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

国际研究奖学金计划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。该计划的奖项提供了联合研究的机会，以及使用国外独特或互补的设施，专业知识和实验条件。该奖项将支持由马修·T博士提供的为期24个月的研究奖学金。Stamps与瑞典斯德哥尔摩的皇家理工学院（KTH）的Svante Linusson博士合作。一个相对较新的数学领域是拓扑组合学，顾名思义，它涉及组合学和代数拓扑学之间的相互作用。 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，又回答了手头的原始问题。 这种方法在过去几十年中取得了许多重大突破;事实上，组合学中有许多深刻的定理，其唯一已知的证明需要拓扑技术。 这个项目考虑了这些技术在拟阵理论和进化生物学中的几个应用。 拟阵是（离散的）数学对象，它捕获了独立性的概念。 它们最常出现在组合数学和最优化中，但它们也来自于球面上的圆的排列（沿着更高维的类似物），这些圆可以从空间图的同伦余限（homotopy colimits）拓扑对象构造。 PI的主要目的是建立一个明确的作用，其中同伦余极限可以提供新的解决方案，以收集问题的拟阵理论，更一般地说，在代数组合学。 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结构，这是一种进化生物学的概率模型，旨在让分类学家处理缺失的信息（这是一个重要的特征，因为在某些地区，新物种经常被发现）。 这一研究路线结合了数学中几个学科的技术，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PI和瑞典斯德哥尔摩的皇家理工学院（KTH）研究小组之间的合作安德斯·比约纳（Anders Björner）博士和斯万特·利努松（Svante Linusson）博士、芬兰赫尔辛基阿尔托大学（亚历山大·恩斯特罗姆（Alexander Engström）博士）和德国自由大学-柏林（Günter齐格勒博士）。

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).