Generalized cohomology theories of flag manifolds, and other manifolds

allen I. Knutson提出的工作是关于旗帜流形上的舒伯特微积分，其组合是关于计算满足一系列与其他旗帜相交条件的旗帜的（明显非负的）数量，并且可以作为旗帜流形上的（非明显非负的）积分计算。该领域的主要问题是找到一个完全正的组合公式，而不是交替和。他与Terence Tao博士在单子空间问题上的工作，改进了已知的全正公式，解决了在这种相对简单的情况下长期存在的猜想，为解决一般问题提供了新的思路。给定在空间中沿一般方向画的四条直线（用蓝色表示），有多少其他直线（用红色表示）与这四条直线相交？在特殊情况下有很多，但一般来说只有两条这样的红线。这是计算平面子空间（在这种情况下是一维的）与许多其他子空间（甚至可以是弯曲的）相交的数量的一般问题的第一个有趣的例子。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，但要做到这一点，需要将许多大的正数和负数加在一起——特别是很难轻易确定这个数字是否为零。此外，许多有趣的工程案例是计算机无法处理的。克努森博士的工作是为这些问题寻找一个明显的正公式（没有抵消），这尤其会使确定这些问题是否有任何解决方案更加直接。在之前国家科学基金会的支持下，他和Terence Tao博士已经在一般问题的一个著名子案例中确定了这个正性标准。

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.