Introductive Backgrounds of Modern Quantum Mathematics with Application to Nonlinear Dynamical Systems

数学物理中的几个漂亮方程

Some beautiful equations of mathematical physics

概述了规范原理的基本思想及其在现代基本粒子物理中的重要作用。量子场论中有三个理论上一致的规范原理：电磁学和标准模型中的自旋1规范原理，广义相对论中的自旋2规范原理，超引力中的自旋3/2规范原理。（狄拉克奖演讲，1993年11月。两个数字是不可用的，但不应该是必要的。

皮埃尔·卡蒂埃：一位富有远见的数学家

Pierre Cartier: A Visionary Mathematician

本文回顾了20世纪至21世纪数学界杰出人物皮埃尔·卡蒂埃的生平与贡献，他作为布尔巴基学派核心成员推动数学形式化现代化，研究跨多领域且以知识慷慨和人文主义方法影响数学思想与文化理解。

关于 Fibonacci 镶嵌及其现代衍生

On the Fibonacci tiling and its modern ramifications

本文从著名的 Fibonacci 链出发，解释非周期序数学理论的新镶嵌与性质，并指出其各种推广及高维现象与结果，为该主题的现代文献提供切入点。

$p$-Adic Mathematical Physics: The First 30 Years

$p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future …

交互空间：迈向复杂系统的通用数学理论

Interaction spaces: towards a universal mathematical theory of complex systems

本文提出交互空间理论的初步框架，用统一数学概念涵盖元胞自动机、智能体模型、神经网络等多种复杂系统建模方法。

Undergraduate Lecture Notes in De Rham-Hodge Theory

These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The only necessary background for comprehensive reading of these notes is Green's theorem from multivariable calculus.

Quantum gravity: unification of principles and interactions, and promises of spectral geometry

Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in modern mathematics. It is however unclear whether it will ever become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss the general opinion that the mere mathematical complexity of …

Introduction to Modern Canonical Quantum General Relativity

This is an introduction to the by now fifteen years old research field of canonical quantum general relativity, sometimes called "loop quantum gravity". The term "modern" in the title refers to the fact that the quantum theory is based on formulating classical general relativity as a theory of connections rather than metrics as compared to in original version due to Arnowitt, Deser and Misner. Canonical quantum general relativity is an attempt to define a mathematically rigorous, non-perturbative, background independent theory of Lorentzian quantum gravity in four spacetime dimensions in the c…