Equivariant symplectic and algebraic geometry of flag and spherical varieties

旗形簇和球簇的等变辛几何和代数几何

• 批准号: RGPIN-2019-06567
• 负责人: Harada, Megumi
• 金额: $ 2.33万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738976
• 关键词: Equivariant; symplectic; algebraic; geometry; flag

My research is in equivariant symplectic and algebraic geometry and its relation to combinatorics and representation theory. This research proposal addresses 2 topics within this broad area: the theory of Newton-Okounkov bodies, and the theory of Hessenberg varieties.  Newton-Okounkov bodies are a far-reaching generalization of the theory of toric varieties, which connects the combinatorics of convex integral polytopes with the geometry of toric varieties. I have substantively contributed to this research area in a joint paper with Kaveh which builds toric degenerations and integrable systems in a very general setting using the theory of Newton-Okounkov bodies. My long-term research program is to  (i) pursue more links between Newton-Okounkov bodies and symplectic geometry, and (ii) compute and understand concrete examples of Newton-Okounkov bodies and associated geometric invariants, with an emphasis on examples from Combinatorial Algebraic Geometry such as spherical varieties.  Hessenberg varieties and subvarieties of the full flag variety G/B for a complex connected reductive algebraic group G. Hessenberg varieties occupy a central place in the fruitful intersection of algebraic geometry, combinatorics, and representation theory. A fundamental contribution a decade ago was Tymoczko's construction of a permutation-group representation on the cohomology rings of regular semisimple Hessenberg varieties. Crucially, recent work of Shareshian-Wachs, Brosnan-Chow and others have shown that Hessenberg varieties are intimately connected with the theory of symmetric functions, and in particular the deep unsolved Stanley-Stembridge conjecture. I have contributed substantively to this area in recent work with Precup where we use Hessenberg varieties to partially solve the Stanley-Stembridge conjecture. My long-term program in this area is to: (i) further illuminate the theory of symmetric and quasi-symmetric functions using the geometry and combinatorics of Hessenberg varieties, including but not limited to a proof of the Stanley-Stembridge conjecture, (ii) further develop and connect the theories of Newton-Okounkov bodies, Hessenberg varieties, and Schubert calculus, and (iii) develop a systematic theory of integrable systems on Hessenberg varieties.  My proposed research objectives have potential for wide and varied impact. Firstly, it is well-known to be difficult to explicitly construct integrable systems. Therefore, my long-term goal of using my past work with Kaveh (mentioned above) to build new integrable systems has the potential to transform the field and bring rich new applications to other areas, such as representation theory. Secondly, my program of introducing geometric Hessenberg-variety techniques to the study of symmetric and quasisymmetric functions is a hot new research topic and has the potential to become a new toolkit for answering combinatorial questions.

我的研究方向是等变辛几何和代数几何，以及它与组合学和表示论的关系。这一研究方案涉及这一广泛领域内的两个主题：牛顿-奥孔科夫体理论和Hessenberg簇理论。牛顿-奥库科夫体是环簇理论的深远推广，它将凸整多面体的组合学与环簇的几何联系起来。我在与Kaveh的一篇联合论文中对这一研究领域做出了实质性的贡献，该论文使用牛顿-奥孔科夫体理论在非常一般的背景下建立了环形退化和可积系统。我的长期研究计划是：(I)寻求牛顿-Okounkov体与辛几何之间的更多联系，(Ii)计算和理解牛顿-Okounkov体及其相关几何不变量的具体例子，重点是组合代数几何的例子，如球体簇。对于复杂的连通约化代数群，Hessenberg簇及其子簇G/B的全旗簇G/B在代数几何、组合学和表示论的丰硕的交集中占有中心地位。十年前的一个基本贡献是Tymoczko在正则半单Hessenberg簇的上同调环上构造了置换群表示。最重要的是，Shareshian-Wachs、Brosnan-Chow和其他人最近的工作表明，Hessenberg簇与对称函数理论密切相关，特别是与深度未解的Stanley-Stembridge猜想密切相关。在最近与Precup的工作中，我对这一领域做出了实质性贡献，其中我们使用Hessenberg变元来部分解决Stanley-Stembridge猜想。我在这个领域的长期计划是：(I)利用Hessenberg簇的几何和组合学进一步阐明对称和准对称函数的理论，包括但不限于Stanley-Stembridge猜想的证明，(Ii)进一步发展和联系牛顿-Okounkov体理论、Hessenberg簇和Schubert演算，以及(Iii)发展关于Hessenberg簇的可积系的系统理论。首先，显式构造可积系统是众所周知的困难。因此，我的长期目标是利用我过去与Kaveh(如上所述)的工作来建立新的可积系统，这一目标有可能改变这一领域，并为其他领域带来丰富的新应用，如表示理论。其次，我将几何Hessenberg变换技术引入对称和拟对称函数的研究是一个热门的新研究课题，并有可能成为回答组合问题的新工具包。