Counting Meromorphic Functions with Critical Points of Large Multiplicities

Counting joints with multiplicities

Let $\mathfrak{L}$ be a collection of $L$ lines in $\R^3$ and $J$ the set of joints formed by $\mathfrak{L}$, i.e. the set of points each of which lies in at least 3 non-coplanar lines of $\mathfrak{L}$. It is known that $|J| \lesssim L^{3/2}$ (first proved by Guth and Katz). For each joint $x \in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that $\sum_{x \in J}N(x)^{1/2} \lesssim L^{3/2}$, while in the last section we extend this result to real algebraic curves of uniformly bounded degree in $\R^3$, as well as to curves in $\…

张量积分解、优秀滤过中的极限、仿射外尔群轨道与杨表计数

Tensor Product Decompositions, Limits in Excellent Filtrations, Affine Weyl Group Orbits, and Tableaux Counting

该研究将A型仿射Kac-Moody代数基本单模张量积外重数的相关结果从秩1推广到所有秩，结合优秀滤过稳定极限与2级仿射外尔群轨道的组合描述，建立了Misra-Wilson杨表集与多划分集不交并的双射关系。

通过词编码和生成函数计数脊柱树-子网络

Counting Spinal Tree-Child Networks via Word Encodings and Generating Functions

研究脊柱树-子网络的计数问题，提出词模型和符号方法两种框架，通过生成函数和递归规范实现闭式计数。

Plethysm and lattice point counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition $μ$ of 3,4, or 5 we obtain an explicit formula in $λ$ and $k$ for the multiplicity of $S^λ$ in $S^μ(S^k)$.

关于极值多重图理论中的若干问题

On problems in extremal multigraph theory

本文研究 (s,q)-多重图中边重数和与积的极值问题，解决了 Day、Falgas-Ravry、Treglown 等人的猜想，揭示了其复杂行为并指出一般情形可能计算难解。

Expectations of Tutte-related functions of random ranked sets with multiplicities

Employing two models, we show that various counting functions of a random variable defined by restriction or contraction of a ranked set with multiplicity (e.g., classical and arithmetic matroids) have expectations given by the corresponding multivariate Tutte polynomial. The first model is based on a generalization of a convolution formula of Kung (2010), extending from matroids to ranked sets with multiplicities. This model enables us to compute the expectations of many familiar polynomials, such as the chromatic, flow and Ehrhart polynomials, generalizing the classical results of Welsh (199…