The PI will study combinatorial, algebraic and computational questions about the geometry of Schubert varieties in Grassmannians and flag manifolds. The two main projects explore central problems in the subject. The first project seeks uniformly stated combinatorial rules in Schubert calculus, via an expanded theory of Young tableaux. The famous Littlewood-Richardson coefficients arise in this context, but also in the theory of representations of general linear groups and of symmetric groups, eigenvalues of sums of Hermitian matrices and short exact sequences of finite abelian p-groups. The PI will find combinatorial rules for these coefficients and their Schubert calculus extensions, building on joint work with H. Thomas. The second project aims to further develop a combinatorial and computational framework, introduced jointly with A. Woo, to understand the singularities of Schubert varieties.
The main tools applied and further developed in this project come from combinatorics, including algebraic and geometric combinatorics, and combinatorial commutative algebra. Combinatorics concerns discrete objects such as permutations, partial orders and graphs, as well as techniques of enumeration. In many instances, such as with Schubert varieties, continuous objects can be parameterized by discrete data, opening the door for combinatorial analysis. Moreover, one often witnesses that the same combinatorial objects govern a priori different mathematical settings. The Littlewood-Richardson coefficients are a prime example of this. The project seeks to similarly find further cross-flow of ideas between areas of mathematics, as well as with other scientific disciplines.
The project's main concern was to find governing combinatorial laws for objects that arise in the study of symmetries, geometry and algebra. Most of the objects involved the subject of Schubert calculus, and a number of successful projects were undertaken and completed, including, a connection to the study of longest increasing subsequences of random words. Such analysis can be used to model airplane boarding times. Another project made a new connection to eigenvalues of Hermitian matrices. These were all possible because of new combinatorial rules that were discovered during the duration of the project by the principal investigator and his collaborators, students and postdoctoral scholars. Two additional general audience/pedagogical projects were completed which may help the layman understand the kind of mathematical reasoning involved in the more technical and theoretical projects alluded to above. The first concerned the language classification of the indigeneous languages of the Americas. At first contact there were, perhaps (and no one knows for sure), about 2000 languages. Now there are somewhere around 1000, split into numerous mutually unintelligible families. The question is, how many such families are there? The principal investigator was drawn into this subject by a beautiful BBC/Nova documentary, ``Before Babel: in search of the first language'', where he learned about a debate. On the one hand, most scholars agreed there were 150-180 such families, whereas the eminent linguist Joseph Greenberg disagreed, arguing there were only three, Eskimo-Aleut, Na-Dene and a superfamily he called Amerind. What we did was model language dispersal in the Americas using the classical Bell numbers (not to be confused with a Bell curve distribution) and observed that the numbers that came out of heavy (desktop) computation agree with the conventional view, thus lending support to it. We call this kind of analysis a ``combinatorial Fermi problem'', alluding to Enrico Fermi, whose ability to make surprisingly accurate estimates with little data is legendary. Another combinatorial Fermi problem we investigated concerns how we rank scholarly productivity. A physicist Jorge Hirsch argued for what we now call the h-index: a scholar has an h-index of H if he/she has at least H papers with H citations, and all other papers have no more than H citations. The h-index is now widely used, and appears, for example, in Google scholar and many other bibliometric databases. In this situation, we modelled citation profiles using combinatorial objects called Young diagrams. We then used an old identity due to Leonhard Euler and Friedrich Gauss to estimate the expected range of h-index for a person with C many citations. We thereby reinterpret a mathematical theorem of Rodney Canfield-Sylvie Corteel-Carla Savage as the rule of thumb of h-index: H is approximately 0.54 x square root of C. We tested this to be a very accurate model for mathematicians, and we believe it is informative for many other fields as well. This provides a specific critique about the basic premise of the h-index and its use. These projects, both of theoretical and applied nature, were used in the training of graduate students, college students and high school students. This represents this project's efforts to help strengthen the American educational infrastructure.
