Random curves and surfaces with conformal symmetries

本研究涉及随机共形几何，特别是随机曲面和随机曲线的研究。这些对象对于物理现象的建模具有重要意义，它们为物理学的基本理论提供了坚实的数学基础。由于其自然和固有的配方，它们在各种环境和应用中出现。PI将指导各级学生和早期职业研究人员，将组织适合初级研究人员的研讨会和讲习班，她将针对包括初级研究人员和非概率学家在内的广泛受众进行演讲，其特别目标是激励年轻女性从事数学和科学事业。两个特别感兴趣的对象是称为Schramm-Loewner演化（SLE）的随机分形曲线和称为Liouville量子引力（LQG）表面的随机分形表面。1999年SLE的发现彻底改变了对统计力学中关键现象的理解，而LQG的研究可以追溯到80年代的物理文献，它与SLE和被称为平面图的离散曲面有着密切的关系。PI将研究SLE和LQG的基本性质，将定义和研究它们的推广，这是应用的主要兴趣，并将证明缩放极限的结果。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。

This research concerns the study of random conformal geometry, particularly random surfaces and random curves. These objects are of major interest for the modelling of physical phenomena and they provide firm mathematical ground for fundamental theories of physics. They arise in a wide variety of settings and applications due to their natural and intrinsic formulation. The PI will mentor students at all levels and early-career researchers, will organize seminars and workshops suitable for junior researchers, she will give talks aimed at a broad audience including junior researchers and non-probabilists, with a particular goal to inspire young women to pursue careers in mathematics and the sciences.Two particular objects of interest are the random fractal curves known as Schramm-Loewner evolutions (SLE) and the random fractal surfaces known as Liouville quantum gravity (LQG) surfaces. The discovery of SLE in 1999 revolutionized the understanding of critical phenomena in statistical mechanics, while LQG, whose study goes back to the physics literature in the 80s, has intimate relationships with SLE and discrete surfaces known as planar maps. The PI will study fundamental properties of SLE and LQG, will define and study generalizations of them which are of major interest for applications and will prove scaling limit results.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.