Combinatorics in geometry and algebra with applications to the natural sciences

在数学上，研究重点是组合数学，几何和代数之间的相互作用。 这些项目围绕自然科学的具体应用，包括进化生物学，医学成像，化学反应动力学，算法和计算复杂性以及理论高能物理。 更具体地说，这些项目包括：（1）在系统发育树空间中寻找质心的算法，以及多面体空间上分段可微函数的一般梯度优化方法，并将其应用于进化生物学和医学成像，以及凸多面体球体的计算复杂性;（2）二项式交换代数在化学中质量作用动力学的动力学和稳定性方面的应用;（3）组合博弈论的有效算法，它是基于涉及生成函数的技术，利用与交换代数的联系;（4）在高能物理中出现的具有大群作用的代数簇的交换代数和代数几何。近年来，现代组合学、几何学和代数学之间的相互作用越来越丰富。 这项研究将探索和深化这些连接在他们的接口与其他一些学科，包括生物学，化学，物理学，计算机科学，统计学和运筹学。 具体的数学项目集中在计算几何和优化;多项式代数;和对称性存在下的光滑几何。 尽管在数学及其应用中提出的智力活动范围很广，但它们都是通过统一几何和代数的主题而结合在一起的。 通过支持人员（本科生、研究生和博士后研究人员），该项目将通过上述领域的研究对教育产生实质性的积极影响。 促进这些领域之间的联系，在人员之间以及在科学发现中，是一个明确的中心目标。

Mathematically, the research focuses on interactions between combinatorics, geometry, and algebra. The projects are centered around specific applications to natural sciences, including evolutionary biology, medical imaging, dynamics of chemical reactions, algorithms and computational complexity, and theoretical high-energy physics. More specifically, the projects include: (1) algorithms for finding centroids in spaces of phylogenetic trees, and general gradient optimization methods for piecewise differentiable functions on polyhedral spaces, with applications to evolutionary biology and medical imaging, and to computational complexity of convex polyhedral spheres; (2) applications of commutative algebra of binomials to dynamics and stability of mass action kinetics in chemistry; (3) efficient algorithms for combinatorial game theory using connections to commutative algebra, based on techniques involving generating functions; and (4) commutative algebra and algebraic geometry of algebraic varieties, with large group actions, arising in high-energy physics.Interactions between modern combinatorics, geometry, and algebra have become increasingly rich in recent years. This research will explore and deepen these connections at their interfaces with a number of other disciplines, including biology, chemistry, physics, computer science, statistics, and operations research. The specific mathematical projects focus on computational geometry and optimization; the algebra of polynomials; and smooth geometry in the presence of symmetry. Despite the range of the proposed intellectual activities in mathematics and its applications, they are drawn together by unifying themes in geometry and algebra. By supporting personnel (undergraduates, graduate students, and postdoctoral researchers), this project will have substantial positive effects on education through research in the aforementioned areas. Fostering ties between these areas, among personnel as well as in scientific discovery, is an explicit, central objective.