Algebraic Combinatorics and its Applications

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The proposed research project is concerned with three classes of problems of algebraic combinatorics. They are related to several aspects of Schubert calculus and its links with representation theory, inverse boundary problems, algebraic geometry, and physics. The first part is devoted to the inverse boundary problem for certain class of networks. This problem emerged in an attempt to explain, generalize, and simplify algebraic and combinatorial constructions related to representation theory of general linear groups and canonical bases. It is directly linked to the study of total positivity on Grassmann manifolds. The second part focuses on combinatorial and algebraic problems came up in quantum Schubert calculus, which deals with quantum cohomology of complex flag manifolds and corresponding Gromov-Witten invariants. The third part is devoted to a new approach to smoothness of Schubert varieties. The investigator and his colleague suggest how to extend the notion of pattern avoidance in a general context of root systems. The main goal of the proposed research project is to investigate several problems that originally came from various areas, such as geometry and physics, and all share a discrete nature. The investigator suggests an approach to the problem of identification of networks by boundary measurements. These networks present a simple model for computer microchips. The problem can be rephrased as follows: "How to identify a microchip by external examination?" Another problem is related to certain geometric invariants that play a role in mathematical physics and algebraic geometry. These invariants are usually extremely hard to calculate. The investigator and his colleagues suggest a new efficient technique for computing invariants of this kind. The last part concerns with a new approach to the classical problem of smoothness.