Combinatorial Commutative Algebra and Integer Programming

这个研究项目涉及交换代数、整数规划和离散几何的交叉问题。这些领域之间的联系是通过历史理想和它们的groebnerbase建立起来的。第一组问题涉及无嵌入素数的环形初始理想的表征和构造。这些问题的答案将通过Gomory的群预化的经典技术在整数规划理论中提供新的结构结果，并将扩展serkan Hosten和该提议者最近在该领域的工作。第二组问题涉及到环面希尔伯特格式，这是一个参数空间，所有多项式理想具有相同的多阶希尔伯特函数作为给定的环面理想。关于这些计划的一个悬而未决的问题是，它们之间是否存在联系。提议者将与Diane Maclagan一起研究这个连通性问题，建立在他们最近关于方案的单项式理想的构建图上，当且仅当方案是连接的。在过去的十年中，他利用代数技术为离散优化的几个新理论的发展做出了贡献，并致力于将所得的非传统算法应用于实际问题。为这些理论做出贡献的研究带来了代数、组合学、优化和离散几何的工具，以解决所有这些领域的问题。代数技术的应用使人们对离散优化的一个分支整数规划领域有了新的认识，而优化的工具和思想也使代数有了新的结果。这种思想和技术的相互作用揭示了这些领域的问题，并有望带来更大的洞察力和进步。该提案与学生在研究的一些计算方面进行合作，并计划让学生参与实现这些算法的软件开发。申请人还对华盛顿大学与本研究相关的学科领域的课程开发感兴趣。

This research program concerns questions at the intersection ofcommutative algebra, integer programming, and discrete geometry. Thelink between these fields is made via toric ideals and their Groebnerbases. The first set of questions concern the characterization andconstruction of toric initial ideals without embedded primes. Answersto these questions will provide new structural results in the theoryof integer programming via the classical technique of grouprelaxations due to Gomory and will extend recent work in this area bySerkan Hosten and the proposer. The second set of questions concern the toric Hilbert scheme, a parameter space for all polynomial idealswith the same multi-graded Hilbert function as a given toric ideal. Acentral open question about these schemes is whether they areconnected. The proposer will work with Diane Maclagan on this connectivity question, building on their recent construction of agraph on the monomial ideals of the scheme which is connected if andonly if the scheme is connected. Over the past decade, the proposer has contributed to the developmentof several new theories in discrete optimization using algebraictechniques and has worked on the application of the resulting,non-traditional algorithms to practical problems. The research thathas contributed to these theories brings tools from algebra,combinatorics, optimization and discrete geometry to bear on problemsfrom all of these fields. The application of algebraic techniques hasled to new understanding in the field of integer programming, a branchof discrete optimization, but tools and ideas from optimization havealso led to new results in algebra. This interplay of ideas andtechniques has shed light on questions from each of these areas andpromises to lead to greater insight and advancement. The proposerworks with students on some of the computational aspects of theresearch and plans to involve students in the development of softwarethat implements these algorithms. The proposer is also interested incurriculum development in subject areas related to this research atthe University of Washington.