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 提出的工作是关于标志流形上的舒伯特演算，它以组合方式计算满足与其他标志的交集条件列表的标志（显然非负）数量，并且可计算为（不是明显非负）旗形流形上的积分。 该领域的主要问题涉及找到一个完全正的组合公式而不是交替和。 他与陶哲轩博士在单一子空间的情况下进行了合作，提炼了已知的全正公式并在相对简单的情况下解决了长期存在的猜想，为一般问题提出了新的攻击路线。给定四条直线（图在空间中，在一般方向上绘制它们（它们为蓝色），有多少其他直线（将它们描绘为红色）接触所有四个？ 特殊情况下有很多条，但一般情况下只有两条这样的红线。这是计算与许多其他子空间（甚至可以是弯曲的）相交的平坦子空间（在本例中为一维）的数量的一般问题的第一个有趣的情况。 在任何给定情况下，计算机都可以确定这个明显的非负数，但这是通过将许多大的正数和负数相加来实现的——特别是很难轻松确定该数字是否为零。 此外，工程中许多有趣的案例也超出了计算机的能力范围。 克努森博士的工作涉及为这些问题寻找一个明显积极的公式（无取消），这尤其会使确定哪些此类问题有解决方案变得更加简单。在之前的 NSF 支持下，他和陶哲轩博士已经在一般问题的一个著名子案例中确定了这一积极性标准。