Two Spring Lecture Series in Geometric Analysis

第35届和第36届阿肯色州数学科学(SLS)春季系列讲座将于2010年和2011年在阿肯色州费耶特维尔的阿肯色大学举行。重点主题如下。对于SLS 2010：“最小曲面和平均曲率流。”会议将于2010年4月15日至17日举行。主要演讲人将是约翰霍普金斯大学的威廉·米尼科齐教授。另外，米尼科齐教授还将邀请10位演讲者进行一小时的演讲，并由研究生和新近获得博士学位的S做20分钟的演讲。米尼科齐教授将介绍极小曲面和平均曲率流研究的最新进展。特别是，他将讲述平均曲率流的弱解的尖锐部分正则性定理，以及随着Ricci流最近的巨大进展而增加对几何流的理解的其他实例。许多最初为极小曲面发明的正则性理论工具已被广泛应用于其他几何问题。在平均曲率流的情况下，这包括惠斯肯的单调性公式，布莱恩·怀特的费德勒降维的抛物线版本，以及布拉克的正则性定理(它是平稳变量的阿拉德定理的抛物线版本)。2011年SLS：“共形微分几何及其与表示论的相互作用”。会议将于2011年4月7日至9日举行，主要演讲人是阿德莱德大学的迈克尔·伊斯特伍德教授。伊斯特伍德教授还将邀请10位演讲者作一小时的演讲，并由研究生和新近获得博士学位的S教授作20分钟的演讲。伊斯特伍德教授的演讲将讨论共形微分几何和伯恩斯坦-盖尔芬德-盖尔芬德(BGG)机械之间的相互作用。保角情形本身很重要，也是更广泛的抛物线几何理论的极好的试验台。特别是，这些会谈将为BGG机械的猜想的巨大扩展和适当的几何延长理论奠定基础。Branson的q-曲率及其公式问题将在本文中讨论。Oshima最近的工作发现了平坦模型上所有共形不变微分算子的对称代数。这些讲座将讨论目前完全开放的曲线型案例。最后，BGG机器可以用于复杂分析，以解决适合的齐次空间上的积分几何的重建问题。讲座将介绍这方面的最新研究和展望。数学会议对这一领域的发展至关重要。它们为来自不同分领域的科学家提供了独特的会面机会，以便(A)总结这些领域的最新进展及其相互作用，(B)以足够详细的方式提出新的研究方向，以制定和制定未决问题清单，以及(C)提供机会，促进新的合作和交流意见，以解决重要的未决问题。按照阿肯色州春季系列讲座的传统，这些会议将为年轻的研究人员和研究生提供与各自研究领域的知名专家互动的独特机会。丹尼尔·罗克莫尔教授(达特茅斯学院)和伊斯特伍德教授(阿德莱德)分别于2010年和2011年向高中生和广大公众发表公开演讲。

The 35th and 36th Arkansas Spring Lectures Series in the Mathematical Sciences (SLS) will take place in 2010 and 2011 at the University of Arkansas, in Fayetteville, AR. The focus topics are the following. For SLS 2010: "Minimal Surfaces and Mean Curvature Flow." The conference will take place April 15-17, 2010. The principal speaker will be Professor William Minicozzi from Johns Hopkins University. There will be additional one-hour talks by ten invited speakers to be chosen by Professor Minicozzi, and several twenty-minute contributed talks by graduate students and recent Ph.D.'s. Professor Minicozzi will address recent progress on the study of minimal surfaces and mean curvature flow. In particular, he will speak about sharp partial regularity theorems for weak solutions of the mean curvature flow and other instances of the increased understanding of geometric flows that have accompanied recent dramatic progress on Ricci flow. Many of the tools of regularity theory that were originally invented for minimal surfaces have been widely applied to other geometric problems. In the case of mean curvature flow, this includes the monotonicity formula of Huisken, Brian White's parabolic version of Federer's dimension reducing, and Brakke's regularity theorem (which is a parabolic version of Allard's theorem for stationary varifolds). For SLS 2011: "Conformal Differential Geometry and Its Interaction with Representation Theory." The conference will take place April 7-9, 2011, and feature principal speaker Professor Michael Eastwood of the University of Adelaide. There will be additional one-hour talks by ten invited speakers to be chosen by Professor Eastwood, and several twenty-minute contributed talks by graduate students and recent Ph.D.'s. Professor Eastwood's talks will address the interplay between conformal differential geometry and Bernstein-Gelfand-Gelfand (BGG) machinery. The conformal case, important in itself, also acts as an excellent testbed for the more general theory of parabolic geometry. In particular, these talks will set the scene for a conjectured vast extension of the BGG machinery and a properly geometric theory of prolongation. Branson's Q-curvature and its prescription problem will be discussed in this context. Recent work of Oshima finds the symmetry algebras of all conformally invariant differential operators on the flat model. These lectures will discuss the curved cases, which are completely open at present. Finally, the BGG machinery can be used with complex analysis to solve reconstruction problems for integral geometry on suitable homogeneous spaces. The talks will present the current research and outlook in this direction.Mathematics conferences are vital to the development of the field. They provide unique opportunity for scientists from different subfields to meet so as to (a) summarize the most recent progress in these fields, and their interactions, (b) present new directions of research with sufficient detail to formulate and develop a list of open problems, and (c) provide an opportunity to foster new collaborations and exchange ideas toward the solution of important open questions. In the tradition of the Arkansas Spring Lectures Series, these meetings will provide unique opportunities for young researchers and graduate students to interact with prominent experts in their research areas. Public talks by Professor Daniel Rockmore (Dartmouth College) in 2010 and by Professor Eastwood (Adelaide) in 2011 will address audiences of high school students and the public at large.