Studies in Algebraic Combinatorics

该提案是在代数组合领域，有六个主要的研究课题。第一个主题是关于对称群不可约字符值的一个新公式。这个公式与Kerov的另一个公式联系在一起，这个公式不是很好理解，尽管它在自由概率论和其他领域有应用。第二个主题涉及平面上晶格路径的经典理论的推广，现在路径位于黎曼曲面上。第三个主题涉及符号平衡的主题，即在某些排列集合中偶数排列和奇数排列的数量之差。主要的焦点是Eremenko和Gabrielov的一个猜想，这个猜想可能与最近关于带舒尔函数的研究有关。第四个主题涉及Littlewood-Richardson系数的饱和猜想，最近由Knutson-Tao等人证明。最近在这一领域的工作开辟了许多新的研究途径。第五个主题是k-三角剖分理论，这是对普通多边形三角剖分的推广。Jakob Jonsson最近的一项突破提出了几个新的开放问题和猜想。最后，申请人计划继续他的增加和减少子序列的研究，这是最近的工作提出了许多新的研究方向的另一个主题。该提案涉及代数组合学中的许多主题，代数组合学是一个将排列和模式（如拼图、计算机芯片设计和飞机登机系统）与复杂的抽象技术联系起来的领域。这种简单、具体的对象与强大的抽象推理的结合导致了许多重要的突破和应用。提议者计划在六个具体领域开展工作，最近的工作表明有可能取得更大进展。这些领域包括使用对称来简化复杂的物体，扩展平面上的路径概念，将几何图形分解成更简单的部分，以及在物体列表中找到模式。在这些非常自然的问题上取得的进展应该有许多应用，无论是在数学领域，还是在调度、排名、优化等实际问题上。

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.