Topological Data Analysis (TDA) has emerged as a powerful framework for uncovering meaningful structure in high-dimensional, complex datasets. In this talk, we present two applications of TDA in analyzing patterns, one in the tumor microenvironment (TME) and the other in high-resolution chemical profiling. In the first case, we develop a TDA-based framework to quantify malignant-immune cell interactions in Diffuse Large B Cell Lymphoma using multiplex immunofluorescence imaging. By introducing Topological Malignant Clusters (TopMC) and leveraging persistence diagrams, we capture both global infiltration patterns and local density-based features. This robust approach enables consistent prognostic assessment regardless of tumor region heterogeneity and reveals correlations with patient survival. In the second application, we utilize the Ball Mapper algorithm to simplify and visualize high-dimensional data obtained from 2D Chromatography with high-resolution mass spectrometry. This enables interpretable chemical profiling of complex mixtures and supports tasks such as sample authentication and environmental analysis. Together, these studies demonstrate the versatility and interpretability of TDA for extracting biologically and chemically meaningful information.

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https://scholar.google.com/citations?user=4w2vNhcAAAAJ&hl=en

This talk is based on joint work with Sungkyung Kang and JungHwan Park. We show that the (2n,1)-cable of the figure-eight knot has infinite order in the smooth concordance group, for any n≥1. The proof relies on the real κ-invariant, which satisfies a real version of the 10/8-inequality, in combination with techniques involving higher-order branched covers of knots and surfaces. Together with earlier work by Hom, Kang, Park, and Stoffregen, this result implies that any nontrivial cable of the figure-eight knot has infinite order in the smooth concordance group.

TBA