Studies in Algebraic Combinatorics

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The proposal is in the area of algebraic combinatorics, with six main research topics. The first topic concerns a new formula for the values of irreducible characters of the symmetric group. This formula is connected with another formula of Kerov which is not well understood, though it has applications to free probability theory and other areas. The second topic deals with a generalization of the classical theory of lattice paths on the plane, where now the paths lie on a Riemann surface. The third topic concerns the subject of sign-balance, i.e., the difference between the number of even and number of odd permutations in certain sets of permutations. The primary focus is on a conjecture of Eremenko and Gabrielov that may be connected with recent work on ribbon Schur functions. The fourth topic concerns the saturation conjecture for Littlewood-Richardson coefficients, recently proved by Knutson-Tao and others. There are many new avenues of investigation opened up by recent work in this area. The fifth topic is the theory of k-triangulations, a generalization of ordinary triangulations of a polygon. A recent breakthrough of Jakob Jonsson suggests several new open problems and conjectures. Finally the proposer plans to continue his research on increasing and decreasing subsequences, another subject for which recent work has suggested a host of new directions of research.The proposal deals with a number of topics in algebraic combinatorics, a field which connects arrangements and patterns (such as jigsaw puzzles, computer chip design, and airplane boarding systems) with sophisticated abstract techniques. This combination of both simple, concrete objects with powerful, abstract reasoning has led to many important breakthroughs and applications. The proposer plans to work in six specific areas in which recent work points to the possibility of much further progress. These areas involve such ideas as using symmetry to simplify complicated objects, extending the notion of paths on a plane surface, decomposing a geometric figure into simpler pieces, and finding patterns in a list of objects. Progress on these very natural questions should have many applications, both within mathematics and to practical problems of scheduling, ranking, optimization, etc.