权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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.

本项目属于随机拓扑学的范畴，主要研究大型随机空间的性质，并预测其几何、拓扑和组合性质。随机拓扑学与经典拓扑学的关系类似于统计力学与经典力学的关系，当随机空间变得“大”（在某种意义上）时，即当经典拓扑学的方法变得不足时，随机拓扑学的预言就变得越来越精确。随机拓扑学的工具可用于各种实际应用中的大型复杂系统的建模。随机拓扑学的方法和结果也可能在纯数学中用于拓扑学中的非构造性存在性证明。随机单纯复形和光滑紧致流形的第一个模型出现在2006年左右，是当前深入研究的对象。我们可能会提到随机表面，随机三维流形，随机机构的配置空间，以及几种不同的随机单纯复形模型。拟议的博士项目将集中在新的多参数模型中的随机单纯复形的拓扑性质。我们对随机单纯复形的同调性质和发生在概率参数的某些临界值处的相变感兴趣。该项目涉及数学各个领域的工具，如代数拓扑学、组合群论、谱分析和概率论元素。