Generalized cohomology theories of flag manifolds, and other manifolds

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

DMS-0072667 Allen I. Knutson Allen 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 two such 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 straightforward to determine which such problems have any solutions at all. Under previous NSF support he and Dr. Terence Tao already determined this positivity criterion in a famous subcase of the general problem.