Some Problems on Pseudo-Random Structures in Discrete Mathematics

图论和数论中的各种结构通过小的排除子结构来捕捉伪随机性。相反，真正的随机矩阵引入了统计结构(例如，维格纳半圆律)。建议通过三类特定的问题来解决这些普遍现象：具有某些排除子图的图问题(小团或树)，关于整数的无和序列的数论问题，以及关于随机矩阵的各种结构定理。所提出的主题适用于数学和科学。尤其是，图问题可以在傅里叶分析(因此在通信技术中)、离散几何(因此在机器人学中)、数论(因此在编码理论中)中得到应用在设计和分析高效的计算机算法(复杂性理论)中，随机矩阵在数学物理中有其最显著的应用，但在从多元统计到运筹学、生物信息学到学习理论等广泛的科学和数学分支中，它们也是非常重要的--仅举几个重要的例子。

Various structures in graph theory and number theorycapture pseudo-randomness by small excluded substructures.Conversely, truly random matrices introducestatistical structure (e.g., Wigner's Semi-Circle Law).The proposal addresses these general phenomenathrough three specific classes of problems:graph problems with certain excluded subgraphs(small cliques or trees), number theory problemsabout sum-free sequences of integers,and various structural theorems for random matrices.The proposed topics are applicable in mathematics and the sciences.In particular, the graph problems may find applicationsin Fourier analysis (and hence in Communications Technology),in Discrete Geometry (and hence in Robotics),in Number Theory (and hence in Coding Theory),and in designing and analysing efficientcomputer algorithms (Complexity Theory).Random matrices have their most notable applicationsin Mathematical Physics, but they are also very importantin a broad range of sciences and several branches of mathematicsfrom Multivariate Statistics to Operations Researchto Bioinformatics to Learning Theory-- just to mention some important ones.