Bridgeland Stability, Moduli Spaces, and Applications

多项式系统控制着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，即研究多项式系统解的领域。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系统的性质。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几何，即向量束的模空间。利用一种叫做桥地稳定性的新技术，PI将研究这些空间的基本几何性质。研究结果将在代数几何、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积极指导众多博士生和博士后以及高中生和本科生进行相关研究。助学金将为研究生提供部分资助。曲面上轴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Donaldson的四流形理论、组合学、表示理论和数学物理的关键人物。在过去的十年里，布里奇兰稳定性已经彻底改变了我们对轮轴模空间的理解。使用这种新技术，PI将推进对轮轴模空间的理解。具体来说，PI将使用桥地稳定性和过壁计算表面上一般稳定层的上同性。在一般上同调已经被理解的情况下，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束张量积的上同调。此外，PI旨在证明由于PI和Woolf的一个猜想，即当判别式趋于无穷时，轮轴模空间的Betti数趋于稳定，并且稳定的Betti数与秩和极化无关。最后，本文还将从双曲性、Lang猜想和可分有理连通性的应用出发，研究曲线正规束在代数变量上的稳定性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.