Equivariant symplectic and algebraic geometry of flag and spherical varieties

我的研究是在等变辛和代数几何和它的关系，组合和代表性理论。本研究计划在这一广泛的领域内解决了两个主题：牛顿-奥肯科夫体理论和海森伯格簇理论。** 牛顿-奥肯科夫体是复曲面簇理论的一个意义深远的推广，它将凸积分多面体的组合学与复曲面簇的几何学联系起来。我有实质性的贡献，这一研究领域的联合文件与Kaveh建立环面退化和可积系统在一个非常一般的设置使用理论的牛顿Okounkov机构。我的长期研究计划是（i）追求牛顿-奥肯科夫机构和辛几何之间的更多联系，（ii）计算和理解牛顿-奥肯科夫机构和相关几何不变量的具体例子，重点是组合代数几何的例子，如球形品种。复连通约化代数群G的满旗簇G/B的 **Hessenberg簇和子簇.海森堡簇在代数几何、组合数学和表示论的丰硕交叉中占有中心地位。一个根本性的贡献十年前是Tymoczko的建设置换群表示上同调环的经常半单Hessenberg品种。重要的是，Shareshian-Wachs、Brosnan-Chow等人最近的工作表明，Hessenberg簇与对称函数理论密切相关，特别是尚未解决的Stanley-Stembridge猜想。我在最近的Precup工作中为这一领域做出了重大贡献，我们使用Hessenberg品种部分解决了Stanley Stembridge猜想。我在这一领域的长期计划是：（i）使用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）进一步发展和连接Newton-Okounkov体，Hessenberg簇和Schubert演算的理论，（iii）发展一个系统的关于Hessenberg簇的可积系统理论。**首先，众所周知，显式构造可积系统是困难的。因此，我的长期目标是利用我过去与Kaveh的工作（如上所述）来构建新的可积系统，这有可能改变该领域，并为其他领域带来丰富的新应用，例如表示论。其次，我的程序引入几何Hessenberg-簇技术的对称和拟对称函数的研究是一个热门的新的研究课题，并有可能成为一个新的工具包回答组合问题。

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.