CAREER: Matroids, polytopes, and their valuations in algebra and geometry

职业：拟阵、多面体及其在代数和几何中的赋值这一工作是由这样一种哲学驱动的，即在纯数学和应用数学中，许多对象、关系和程序都是通过研究它们背后丰富的离散结构来最好地理解的。PI将研究数值分析（zonotopal空间），不变理论和代数几何（Cox-Nagata环），分枝覆盖的枚举几何（Hurwitz数）和热带几何（热带线性空间和热带齐性空间）中出现的对象。这些对象的几个重要特征被编码在（Coxeter）拟阵、多面体和赋值的组合学中。所获得的结果将在盒样条理论和遗传学的应用。这个研究计划是学术骨干的旧金山弗朗西斯科州立大学-哥伦比亚组合动力学倡议，教师和学生之间的新兴合作，在这两个位置，其中大部分是数学代表性不足的群体的成员。这项倡议的目的是提供有影响力的研究和教学经验，在数学两个服务不足的社区。通过联合课程和研究项目，学生参与他们的第一次国际学术经验，同时作出认真的科学贡献。该提案由组合学和国际科学与工程办公室共同资助。

CAREER: Matroids, polytopes, and their valuations in algebra and geometryThis work is driven by the philosophy that many objects, relationships, and procedures in pure and applied mathematics are best understood by studying the rich discrete structures underlying them. The PI will study objects arising in numerical anaylsis (zonotopal spaces), invariant theory and algebraic geometry (Cox-Nagata rings), enumerative geometry of ramified covers (Hurwitz numbers), and tropical geometry (tropical linear spaces and tropical homogeneous spaces). Several important features of these objects are encoded in the combinatorics of (Coxeter) matroids, polytopes, and valuations. The results obtained will have applications in box spline theory and phylogenetics.This research program is the academic backbone of the San Francisco State University-Colombia Combinatorics Initiative, an emerging collaboration between faculty and students in these two locations, most of whom are members of underrepresented groups in mathematics. The purpose of this initiative is to provide influential research and teaching experiences to two underserved communities in mathematics. Through joint courses and research projects, students participate in their first international academic experience, while making serious scientific contributions.This proposal is being funded jointly by Combinatorics and the Office of International Science and Engineering.