Collaborative Research: Midwest Geometry Conference 2005-2007

AbstractAward：DMS 0509068、0508661、0509714主要研究者：托马斯P. Branson、Shihshu Walter Wei、Andrzej J. Derdzinski机构：爱荷华州大学、俄克拉荷马州大学、俄亥俄州州立大学自1991年以来，中西部中西部几何会议每年举行一次。 它的目标是目前在微分几何，几何分析，和积分几何的最新进展，从中西部的geometers，包括初级水平的研究人员和一些研究生。 该计划的主要部分包括由特定子领域的专家（不需要是中西部的）提供的几个全体会议讲座;也为一些参与者提供了一个机会，以提供有贡献的演讲。 目前的项目提供了会议在2005年，2006年和2007年春季在俄亥俄州州立大学，俄克拉何马大学和，分别在爱荷华州大学。 俄亥俄州会议，计划于2005年4月29日至5月1日，将集中在三个主题：正弯曲流形;里奇流，孤子和爱因斯坦度量;几何群论。 其他两个会议的主题列表仍然是暂定的，包括：对称临界性; p-调和几何;凸性，正曲流形和Ricci流（俄克拉荷马州，2006年），以及：偏微分方程的对称性和超定系统; AdS/CFT对应;非线性双曲演化方程;几何散射理论（爱荷华州，2007年）。中西部几何会议致力于中西部几何界成员之间的思想交流，以及重要研究进展和思想的传播，在全国和世界范围内，在微分几何和相关领域。 会议的一个关键目标是试图鼓励研究人员和潜在的研究人员在中西部，特别是那些谁的工作一些距离大型研究中心，以及年轻人，妇女和成员ofundrepresented组。 为研究生和年轻研究人员准备的“热身”会议将在会议前几天举行。 每次会议上的三四个焦点问题中，至少有一个与几何的应用有关。从历史上看，这样的应用主要存在于物理学中，爱因斯坦的广义相对论和目前被接受的基本粒子及其相互作用的标准模型提供了突出的例子。 微分几何的其他众所周知的应用是在工程，例如机器人。 另一个例子涉及到演化方程，这是目前微分几何学家非常感兴趣的一个主题，同时，具有描述某些材料或空间本身如何随时间发展的应用方面。 平均曲率流是后一种情况的一个非常具体的例子。 一个熔融塑料球的表面会随着时间的推移而演变或适应，以具有尽可能小的曲率;这种演变完全由球的曲率决定。 这一基本的观察提供了对广泛的流动问题的理解，并允许一种预测流动的方法。 这与材料的注塑成型直接相关，例如汽车的仪表板和保险杠。 但是，平均曲率流背后的基本原理也为像里奇流这样的问题提供了一个范例，在里奇流中，空间本身的度量性质是不断演化的量。 后一种问题将是中西部几何会议即将召开的会议的中心议题。

AbstractAward: DMS 0509068, 0508661, 0509714Principal Investigator: Thomas P. Branson, Shihshu Walter Wei, Andrzej J. DerdzinskiInstitutions: University of Iowa, University of Oklahoma,Ohio State UniversityThe Midwest Geometry Conference has been meeting annually since1991. Its goal is to present recent advances in differentialgeometry, geometric analysis, and integral geometry to geometersfrom the Midwest, including junior level researchers and somegraduate students. The main part of the program consists ofseveral plenary lectures given by experts in specific sub-fields(who need not be Midwest-based); an opportunity is also providedfor some of the participants to give contributed talks. Thecurrent project provides for conferences in the spring of 2005,2006, and 2007 at the Ohio State University, the University ofOklahoma and, respectively, the University of Iowa. The OhioState conference, planned for April 29 to May 1, 2005, will focuson three topics: positively curved manifolds; Ricci flow,solitons and Einstein metrics; geometric group theory. The listsof topics for the other two conferences are still tentative, andinclude: symmetric criticality; p-harmonic geometry; convexity,positively curved manifolds, and Ricci flow (Oklahoma, 2006), aswell as: symmetries and overdetermined systems of PDEs; theAdS/CFT correspondence; nonlinear hyperbolic evolution equations;geometric scattering theory (Iowa, 2007).The Midwest Geometry Conference is dedicated to the exchange ofideas among members of the geometry community in the Midwest, aswell as dissemination of important research advances and ideas,nationwide and worldwide, in differential geometry and relatedfields. A key objective of the conference is the attempt toencourage researchers and potential researchers in the Midwest,especially those who work some distance from the large researchcenters, as well as young people, women, and members ofunderrepresented groups. "Warm-up" sessions for graduatestudents and young researchers are planned for the days leadingup to the conference. At least one of the three or four focustopics at each conference is related to applications of geometry.Historically, such applications lie primarily in physics, withprominent examples provided by Einstein's general theory ofrelativity and the currently accepted Standard Model ofelementary particles and their interactions. Other well-knownapplications of differential geometry are in engineering, forexample robotics. A further example pertains to evolutionequations, which are currently a subject of keen interest todifferential geometers and, at the same time, have the appliedaspect of describing how some materials, or the space itself,develop in time. A very specific example of the latter kind isprovided by the mean curvature flow. The surface of a globule ofmolten plastic evolves, or adapts in time, to have the minimumpossible curvature; this evolution is completely determined bythe globule's curvature. This fundamental observation providesan understanding of a wide range of flow problems, and allows anapproach to predicting the flow. This has immediate relevance toinjection molding of materials, for example in the dashboards andbumpers of automobiles. But the fundamental principles behindmean curvature flow have also provided a paradigm for problemslike the Ricci flow, in which the metric properties of spaceitself are the quantities which are evolving. Questions of thelatter kind will be central to the coming meetings of the MidwestGeometry Conference.