Random curves and surfaces with conformal symmetries

本研究主要研究随机共形几何，特别是随机曲面和随机曲线。这些物体对物理现象的建模很有意义，它们为物理学的基本理论提供了坚实的数学基础。由于它们的自然和内在配方，它们出现在各种各样的环境和应用中。PI将指导所有级别的学生和职业早期研究人员，将组织适合初级研究人员的研讨会和工作坊，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，特别目的是激励年轻女性在数学和科学领域追求职业生涯。两个特别感兴趣的对象是被称为Schramm-Loewner演化(SLE)的随机分形曲线和被称为Liouville量子重力(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临界现象的理解，而LQG的研究可以追溯到80年代的物理文献，他与SLE和称为平面映射的离散表面有着密切的关系。PI将研究SLE和LQG的基本属性，将定义和研究它们的概括，这些概括对应用程序具有重大意义，并将证明扩展限制结果。该奖项反映了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 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.