This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

Hamiltonian time-series data are observations derived from a Hamiltonian dynamical system. Our goal is to analyze the time-series data using the topological information of Hamiltonian dynamical systems. Exact Multi-parameter Persistent Homology is one aspect of this analysis, in this case, the Hamiltonian system is composed of uncoupled one-dimensional harmonic oscillators. This is a very simple model. However, we can induce the exact persistence barcode formula from it. From this formula, we can obtain a calculable and interpretable analysis. Filtration is necessary to extract the topological information of data and to define persistent homology. However, in many cases, we use static filtrations, such as the Vietoris-Rips filtration. My ongoing research is on topological optimization, which involves finding a filtration in Exact Multi-parameter Persistent Homology that minimizes the cross-entropy loss function for the classification of time-series data.

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

Geometric and topological structures can aid statistics in several ways. In high dimensional statistics, geometric structures can be used to reduce dimensionality. High dimensional data entails the curse of dimensionality, which can be avoided if there are low dimensional geometric structures. On the other hand, geometric and topological structures also provide useful information. Structures may carry scientific meaning about the data and can be used as features to enhance supervised or unsupervised learning. In this talk, I will explore how statistical inference can be done on geometric and topological structures. First, given a manifold assumption, I will explore the minimax rate for estimating the dimension of the manifold. Second, also under the manifold assumption, I will explore the minimax rate for estimating the reach, which is a regularity quantity depicting how a manifold is smooth and far from self-intersecting. Third, I will investigate inference on cluster trees, which is a hierarchy tree of high-density clusters of a density function. Fourth, I will investigate inference on persistent homology, which quantifies salient topological features that appear at different resolutions of the data.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.