Varieties with group actions

具有集体行动的品种

• 批准号: 0652386
• 负责人: Venkatramani Lakshmibai
• 金额: $ 21.45万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652386&HistoricalAwards=false
• 关键词: Varieties; group; actions

The main theme of this proposal is the study of algebraic varieties with algebraic group actions, typical examples being flag varieties. Homogeneous spaces form basic fundamental objects in Geometry and other related fields. The flag varieties constitute an important class of homogeneous spaces; Schubert subvarieties in flag varieties provide a powerful inductive machinery for the study of flag varieties. Problems of this proposal are related to some interesting and important algebraic varieties - Schubert varieties, affine Grassmannians (and affine Schubert varieties), quiver varieties, nilpotent orbit closures, toric varieties. Lakshmibai, Musili and Seshadri developed a "Standard Monomial Theory" (henceforth abbreviated SMT) for flag varieties and their Schubert subvarieties. This theory has led to very many important geometric & representation-theoretic consequences. This proposal deals first with developing a SMT for the varieties mentioned above, next proposes to use SMT to study such geometric problems as determination of singular locus, determination of the multiplicity at a singular point etc. Lakshmibai's recent research shows that once there is a good SMT for an algebraic variety, much information could be inferred about the variety (using SMT); for instance, SMT throws light on the degenerations of the variety. The degenerations of a variety in turn facilitate the understanding of the geometric aspects of the variety. This technique has been used very recently in the area of Complexity Theory in Computer Science, esp., in the context of the "P not equal to NP-conjecture".Modern Algebraic Geometry (developed in the latter half of the 20-th century) has proved itself (beyond any doubts) to be indispensable in various disciplines within Mathematics as well as in other areas outside Mathematics, such as: Topology, Representation Theory, Combinatorics (within Mathematics), the modern Quantum theory (especially Quantum & Conformal field theories) in Physics; Robotics, Complexity theory in Computer Science. This proposal is at the cross-roads of Commutative Algebra, Algebraic Geometry, Combinatorics & Representation-theory. The varieties studied in this proposal form an important class of varieties in Algebraic Geometry; for example, the theory of Schubert varieties (over finite fields) is closely linked to Coding theory. The principal investigator believes that this proposal is bound to have significant impacts on the above-mentioned disciplines.

这个建议的主题是研究代数簇与代数群的行动，典型的例子是旗品种。齐性空间是几何学和其他相关领域的基本对象。旗簇是一类重要的齐性空间，旗簇中的Schubert子簇为旗簇的研究提供了一个强有力的归纳机制。这一建议的问题涉及到一些有趣的和重要的代数品种-舒伯特品种，仿射格拉斯曼（和仿射舒伯特品种），非线性品种，幂零轨道闭包，环面品种。Lakshmibai、Musili和Seshadri为旗叶变种及其舒伯特亚种发展了一个“标准单式理论”（Standard Monomial Theory，以下简称SMT）。这一理论导致了许多重要的几何表示理论的后果。本文首先提出了一个代数簇的最小均方图，然后提出了利用最小均方图研究诸如奇异轨迹的确定、奇点重数的确定等几何问题。Lakshmibai最近的研究表明，一旦一个代数簇有了一个好的最小均方图，就可以推断出该代数簇的许多信息（使用SMT）;例如，SMT揭示了品种的退化。一个变种的退化反过来又促进了对该变种的几何方面的理解。这种技术最近已被用于计算机科学中的复杂性理论领域，特别是，现代代数几何学（发展于世纪后半叶）已经证明了它在数学中的各个学科以及数学之外的其他领域都是不可或缺的，如：拓扑学，表示论，组合数学（数学），物理学中的现代量子理论（特别是量子共形场论）;机器人学，计算机科学中的复杂性理论。这一建议处于交换代数、代数几何、组合数学表示论的交叉点。在这个建议中研究的变种形成了代数几何中一类重要的变种;例如，舒伯特变种理论（在有限域上）与编码理论密切相关。主要研究者认为，该提案势必会对上述学科产生重大影响。