CAREER: Geometric flows and four-dimensional geometry

这笔赠款将支持一个多方面和相互关联的教育和研究计划，旨在这两个领域取得重大成果。研究目标围绕更深入地了解物理学和几何学中产生的自然进化过程的行为。主要的例子是控制肥皂泡和薄膜演化的方程。 对这些方程的透彻理解需要结合物理，分析和几何的技术，并导致对所有这些领域的更深入的理解。 为了长期实现这些研究目标，我们将开展一些教育活动。其中包括招募当地高中和社区大学学生进入加州大学系统的推广活动，为本科生提供在数学这一领域获得研究经验的机会，以及旨在研究生水平教育和培训研究和教学活动的计划。 该项目将直接支持NSF的目标，即保持在数学研究中的领导地位，并支持卓越的数学教育。几何的中心主题之一是理解流形的拓扑和几何之间的相互作用，几何流已被证明是理解这种关系的强大工具。尽管在三维空间中有着惊人的结果，但仅凭里奇流可能不足以理解四维流形的几何。该项目围绕着更深入地了解长期行为，奇点形成和某些Ricci流概括的几何内容，重点是理解复杂几何和四维几何的各个方面。我们将使用PDE和几何的各种技术来理解这些流动，并结合培养各级学生的目标。这些方程在物理环境中非常自然地出现，因此本提案中概述的问题的进展将产生见解并产生超越几何的影响，特别是对理解PDE，复杂几何和数学物理。 此外，这些方程，虽然本身就值得研究，但原则上有可能解决拓扑学和几何学中长期存在的问题。为了长期实现这些研究目标，我们将开展一些教育活动。我们将追求一个强大的研究生培训计划，包括开发新颖的课程，学习研讨会和研究项目的开发。我们还将创建一个“未来教师计划”，旨在培养研究生成为未来的教师。在南加州几何分析研讨会成功的基础上，我们将在会议之前为研究生开设为期一周的迷你课程，并为研究生和博士后举办海报会议。 此外，我们将制定一系列招聘日，旨在促进加州大学欧文分校本科生和研究生与当地高中和社区大学学生之间的互动，以帮助转移到加州大学系统和其他四年制院校的目标。

This grant will support a multifaceted and interconnected program of education and research aiming at significant results in both areas. The research goals center around a deeper understanding of the behavior of natural evolution processes arising in physics and geometry. Central examples are the equations governing the evolution of soap bubbles and thin films. A thorough understanding of these equations requires incorporating techniques from physics, analysis, and geometry, and results in a deeper understanding of all of those areas. Aiming at the long term fulfillment of these research goals, we will enact a number of educational activities. These include outreach activities to recruit local high school and community college students into the UC system, opportunities for undergraduates to gain research experience in this area of mathematics, and programs aimed at graduate-level education and training in both research and teaching activities. This project will directly support the NSF goals of maintaining a position of leadership in mathematics research as well as supporting excellence in mathematics education.One of the central themes in geometry is to understand the interplay between the topology and geometry of manifolds, and geometric flows have proven to be powerful tools in understanding this relationship. Despite striking results in three dimensions, Ricci flow alone may not be enough to understand the geometry of four-manifolds. This project centers around a deeper understanding of the long term behavior, singularity formation, and geometric content of certain generalizations of Ricci flow, with a focus on understanding aspects of complex geometry and four-dimensional geometry. We will use a variety of techniques from PDE and geometry to understand these flows, in conjunction with the goal of training students at all levels. These equations appear quite naturally in physical contexts, so progress on the questions outlined in this proposal would yield insights and have impact beyond geometry, specifically to understanding PDE, complex geometry, and mathematical physics. Moreover, these equations, while already worthy of study in their own right, in principle have the potential to address long standing conjectures in topology and geometry. Aiming at the long term fulfillment of these research goals, we will enact a number of educational activities. We will pursue a robust program of graduate student training involving the development of novel courses, learning seminars, and the development of research projects. We also will create a "Future Faculty Program" aimed at training graduate students as future teachers. Building on the success of the Southern California Geometric Analysis Seminar, we will initiate a one-week minicourse for graduate students preceding the conference as well as a poster session for graduate students and postdocs accompanying the conference. Moreover, we will enact a series of recruitment days designed to foster interaction between UC Irvine undergraduates and graduate students and local high school and community college students, with the goal of aiding transfer into the UC system and other four-year institutions.