■ ※課程目標及內涵 (Course Objectives and Contents) (限1950中文字)
The course will serve as an introduction to topological data analysis (TDA), with a focus on persistent homology of point clouds for applications to data analysis. TDA is a relatively young field (established within the last 20 years and really picking up steam over the last decade) and can be studied from a theoretical perspective (mainly involving ideas from algebraic topology and metric geometry) or from a computational perspective (involving optimization, machine learning and domain-specific applications). The course will present a balance of both perspectives. More specifically, the course will include introductions to basic ideas of topology, simplicial homology, data clustering, advanced topics from metric geometry (Gromov-Hausdorff distance) and introductory machine learning. Real-world applications to data analysis will be provided. The course will roughly break down into three parts:
Background: The first part of the course will be spent on background material from data science, linear algebra and algebraic topology.
Foundations of TDA: The main part of the course will cover foundational results and techniques from TDA. This will give you the prerequisite knowledge to start working on your final project
Special Topics: We will be spent on exploring some more specialized topics in modern TDA research. The particular topics we cover here may be tailored to the specific research interests of the students.

■ ※Course Objectives and Contents (limited to 1950 Chinese characters)
The course will serve as an introduction to topological data analysis (TDA), with a focus on persistent homology of point clouds for applications to data analysis. TDA is a relatively young field (established within the last 20 years and really picking up steam over the last decade) and can be studied from a theoretical perspective (mainly involving ideas from algebraic topology and metric geometry) or from a computational perspective (involving optimization, machine learning and domain-specific applications). The course will present a balance of both perspectives. More Specifically, the course will include introductions to basic ideas of topology, simplicial homology, data clustering, advanced topics from metric geometry (Gromov-Hausdorff distance) and introductory machine learning. Real-world applications to data analysis will be provided. The course will roughly break down into three parts:
Background: The first part of the course will be spent on background material from data science, linear algebra and algebraic topology.
Foundations of TDA: The main part of the course will cover foundational results and techniques from TDA. This will give you the prerequisite knowledge to start working on your final project
Special Topics: We will be spent on exploring some more specialized topics in modern TDA research. The particular topics we cover here may be tailored to the specific research interests of the students.

拓樸資料分析是一種利用拓樸學的方法來分析資料的技術。有別於一般資料分析依賴傳統的距離或統計量，拓樸資料分析會挖掘資料在高維空間中的形狀與連接方式，並用這些形狀特徵去做分析或分類。

Topological data analysis is a technology that uses topological methods to analyze data. Unlike general data analysis that relies on traditional distances or statistics, topological data analysis mines the shape and connection of data in high-dimensional space, and uses these shape features for analysis or classification.

Carlsson, G., 2014. Topological pattern recognition for point cloud data. Acta Numerica, 23, p.289.
Needham, T., 2020. Introduction to Applied Algebraic Topology.

Carlsson, G., 2014. Topological pattern recognition for point cloud data. Acta Numerica, 23, p.289.
Needham, T., 2020. Introduction to Applied Algebraic Topology.