差异理论：算法与几何视角
Discrepancy Theory: An Algorithmic and Geometric Perspective
AI总结：
该专著介绍差异理论的现代进展，聚焦其核心算法与凸几何思想，为相关成果提供新的简洁分析，涵盖组合差异理论的核心问题、经典成果及近年发展。
AI中文摘要：
组合差异理论是一门植根于组合学、几何学和数论的学科，在数学和计算机科学领域有着众多应用。其核心是将一组对象划分为尽可能平衡的两部分，例如，给定有限论域的一组子集，我们可用两种颜色对元素着色，使每个集合大致均匀拆分；其他差异问题更具几何特性，例如为一组向量分配符号，使带符号向量的和尽可能小。Beck-Fiala定理和Spencer的“六偏差”结果等经典成果表明，通常可以实现极小的差异，往往远小于朴素随机着色所能达到的水平。近年来，差异理论因新算法技术及概率、优化与凸几何之间的丰富相互作用而发生转变，这些进展不仅为基础定理提供了新的构造性证明，还催生了若干新成果和研究方向。本专著旨在为这些现代进展提供易懂且统一的介绍，重点阐述推动这些进展的核心算法和凸几何思想，对部分成果提供了新的更简洁的分析，同时强调证明背后的直观逻辑。
英文摘要：
Combinatorial discrepancy theory is a subject with roots in combinatorics, geometry, and number theory, and with numerous applications to mathematics and computer science. At its core, discrepancy theory is about dividing a collection of objects into two parts that are as balanced as possible. For example, given a collection of subsets of a finite universe, we may wish to color the elements with two colors so that each set is approximately evenly split. Other problems in discrepancy are more geometric in flavor, and ask for example, to assign signs to a collection of vectors, so that the sum of the signed vectors is as small as possible. Classical results, such as the Beck-Fiala theorem and Spencer's "six deviations" result, show that it is often possible to attain remarkably small discrepancy, often far smaller than what naive random colorings achieve. In recent years, discrepancy theory has undergone a transformation, driven by new algorithmic techniques and a rich interplay between probability, optimization, and convex geometry. These developments have led not only to new constructive proofs of foundational theorems, but also to several new results and research directions. This monograph aims to provide an accessible and unified introduction to these modern developments, with a focus on the core algorithmic and convex geometric ideas that have driven them. For several results, we provide new simpler analyses, while highlighting the intuition behind the proofs.
