The Asymptotic Geometry of Moduli Spaces

这个项目是基于几何学和物理学之间的界面的最新发展。关于数学中的中心对象(“希钦模空间”和“K3空间”)，有许多错综复杂的猜想，这些猜想来自于研究特定低能量子系统的理论物理学家。PI使用几何分析中出现的新工具，这些工具非常适合于验证关于Hitchin模空间和K3空间的几何的极其精细的猜想。这一策略经常延伸了目前通过几何分析所能做的事情的极限，同时也带来了对这些物理猜想的新见解。由于Hitchin模空间是代数几何、微分几何、表示论、几何分析等多个数学领域的交叉点，在几何学中占有重要的地位，因此这项工作具有跨学科的影响。主要目的是验证物理学家Gaiotto、Moore和Neitzke对Hitchin模空间上的Hyperkaehler度规的美丽猜想描述。第二个目标是验证物理学家Kachru、Triparsis和Zimet对椭圆纤维K3表面上的Hyperkaehler度量的类似猜想描述。Hitchin模空间和椭圆纤维K3曲面都是具有Hyperkaehler度量的代数完全可积系统。这两种纤维都是在半维基上纤化的，而普通纤维是阿贝尔品种。有些纤维是奇异的，而这些猜想在这些奇异纤维附近是最有趣和最困难的。构造性分析技术和几何微局域分析方法似乎是验证物理学猜想的最合适的方法，因为它们非常适合分析自然出现的奇异微分算子。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。

This project is based on recent developments at the interface between geometry and physics. There are a number of intricate conjectures about central objects in mathematics (“Hitchin moduli spaces” and “K3 spaces”) coming from theoretical physicists studying particular quantum systems at low energies. The PI uses new tools coming out of geometric analysis which are well-suited for verifying the extremely delicate conjectures about the geometry of the Hitchin moduli space and K3 spaces. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these physics conjectures. Because Hitchin moduli spaces occupy a distinguished position in geometry at the crossroads of a number of mathematical fields including algebraic geometry, differential geometry, representation theory, and geometric analysis, this work has cross-disciplinary impact.This project is centered on the asymptotic geometry of Hitchin moduli spaces and families of K3 surfaces. A main goal is to verify the beautiful conjectural description of the hyperkaehler metric on the Hitchin moduli space by the physicists Gaiotto, Moore, and Neitzke. A second goal is to verify a similar conjectural description of hyperkaehler metrics on elliptically-fibered K3 surfaces by the physicists Kachru, Tripathy, and Zimet. The Hitchin moduli spaces and elliptically-fibered K3 surfaces are both algebraic completely integrable systems admitting a hyperkaehler metric. Both are fibered over a half-dimensional base and the generic fibers are abelian varieties. Some fibers are singular, and these conjectures are most interesting and most difficult near these singular fibers. Constructive analytic techniques and the methods of geometric microlocal analysis seem to be the most appropriate way to verify conjectures from physics because they are well-suited to analyzing the singular differential operators that naturally appear.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.