Geometry, Topology and Combinatorics of large random simplicial complexes

本课题属于随机拓扑领域，研究大型随机空间的性质，并预测其几何、拓扑和组合性质。随机拓扑和经典拓扑之间的关系类似于统计力学和经典力学之间的关系。当随机空间变得“大”（在某种意义上），即当经典拓扑方法变得不充分时，随机拓扑的预测变得越来越精确。在各种实际应用中，随机拓扑工具可用于大型复杂系统的建模。随机拓扑的方法和结果也可用于纯数学中拓扑的非构造存在性证明。随机简单复合体和光滑紧致流形的第一个模型出现在2006年左右，是当前研究的热点。我们可能会提到随机曲面、随机3流形、随机机制的构形空间和几种不同的随机简单复合体模型。建议的博士项目将集中在新的多参数模型中随机简单复合物的拓扑性质。我们感兴趣的是随机简单复合体的同调性质和发生在某些概率参数临界值处的相变。该项目涉及数学各个领域的工具，如代数拓扑、组合群论、谱分析和概率论要素。

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.