Algebraic Topology

Aspects of Symmetry, Topology and Anomalies in Quantum Matter

In this thesis, we explore the aspects of symmetry, topology and anomalies in quantum matter with entanglement from both condensed matter and high energy theory viewpoints. The focus of our research is on the gapped many-body quantum systems including symmetry-protected topological states and topologically ordered states. Chapter 1. Introduction. Chapter 2. Geometric phase, wavefunction overlap, spacetime path integral and topological invariants. Chapter 3. Aspects of Symmetry. Chapter 4. Aspects of Topology. Chapter 5. Aspects of Anomalies. Chapter 6. Quantum Statistics and Spacetime Surgery.…

Algebraic topology and the quantization of fluctuating currents

We give a new approach to the study of statistical mechanical systems: algebraic topology is used to investigate the statistical distributions of stochastic currents generated in graphs. In the adiabatic and low temperature limits we will demonstrate that quantization of current generation occurs.

The general structure and ergodic properties of quantum and classical mechanics: A unified C*-algebraic approach

Quantum mechanics is essentially a statistical theory. Classical mechanics, however, is usually not viewed as being inherently statistical. Nevertheless, the latter can also be formulated statistically. Furthermore, a statistical formulation of both can be expressed in terms of unital C*-algebras, in which case it becomes clear that they have the same general structure, with quantum mechanics a noncommutative generalization of the classical case. In purely mathematical terms it is seen that quantum mechanics is a noncommutative generalization of probability theory. The most important insight i…

代数量子场论中的Operads、同伦论与高阶范畴

Operads, homotopy theory and higher categories in algebraic quantum field theory

本章以非技术视角概述代数量子场论（AQFT）与operads、模型范畴、高阶范畴等抽象数学领域近期的交叉互动及研究动因。

Symmetrical laws of structure of helicoidally-like biopolymers in the framework of algebraic topology. I. Root lattice E8 and the closed sequence of algebraic polytopes

In the framework of algebraic topology the closed sequence of 4-dimensional polyhedra(algebraic polytopes) was defined. These polytopes were determined by the second coordination sphere of 8-dimensional lattice E8. The ordered non-crystalline structure is determined by a chain of constructions of algebraic topology: an algebraic polytope, a homogeneous manifold in a 3-dimensional Euclidean space E3, locally-homogeneous manifold, locally minimal surface, one-parameter family of helicoids, bundle (cover) with a base of cell complexes, local-lattice packing of cell complexes into a substructure o…

一种基于单纯集合的代数拓扑量子计算新框架

A new quantum computational set-up for algebraic topology via simplicial sets

本文提出一种基于单纯集合理论的代数拓扑量子计算框架，通过构建有限维单纯Hilbert空间解决同调计算问题，并设计了在有限资源下计算同调空间与Betti数的量子算法方案。