Geometry, Topology and Combinatorics of large random simplicial complexes

本项目属于随机拓扑学领域，研究大型随机空间的性质，并预测其几何、拓扑和组合性质。随机拓扑学与经典拓扑学之间的关系类似于统计力学与经典力学之间的关系，当随机空间变得“大”(在一定意义上)，即当经典拓扑学方法变得不足时，随机拓扑学的预测变得越来越准确。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对拓扑学中的非构造性存在性证明有用。随机单纯形复形和光滑紧流形的第一个模型在2006年左右出现，是当前深入研究的对象。我们可以提到随机曲面、随机3-流形、随机机构的配置空间，以及几种不同的随机单纯形复杂结构模型。我们对随机单纯复形的同调性质和在某些概率参数的临界值上发生的相变感兴趣。这个项目涉及到不同数学领域的工具，如代数拓扑学、组合群论、谱分析和概率论元素。

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.