaces, random 3-manifolds, configuration spaces of randommechanisms, and several different models of random simplicial complexes.The proposed PhD project will be focused on topological properties of randomsimplicial complexes in the new multi-parameter model. We are interestedin homological properties of random simplicial complexes and in phase transitionswhich happen at some critical values of the probability parameters.The project involves tools from various areas of mathematics such as algebraictopology, combinatorial group theory, spectral analysis and elements ofprobability theory.

该项目属于随机拓扑的领域，该领域研究了大型随机空间的特性，并预测了它们的几何，拓扑和组合型。随机拓扑与古典拓扑之间的关系类似于统计和经典力学之间的关系。随机拓扑学的预测在therandom空间变得“大”（从某些意义上）（即，当阶级拓扑的方法变得不足时）变得越来越准确。随机拓扑的工具可以在各种实际应用中使用大型复杂系统。随机拓扑结构的方法和随机拓扑结构的结果也可能在拓扑结构的纯数学中很有用。拓扑结构的证明。首先在2006年左右左右的随机简单复合物和光滑的紧凑型歧管模型，并且是当前研究的强化对象。我们可能随机表面，随机的3个manifolds，randomenanism的配置空间以及几种不同的随机简单络合物模型。提出的PHD项目将集中在新的多参数模型中随机造型复合物的拓扑特性上。我们感兴趣的是随机简单复合物的同源性能，并且在阶段过渡中发生在概率参数的某些临界值中。该项目涉及来自数学领域的工具，例如代数分类学，组合群体理论，光谱分析和探索理论的元素。