Graph Theory: Confluences in Molecular Biology and the Physical Sciences

图在物理学、分子生物学、数论、Teichmueller理论、动力系统、流体动力学和3流形不变量等领域中有一定的融合，这反过来反映了它们的理论基础之间可能存在更深层次的关系，而这些关系尚未得到发展。研究人员组织了一个研讨会，以考虑数学结构在与数学日益相关的领域的具体问题中的应用，例如大分子折叠的全局几何，包括蛋白质和RNA。最近的数学结果允许对完整的空间折叠问题进行处理：大分子的物理化学性质如何决定它们的空间折叠特征？粗略地说，折叠可以被建模成一组加权弧线，每个弧线对应一个化学键，权重是键的玻尔兹曼强度。由这些加权图组合而成的图给出了一个表示所有可能折叠的加权图空间。能否在这样的空间中找到一个合适的能量泛函来研究总能量最小的梯度流？一些参与者使用这种方法来理解动态折叠。在生物学中，图的相关空间是由物种形成的系统发生树的空间。虽然我们了解了这些空间的全局几何，但局部几何的性质却不为人所知；一些参与者研究这些问题。弦理论中最近的一个发现是与图空间相关的“操作符”的奇特结构。研讨会上的一些研究人员研究了这些更高级的有序结构，以及它们在生物学中的可能用途。图论是处理被称为图的对象的数学的一个分支。图由一组点或顶点组成，这些点或顶点由链接或边连接起来。例如，普通地图是一个顶点是城镇，边缘是道路的图形。虽然图形是抽象的对象，但它们已经被用来表示各种各样的实际对象或过程。图表的常见用途是表示一组生物体之间的进化关系——家谱。图形在生物学、经济学和计算机科学等不同领域的应用证明了图形的多功能性和实用性。在生物技术领域，用图形来表示由DNA产生的基本蛋白质的复杂物理结构已经出现了爆炸式的增长。这项工作在理解遗传疾病和设计治疗这些疾病的药物方面具有重要的应用。因为基于图论的方法在许多领域都是独立发展的，所以在不同领域之间共享问题和结果很可能会带来进步。本次研讨会的目的是将致力于图论问题的数学家和生物学家聚集在一起，在跨学科的环境中描述和讨论他们的工作。

Solow0109738 There are certain confluences involving graphs in physics,molecular biology, number theory, Teichmueller theory, dynamicalsystems, fluid dynamics, and 3-manifold invariants, which in turnreflect possibly deeper relations among their theoreticalfoundations which have yet to be developed. The investigatorsorganize a workshop to consider mathematical structures withapplication to specific problems in fields increasingly connectedto mathematics, for example the global geometry of macromolecularfolding, including both proteins and RNA. Recent mathematicalresults allow for a treatment of the full spatial foldingproblem: how do the physico-chemical properties of macromoleculesdetermine their spatial folding characteristics? Roughly,folding can be modeled as a family of weighted arcs, each arccorresponding to a chemical bond, the weight the Boltzmannstrength of the bond. The graphs formed from combining theseweighted ars give a space of weghted graphs representing allpossible foldings. Can an appropriate energy functional be foundon such a space to study the gradient flow minimizing the totalenergy? Several participants use this approach to understanddynamic folding. Related spaces of graphs in biology are those ofphylogenetic trees formed by speciation. Though the globalgeometry of these spaces is understood, the nature of the localgeometry is not known; some of the participants study thesequestions. A recent discovery in string theory is the exoticstructure of an "operad" associated with spaces of graphs. Someof the researchers at the workshop investigate these higher orderstructures, and their possible use in biology. Graph theory is a branch of mathematics dealing with objectscalled graphs. A graph consists of a set of points or verticesthat are connected by links or edges. An ordinary map, forexample, is a graph whose vertices are towns and whose edges areroads. Although graphs are abstract objects, they have been usedto represent a wide variety of real objects or processes. Afamiliar use of graphs is to represent the evolutionaryrelationships between a group of organisms -- family trees. It isa testimony to their great versatility and usefulness that graphshave been used in fields as diverse as biology, economics, andcomputer science. In the field of biotechnology, there has beenan explosion in the use of graphs to represent the complexphysical structure of the basic proteins produced by DNA. Thiswork has important applications in understanding genetic diseasesand in designing medicines to cure them. Because methods based ongraph theory have been developed independently in many fields, itis extremely likely that sharing problems and results betweenfields will lead to advances. The purpose of this workshop is tobring together mathematicians and biologists working on problemsinvolving graph theory to describe and discuss their work in aninterdisciplinary setting.