Abstracts

In various applications of data classification and clustering problems, multiparameter analysis is effective and crucial because data are usually defined in multiparametric space. Multi-parameter persistent homology, an extension of persistent homology of one-parameter data analysis, has been developed for topological data analysis (TDA). Although it is conceptually attractive, multi-parameter persistent homology still has challenges in theory and practical applications. In this study, we consider time-series data and its classification and clustering problems using multiparameter persistent homology. We develop a multi-parameter filtration method based on Fourier decomposition and provide an exact formula and its interpretation of one-dimensional reduction of multi-parameter persistent homology. The exact formula implies that the one-dimensional reduction of multi-parameter persistent homology of the given time-series data is equivalent to choosing diagonal ray (standard ray) in the multi-parameter filtration space. For this, we first consider the continuousization of time-series data based on Fourier decomposition towards the construction of the exact persistent barcode formula for the Vietoris-Rips complex of the point cloud generated by sliding window embedding. The proposed method is highly efficient even if the sliding window embedding dimension and the length of time-series data are large because the method precomputes the exact barcode and the computational complexity is as low as the fast Fourier transformation of O(N log N). Further the proposed method provides a way of finding different topological inferences by trying different rays in the filtration space in no time.

In 1935, Whitney introduced matroids as an abstraction of the linear dependency of vectors. A rank-d matroid with n elements behaves like a d-dimensional subspace of an n-dimensional Euclidean space. The (co)circuits of a matroid interpret vectors of the vector space, and the bases of a matroid interpret its Plücker coordinate. Thus, matroids play a key role not only in combinatorics but also in geometry. There were many attempts to refine the abstract properties of geometry, and one of the big successes was oriented matroids, which can be regarded as a combinatorial abstraction of point configurations and hyperplane arrangements over the reals. Recently, Baker and Bowler presented 'Matroids over partial hyperstructures' (2019), providing the view of Grassmannian over more general algebraic structures, called tracts, and unifying previous research on this topic.

We survey cryptomorphic definitions of matroids compared to properties of vector spaces. We review the historical background and introduce Baker and Bowler’s theory. Further, we investigate (even) delta-matroids arising from symplectic and orthogonal geometry.

Let R be a commutative ring with identity. Then R is a Noetherian ring if every ascending chain of ideals of R satisfies ascending chain condition (equivalently, every ideal of R is finitely generated.) In this talk, we will investigate ∗-Noetherian. To do this, we need to know the definition of star operation and define w-operation, an example of star operation. Also, we will introduce the properties of star operation and some properties of ∗-Noetherian. More precisely, we will focus on the strong Mori domain and strong Mori module, so we will focus on introducing w-operation.