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topological data analysis
Topological data analysis (TDA) is a new and vastly growing branch of applied mathematics.
Data analysis is of extreme importance in almost all areas of modern applied science. However, to extract information from real data which are usually large, high-dimensional, incomplete and noisy is challenging. TDA provides a general framework to analyze data, being successful in coordinate-freeness, insensitive to particular metric, dimension reduction and robustness to noise. Beyond, it inherits functorality, one of the keys to the modern mathematics, from its topological nature, which makes it adaptive to new tools from mathematics.
The initial motivation is to study the shape of data. TDA has combined algebraic topology and other tools from pure mathematics to give mathematically strict and quantitative study of "shape". The main tool is persistent homology, a modified concept of homology group. Nowadays, this area has been proven to be successful in practice. It has been applied to many types of data input, and different data resource and numerous fields. Moreover, its mathematical foundation is also of theoretical importance to mathematics itself. Its unique features make it a promising bridge between topology and geometry.
== Basic theory ==
Basic concepts and theoretical results in TDA will be introduced. Specially, the focus would be on the standard paradigm, namely the barcode of point clouds. These results are widely used in applications. For the basic concepts of algebraic topology, please refer to section 2 of Carlsson for a short introduction, or to the standard textbooks, such as Hatcher.
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