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.

多项式系统管理着从计算机科学到物理学、经济学到生物学的许多不同领域的过程。PI专门研究代数几何，该领域研究多项式系统的解决方案。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，更复杂系统的性质可以从更简单的系统推导出来。PI将应用这一原理来研究在数学和物理中无处不在的空间的几何，即向量丛的模空间。使用一种称为Bridgeland稳定性的新技术，PI将研究这些空间的基本几何性质。所得结果在代数几何、交换代数、拓扑学和数学物理中有广泛的应用。PI还致力于教育下一代数学家，并在美国建立一支强大的STEM劳动力队伍。为了实现这一目标，PI积极监督许多博士生和博士后，以及高中生和本科生的相关研究。这笔补助金将为研究生提供部分资助。曲面层的模空间在数学和物理中起着重要的作用。他们进行基本信息的线性系列和周群和关键球员在唐纳森的理论的四个流形，组合，代表性理论和数学物理。在过去的十年里，Bridgeland稳定性彻底改变了我们对层模空间的理解。使用这种新技术，PI将推进对层模空间的理解。具体来说，PI将使用Bridgeland稳定性和跨壁计算表面上一般稳定层的上同调。 在一般上同调已经被理解的情况下，例如极小有理曲面和K3曲面，PI将启动对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的目的是证明一个猜想，由于PI和伍尔夫，国家的贝蒂数的模空间的层稳定的判别趋于无穷大和稳定的贝蒂数是独立的秩和极化。最后，PI还将研究代数簇上曲线的正规束的稳定性，以期应用于双曲性、Lang拓扑和可分离的理性连通性。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。