IRFP: Topological Representations of Matroids and the Geometry of Phylogenetic Trees

国际研究奖学金项目使美国科学家和工程师能够在国外进行9至24个月的研究。该计划的奖励为联合研究提供了机会，并利用独特或互补的设施、专业知识和国外的实验条件。该奖项将支持Matthew T. Stamps博士与瑞典斯德哥尔摩皇家理工学院（KTH）的Svante Linusson博士进行为期24个月的研究。一个相对较新的数学领域是拓扑组合学，顾名思义，它关注的是组合学和代数拓扑之间的相互作用。其主要思想是将一个真正的组合问题转化为一个拓扑问题，其解决方案既得到了很好的研究，又回答了手边的原始问题。在过去的几十年里，这种方法带来了许多重大突破；事实上，在组合学中有许多深奥的定理，其唯一已知的证明需要拓扑技术。本项目考虑了这些技术在拟阵理论和进化生物学中的几种应用。拟阵是捕获独立性概念的（离散）数学对象。它们最常出现在组合学和优化学中，但它们也出现在球体上的圆的排列（以及高维类似物）中，这些排列可以由称为空间图的同伦极限的拓扑对象构造。PI的主要目的是建立一个明确的角色，在这个角色中，同伦极限可以为矩阵理论中的一系列问题提供新的解决方案，更一般地说，在代数组合学中。该项目的第二个组成部分是探索系统发育树的边积空间的几何形状，这是进化生物学的一个概率模型，旨在使分类学家能够处理缺失的信息（这是一个重要的特征，因为在某些地区，新物种的发现相当频繁）。这条研究路线结合了几个数学学科的技术，包括交换代数、组合学、离散几何、代数拓扑和概率论，同时发展了PI与瑞典斯德哥尔摩皇家理工学院（KTH） （Anders博士Björner和Svante Linusson博士）、芬兰赫尔辛基阿尔托大学（Alexander博士Engström）的研究小组之间的合作。以及德国柏林的自由博物馆Universität （g<s:1> nter Ziegler博士）。

The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four-month research fellowship by Dr. Matthew T. Stamps to work with Dr. Svante Linusson at the Royal Institute of Technology (KTH) in Stockholm, Sweden.A relatively new field of mathematics is topological combinatorics, which concerns, as its name suggests, the interactions between combinatorics and algebraic topology. The main idea is to translate a genuine combinatorial question into a topological problem whose solution is both well studied and answers the original question at hand. This approach has led to many significant breakthroughs over the past several decades; in fact, there are a number of deep theorems in combinatorics whose only known proofs require topological techniques. This project considers several applications of those techniques to matroid theory and evolutionary biology. Matroids are (discrete) mathematical objects that capture the notion of independence. They most frequently appear in combinatorics and optimization, but they also arise from arrangements of circles on spheres (along with higher dimensional analogs) that can be constructed from topological objects called homotopy colimits of a diagram of spaces. The primary aim of the PI is to establish an explicit role in which homotopy colimits can provide novel solutions to a collection of problems in matroid theory and, more generally, in algebraic combinatorics. A secondary component of the project explores the geometry of the edge-product space of phylogenetic trees, a probabilistic model from evolutionary biology aimed at allowing taxonomists to work with missing information (an important feature given that, in some areas, new species are discovered rather frequently). This line of research incorporates techniques from several disciplines in mathematics, including commutative algebra, combinatorics, discrete geometry, algebraic topology, and probability theory, while developing collaboration between the PI and research groups at the Royal Institute of Technology (KTH) in Stockholm, Sweden (Dr. Anders Björner and Dr. Svante Linusson), Aalto University in Helsinki, Finland (Dr. Alexander Engström), and the Freie Universität - Berlin in Germany (Dr. Günter Ziegler).