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一、 課程背景與目的：
Over the past decade, topological data analysis has undergone a rapid development and expansion; it has been proved effective in analyzing data sets, which traditional methods cannot cope with.
The main tool in topological data analysis is persistent homology, which is often defined and computed in terms of simplicial complex structures; namely, for every filtrated simplicial complex, one can compute its associated persistence diagram.
In recent years, new interpretation of persistent homology from a more categorical point of view has been worked out by several authors, and this point of view provides a more general framework and allows us to define multidimentional persistence and could eventually lead to new methods of analyzing data sets that have more than one filtrations.
This goal of the seminar is to understand the approach to multidimensional persistence and its theoretical foundation laid out by J. M. Curry in his thesis “Sheaves, cosheaves, and applications”.
二、 課程之大綱：
• Week 1: Multidimensional Persistence Diagrams via (co)sheaf Theory--Categorical Preliminaries
• Week 2: Sheaves and cosheaves
• Week 3: Examples
• Week 4: Cellular sheaves and cosheaves
• Week 5: Functors associated to maps
• Week 6: Homology and cohomology
• Week 7: Derived category theory
• Week 8: Persistence in terms of (co)sheaves I
• Week 9: Persistence in terms of (co)sheaves II
• Week10: Multidimensional persistence
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