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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715043
• 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及其与组合学和表示理论的关系。本研究计划涉及这一广泛领域内的两个主题：牛顿-奥库科夫体理论和海森伯格变分理论。