Challenges of Applied Algebraic Topology

在这项研究中，我们将讨论理解与新型连杆机构有关的构型空间的拓扑性质的数学挑战，包括带伸缩支腿的连杆机构和固定角度的连杆机构，以及在代数几何和数学物理以及一些统计问题中出现的多维连杆机构。具有伸缩支腿的连杆机构的构形空间(即具有可变长度的支腿)一般是带角的流形，我们计划使用Morse理论技术来寻找连杆机构的度量参数与其构形空间的拓扑不变量之间的关系。定角连接及其构型空间对蛋白质骨架的各种形状提供了一个很好的近似，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。多维连杆机构的构型空间一般是具有奇点的流形；它们的研究代表着重大的数学挑战，需要新的数学工具。我们还计划应用概率-拓扑混合技术，在杆的杆数很大(趋于无穷)的假设下，研究具有随机长度参数的(各种类型)连杆机构的拓扑不变量。我们希望能够将以前获得的这种类型的结果推广到新的重要的链接类。作为这项研究的一部分，我们还将使用应用代数拓扑学的方法和结果来解决一个著名的经典拓扑问题，即怀特黑德猜想。它是由J.H.C.怀特黑德在1941年提出的，尽管数学家们多次尝试，它仍然是开放的。怀特黑德猜想声称，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)证明了这一猜想的概率版本。更准确地说，我们证明了随机产生的非球面2-络合物满足概率趋于1的怀特黑德猜想。在这项研究中，我们将尝试利用这些概率结果，希望获得对该猜想的完全确定解。

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.