Challenges of Applied Algebraic Topology

在这项研究中，我们将解决数学的挑战，理解与新型连杆机构相关的配置空间的拓扑性质，包括伸缩腿和固定角度的连杆机构，以及在代数几何和数学物理中出现的多维连杆机构和一些统计问题。具有伸缩腿（即具有可变长度的腿）的连杆机构的位形空间是具有角的一般流形，并且我们计划使用莫尔斯理论技术来寻找连杆机构的度量参数与其位形空间的拓扑不变量之间的关系。固定角连接和它们的构型空间提供了对蛋白质骨架的各种形状的良好近似，因此关于这些空间的拓扑性质的信息可以潜在地用于蛋白质折叠问题。多维连杆机构的位形空间一般是具有奇点的流形，它们的研究代表了重大的数学挑战，需要新的数学工具。我们还计划应用混合概率拓扑技术和研究拓扑不变量的连杆机构（各种类型）与随机长度参数的假设下，连杆机构的杆数是大的（趋于无穷大）。我们希望能够概括以前获得的结果，这种类型的新的重要类的联系。作为这项研究的一部分，我们还将使用应用代数拓扑的方法和结果来解决一个著名的经典拓扑问题，称为怀特海猜想。是J.H.C.提出的怀特黑德在1941年和仍然开放，尽管多次尝试的数学家。怀特海猜想声称非球面二维复形的子复形也是非球面的。我们最近的结果（2012）证明了该猜想的概率版本。更准确地说，我们证明了随机产生的非球面2-复合物满足Whitehead猜想，概率趋于1。在这项研究中，我们将试图利用这些概率结果，希望获得一个完整的确定性解决方案的猜想。

In this research we shall address mathematical challenges of understanding the topological properties of configuration spaces associated with linkages of new types, including linkages with telescopic legs and fixed-angle linkages, as well as multidimensional linkages which arise in algebraic geometry and mathematical physics and in some problems of statistics. The configuration spaces of linkages with telescopic legs (i.e. legs having variable lengths) are generically manifolds with corners and we plan using Morse theory techniques for finding relations between the metric parameters of linkages and the topological invariants of their configuration spaces. The fixed-angle linkages and their configuration spaces provide a good approximation to the variety of shapes of protein backbones and therefore information about topological properties of these spaces can be potentially used in the protein folding problem. Configuration spaces of multidimensional linkages are generically manifolds with singularities; their study represents significant mathematical challenges and requires new mathematical tools. We also plan to apply mixed probabilistic-topological techniques and study topological invariants of linkages (of various types) with random length parameters under the assumption that the number of bars of the linkage is large (tends to infinity). We hope to be able to generalise the previously obtained results of this type to new important classes of linkages. As part of this research we will also use the methods and results of applied algebraic topology to tackle a well-known classical topological problem known as the Whitehead conjecture. It was raised by J.H.C. Whitehead in 1941 and remains open despite multiple attempts of mathematicians. The Whitehead conjecture claims that a subcomplex of an aspherical 2-dimensional complex is also aspherical. Our recent results (2012) prove a probabilistic version of the conjecture. More precisely, we showed that aspherical 2-complexes produced randomly satisfy the Whitehead conjecture with probebility tending to one. In this research we shall try to exploit these probabilistic results hoping to obtain a full deterministic solution to the conjecture.