Topological data analysis (TDA) is an emerging concept in applied mathematics, by which we can characterize shapes of massive and complex data using topological methods. In particular, the persistent homology and persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems. In this talk, I will survey our recent research on persistent homology from three interrelated perspectives; quiver representation theory, random topology, and applications on materials science. First, on the subject of quiver representation theory, I will talk about our recent challenges to develop a theory of multiparameter persistent homology on commutative ladders. By applying interval decompositions/approximations on multiparameter persistent homology (Asashiba et al, 2022) to our setting, I will introduce a new concept called connected persistence diagrams, which properly possess information of multiparameter persistence, and show some properties of connected persistence diagrams. Next, about random topology, I will show our recent results on limit theorems (law of large numbers, central limit theorem, and large deviation principles) of persistent Betti numbers and persistence diagrams defined on several stochastic models such as random cubical sets and random point processes in a Euclidean space. Furthermore, I will also explain a preliminary work on how random topology can contribute to understand the decomposition of multiparameter persistent homology discussed in the first part. Finally, about applications, I will explain our recent activity on materials TDA project. By applying several new mathematical tools introduced above, we can explicitly characterize significant geometric and topological hierarchical features embedded in the materials (glass, granular systems, iron ore sinters etc), which are practically important for controlling materials funct

Modern datasets are often characterized by high-dimensionality and heterogenous environments under distribution shifts, posing significant challenges in terms of signal recovery, robustness, and interpretability. In this talk, I will present three research contributions to address these challenges. First, I will introduce the notion of local concavity coefficients, a novel tool for quantifying the concavity of a set. I will demonstrate its effectiveness in analyzing optimization problems for signal recovery in high-dimensional settings. Second, I will discuss recent advancements in machine learning to handle distribution shifts and emphasize the critical role of invariant features in achieving robust predictions. Finally, I will show how wavelets allow for interpreting feature spaces learned by deep neural networks. Motivated by applications in cosmology, I'll showcase how this tool can be applied to the problem of cosmological parameter inference.

A perfect field is said to be Kummer-faithful if the Kummer maps for semiabelian varieties over the field are injective. This notion originates in the study of anabeian geometry. At the same time, our study is also motivated by a conjecture of Frey and Jarden on the structure of Mordell-Weil groups over large algebraic extensions of a number field. I will begin with a review of known results in this direction, as well as a brief discussion on anabelian geometry. Then I will introduce some recent results on the construction of "large" Kummer faithful fields. This is a joint work with Takuya Asayama.

In this talk, I will discuss the expansion of the free energy of two-dimensional Coulomb gases as the size of the system increases. This expansion plays a central role in proving the law of large numbers and central limit theorems. In particular, I will explain how potential theoretic, topological, and conformal geometric information of the model arises in this expansion and present recent developments.