Unlike the classical tools and methods that data scientists have historically employed to analyze data, topological data analysis (TDA) provides a tool for analyzing the shape of data. One of the central tasks in TDA is finding the proper representation of a given data cloud. Such representation contains algebraic and geometric features of the data cloud, and one can understand the shape of the data using those features. Moreover, one can evaluate the difference or similarity between two data clouds by measuring the similarity of their barcodes.
Although the theory of 1-parameter TDA is a powerful tool helping people understand data, the data cloud itself may naturally contain more than one parameter. Analyzing one parameter at a time is not only time-consuming but also information-losing because taking one parameter at a time only gives a slice of the feature by fixing the other parameters. Multi-parameter topological data analysis has promising application potential in dealing with higher-dimensional data clouds; however, the theory of multi-parameter TDA is currently under development. It is known that when there are multiple parameters, it is impossible to obtain a classification theorem, and such a theorem is crucial in order to generalize the notion of barcodes to the higher dimensions.
I am interested in both theoretical and applied aspects of Topological Data Analysis and Topological Robotics. On the theoretical side, in particular, I am currently developing my understanding and pursuing my research in the following topics:
projective and injective modules over the incidence algebra on a (locally-finite) poset;
exact structures of persistence modules over a (locally-finite) poset.