Generalized cohomology theories of flag manifolds, and other manifolds

DMS-0072667 Allen I.Knutson提出的工作是关于旗形上的Schubert演算，它结合起来是关于计数满足一列相交条件的旗子与其他旗子的(明显非负的)数目，并且可以作为旗子流形上的(不明显非负的)积分来计算。这个领域的主要问题是寻找一个完全正的组合公式，而不是交错和。他和陶渊明博士在单个子空间的情况下，提炼了已知的全正公式，并在那个相对简单的情况下解决了长期存在的猜想，提出了对一般问题的新的攻击路线。给出在空间中绘制的四条直线(想象为蓝色)，在一般方向上，有多少条直线(想象为红色)与这四条直线相交？在特殊情况下，红线很多，但一般来说，恰好有两条这样的红线。这是计算平坦子空间(在这种情况下是一维的)与许多其他子空间(甚至可以是曲线的)相交的子空间数量的第一个有趣的一般问题。在任何给定的情况下，计算机都可以确定这个明显的非负数，但通过将许多大的正数和负数相加来实现这一点--特别是很难容易地确定该数字是否为零。此外，工程学中的许多有趣的案例都超出了计算机的能力范围。克努森博士的工作关注的是为这些问题寻找一个明显积极的公式(不取消)，这尤其会让我们更直接地确定哪些此类问题有任何解决方案。在NSF以前的支持下，他和Terence陶博士已经在一般问题的一个著名的子例中确定了这个正性准则。

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.