Bridgeland Stability, Moduli Spaces, and Applications

Bridgeland 稳定性、模空间和应用

• 批准号: 2200684
• 负责人: Izzet Coskun
• 金额: $ 20万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200684&HistoricalAwards=false
• 关键词: Bridgeland; Stability; Moduli; Spaces; Applications

Polynomial systems govern processes in many diverse fields ranging from computer science to physics and economics to biology. The PI specializes in algebraic geometry, the field which studies solutions of polynomial systems. Such systems can often be simplified by varying the coefficients of the polynomials appropriately. As a result, properties of a more complicated system can be deduced from the simpler system. The PI will apply this principle to study the geometry of spaces that are ubiquitous in mathematics and physics, namely moduli spaces of vector bundles. Using a novel technique called Bridgeland stability, the PI will investigate fundamental geometric properties of these spaces. The results will have a wide range of applications in algebraic geometry, commutative algebra, topology, and mathematical physics. The PI is also dedicated to educating the next generation of mathematicians and building a strong workforce in STEM in the US. Towards this goal, the PI actively supervises numerous PhD students and postdocs, as well as high school students and undergraduate students in related research. The grant will provide partial support to the graduate students. The moduli spaces of sheaves on surfaces play a fundamental role in mathematics and physics. They carry essential information about linear series and Chow groups and are key players in Donaldson’s theory of four manifolds, combinatorics, representation theory and mathematical physics. In the last decade, Bridgeland stability has revolutionized our understanding of moduli spaces of sheaves. Using this novel technique, the PI will advance understanding of moduli spaces of sheaves. Specifically, the PI will compute the cohomology of the general stable sheaf on a surface using Bridgeland stability and wall-crossing. In cases where the generic cohomology is already understood, such as minimal rational surfaces and K3 surfaces, the PI will initiate a systematic study of the cohomology jumping loci and compute the cohomology of the tensor product of two general stable sheaves. In addition, the PI aims to prove a conjecture due to the PI and Woolf that states that the Betti numbers of the moduli spaces of sheaves stabilize as the discriminant tends to infinity and the stable Betti numbers are independent of the rank and polarization. Finally, the PI will also study the stability of normal bundles of curves on algebraic varieties with a view towards applications to hyperbolicity, Lang conjectures and separable rational connectedness.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

多项式系统管理许多不同领域的过程，从计算机科学到物理学，从经济学到生物学。PI专门研究代数几何，该领域研究多项式系统的解。这样的系统通常可以通过适当地改变多项式的系数来简化。因此，可以从较简单的系统推导出较复杂的系统的性质。PI将应用这一原理来研究数学和物理中普遍存在的空间的几何，即向量丛的模空间。利用一种名为布里奇兰稳定性的新技术，PI将研究这些空间的基本几何性质。这些结果将在代数几何、交换代数、拓扑学和数学物理中有广泛的应用。PI还致力于培养下一代数学家，并在美国的STEM建立一支强大的劳动力队伍。为实现这一目标，该协会积极指导众多博士后和博士后，以及高中生和本科生进行相关研究。这笔助学金将为研究生提供部分支持。曲面上的滑轮的模空间在数学和物理中起着重要的作用。它们携带着关于线性级数和Chow群的基本信息，是唐纳森的四种流形理论、组合学、表示论和数学物理的关键参与者。在过去的十年中，Bridgeland稳定性彻底改变了我们对滑轮的模空间的理解。使用这一新技术，PI将促进对滑轮模数空间的理解。具体地说，PI将使用Bridgeland稳定性和墙交叉来计算曲面上一般稳定层的上同调。在已了解上同调的情况下，例如极小有理曲面和K3曲面，PI将启动上同调跳跃轨迹的系统研究，并计算两个一般稳定层的张量积的上同调。此外，PI的目的是证明由PI和Woolf提出的一个猜想，即当判别式趋于无穷大时，轮的模空间的Betti数稳定，且稳定的Betti数与阶和极化无关。最后，PI还将研究代数簇上正常曲线丛的稳定性，以期应用于双曲性、Lang猜想和可分有理连通性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。