Combinatorial K-theory

研究了部分旗簇和半旗簇的上同调理论和K-理论。 特别是他将寻找forK理论的平行已知结果的上同调。 第一个目标是上同调和Grothendieck环的部分旗品种。 由于任何部分旗簇都有一个Schubert胞腔分解，所以它的上同调环有一个由Schubert簇的上同调类组成的基。 同样，代数向量丛的Grothendieck环也有Schubert簇的结构层的基。 研究者将研究这些环的结构常数与它们的基地索引的舒伯特品种。 主要目标是找到这些常数的显式公式，但积极性问题也很有趣。 在上同调中，由于几何原因，已知结构常数为正值，但目前还没有这一事实的组合证明。 在K-理论中，研究者已经证明，结构常数具有交替的符号，即它们在偶数度上是非负的，在奇数度上是非正的。 找到一个证明这一点将是非常有趣的。 另一个目标是找到有效的计算机算法来计算这些结构常数。 本章还将试图找到一般群簇的结构层的K-理论类的公式。 这样一个公式将推广一个公式的上同调类的一个品种，其中研究者已证明较早的富尔顿。 研究者希望证明这样一个公式将有助于构造一个簇的结构层的显式分解。 这将推广经典的结构，如Koszul复形，这是基础的重要性，在同调代数。发展的上同调理论的部分动机是问题的分类拓扑空间。 这对于解决诸如“我们宇宙的形状是什么？”之类的问题很重要。". 上同调理论也是解决计数几何问题的一个重要工具，在计数几何中，人们寻求确定和计数几何问题的所有解。 例如，如果几何问题是要找到与给定的固定几何图形集合相交或相切的直线，则需要确定这些直线的数量。 一个非常强大的技术forsolving这类问题是构造一个空间组成的allobboundary这可能是一个潜在的解决方案。 计算一个问题的解的个数通常等价于在这个潜在解空间的上同调环中进行计算。 在许多情况下，被计数的对象可以用给定向量空间中的子空间的标志来标识。 因此，旗形簇是潜在解空间的典型候选者，其点对应于子空间的旗形。 这使得能够在旗簇的上同调环内进行有效的计算变得重要，并给出了为什么几何学家和组合学家几十年来一直想要这个环的结构常数的原因。 簇的K-理论或Grothendieckr环可以看作是上同调环的推广，理解K-理论将使上同调理论更加完整。 同时，K-理论对于簇上向量丛的研究和同调代数的研究都有重要意义. 这使得将上同调理论的已知结果推广到K-理论是很自然的。

The investigator will study the cohomology theory and K-theory of partialflag varieties and quiver varieties. In particular he will look forK-theory parallels of known results in cohomology. A first target is thecohomology and Grothendieck rings of a partial flag variety. Since anypartial flag variety has a cell decomposition into Schubert cells, itscohomology ring has a basis consisting of the cohomology classes ofSchubert varieties. Likewise the Grothendieck ring of algebraic vectorbundles has a basis of structure sheaves of Schubert varieties. Theinvestigator will study the structure constants for these rings withrespect to their bases indexed by Schubert varieties. The main goal is tofind explicit formula for these constants, but also positivity questionsare of interest. In cohomology the structure constants are known to bepositive for geometric reasons, but no combinatorial proof of this fact isknown. In K-theory the investigator has conjectured that the structureconstants have alternating signs, i.e. they are non-negative in evendegrees and non-positive in odd degrees. Finding a proof of this would bevery interesting. An additional goal is to find efficient computeralgorithms for calculating these structure constants. The investigatorwill also try to find a formula for the K-theory class of the structuresheaf of a general quiver variety. Such a formula will generalize aformula for the cohomology class of a quiver variety, which theinvestigator has proved earlier with Fulton. The investigator hopes thatproving such a formula will be of help for constructing an explicitresolution of the structure sheaf of a quiver variety. This wouldgeneralize classical constructions such as the Koszul complex, which is offundamental importance in homological algebra.The development of cohomology theory was motivated in part by the problemof classifying topological spaces. This is important for addressingquestions such as "what is the shape of our universe?". Cohomology theoryis also an important tool for solving problems in enumerative geometry, inwhich one seeks to determine and count all the solutions to a geometricproblem. For example, if the geometric problem is to find lines whichintersect or are tangent to a given collection of fixed geometric figures,then the number of such lines is desired. A very powerful technique forsolving this type of problems is to construct a space consisting of allobjects which could potentially be a solution. Counting the number ofsolutions to a problem is often equivalent to performing a calculation inthe cohomology ring of this space of potential solutions. The objects tobe counted can in many situations be identified with flags of subspaces ina given vector space. Flag varieties, whose points correspond to suchflags of subspaces, are therefore typical candidates to act as the space ofpotential solutions. This makes it important to be able to do efficientcomputations within the cohomology ring of a flag variety, and gives thereason why the structure constants for this ring has been wanted bygeometers and combinatorialists for decades. The K-theory or Grothendieckring of a variety can be seen as a generalization of the cohomology ring.A good understanding of K-theory will therefore give a more completepicture of cohomology theory. At the same time K-theory is important forthe study of vector bundles on a variety and for homological algebra. Thismakes it very natural to try to generalize the known results aboutcohomology theory to K-theory.