Two Spring Lecture Series in Geometric Analysis

第35届和第36届阿肯色州春季数学科学系列讲座（SLS）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。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。主讲人将是约翰霍普金斯大学的William Minicozzi教授。此外，Minicozzi教授还将邀请10位演讲者进行1小时的演讲，并邀请研究生和刚毕业的博士进行20分钟的演讲。Minicozzi教授将介绍最小曲面和平均曲率流的最新研究进展。特别是，他将谈到平均曲率流弱解的尖锐部分正则性定理，以及最近在里奇流方面取得重大进展的对几何流的进一步理解的其他实例。正则性理论的许多工具最初是为最小曲面而发明的，现在已经广泛应用于其他几何问题。在平均曲率流的情况下，这包括Huisken的单调公式，Brian White的抛物线版本的费德勒降维，以及Brakke的正则定理（这是平稳变分的Allard定理的抛物线版本）。2011年SLS：“共形微分几何及其与表示理论的相互作用。”会议将于2011年4月7日至9日举行，阿德莱德大学的迈克尔·伊斯特伍德教授将担任主要发言人。此外，伊斯特伍德教授还将邀请10位演讲者进行1小时的演讲，并邀请研究生和刚毕业的博士进行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-盖尔芬（BGG）机械之间的相互作用。共形情形本身就很重要，它也为更一般的抛物线几何理论提供了一个很好的试验台。特别是，这些会谈将为猜测的BGG机制的大规模扩展和适当的几何延长理论奠定基础。布兰森的q曲率及其处方问题将在此背景下进行讨论。Oshima最近的工作发现了平面模型上所有共形不变微分算子的对称代数。这些讲座将讨论曲线情况，这是目前完全开放的。最后，BGG机械可以与复杂分析相结合，在合适的齐次空间上解决积分几何的重构问题。讲座将介绍这一方向的研究现状和展望。数学会议对该领域的发展至关重要。他们为来自不同子领域的科学家提供了独特的机会，以便(a)总结这些领域的最新进展，以及他们之间的相互作用，(b)提供新的研究方向，并提供足够的细节来制定和发展一系列开放问题，(c)提供机会来促进新的合作和交流思想，以解决重要的开放问题。在阿肯色春季系列讲座的传统中，这些会议将为年轻的研究人员和研究生提供独特的机会，与他们研究领域的杰出专家进行互动。丹尼尔·洛克莫尔教授（达特茅斯学院）和伊斯特伍德教授（阿德莱德学院）将分别于2010年和2011年分别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.