Generalized cohomology theories of flag manifolds, and other manifolds

标志流形和其他流形的广义上同调理论

• 批准号: 0072667
• 负责人: Allen Knutson
• 金额: $ 7.5万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072667&HistoricalAwards=false
• 关键词: Generalized; cohomology; theories; flag; manifolds

DMS-0072667Allen I. KnutsonAllen Knutson's proposed work is about Schubert calculus on flag manifolds, which combinatorially is about counting the (obviously nonnegative) number of flags satisfying a list of intersection conditions with other flags, and is computable as a (not obviously nonnegative) integral over a flag manifold. The main question in the field concerns finding a totally-positive combinatorial formula rather than the alternating sum. His work with Dr. Terence Tao on the case of single subspaces, refining totally-positive formulae already known and solving long-standing conjectures in that relatively simple case, has suggested new lines of attack on the general problem.Given four straight lines (picture them as blue) drawn in space, in generic directions, how many other straight lines (picture them as red) touch all four? In special cases there are many, but generically there are exactly twosuch red lines. This is the first interesting case of a general problem counting the number of flat subspaces (in this case one-dimensional) intersecting a number of other subspaces (which can even be curved). Computers can determine this obviously nonnegative number in any given case, but do so by adding many large positive and negative numbers together -- in particular it is difficult to determine easily if the number is zero. Also many interesting cases in engineering are beyond the reach of computers. Dr. Knutson's work concerns the search for a manifestly positive formula for these (no cancelation), which in particular would make it much more straightforwardto determine which such problems have any solutions at all. Under previous NSF support he and Dr. Terence Tao alreadydetermined this positivity criterion in a famous subcase of the general problem.

DMS-0072667ALLEN I. KNUTSONALLEN KNUTSON的拟议工作是关于标志歧管上的Schubert Cicculus，这是关于计数（显然是非负）的标志，这些标志数量满足与其他标志的交叉点列表，并且可以计算为A（显然是非阴性的）整体上的旗帜。 该领域的主要问题是找到一个完全阳性的组合公式，而不是交替的总和。 他与Terence Tao博士在单个子空间中的合作，在相对简单的情况下精炼了已经已知的完全阳性公式，并解决了长期存在的猜想，提出了对一般问题的新攻击线。赋予四条直线（图片为蓝色），在太空中绘制，在通用方向上绘制，以通用的方向，其他直线（图片为红色），触摸了所有四个触摸？ 在特殊情况下，有很多，但通常有两条红线。这是一个普遍问题的第一个有趣的情况，即计算平面子空间的数量（在这种情况下为一维）与许多其他子空间相交（甚至可以弯曲）。 在任何给定的情况下，计算机都可以确定这一明显的非负数，但是通过将许多大的正和负数添加在一起 ​​- 特别是很难轻松确定数字是否为零。 同样，工程中许多有趣的案例也超出了计算机的影响力。 诺特森博士的工作涉及寻找这些明显的积极公式（无取消），这尤其会使它更加直接地确定哪些问题完全有任何解决方案。在以前的NSF支持下，他和Terence Tao博士已经确定了这一积极的标准，这是一个著名的一般问题。