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Title: Topology in Motion: Mathematical Foundations of Topological Flow Data Analysis

Abstract: Topological Flow Data Analysis (TFDA) is based on a classification theory of topological structures of particle orbits generated by two-dimensional vector fields. It enables the extraction of both qualitative and quantitative information from complex flow data obtained through numerical simulations and laboratory experiments. This presentation will introduce the mathematical framework of TFDA, bridging computational topology and fluid dynamics. I will explain how TFDA uniquely converts the topological structure of a flow pattern into a plane tree (COT), and its corresponding string expression (COT representation). By translating continuous vector fields into symbolic alphabets, we can systematically partition the infinite-dimensional phase space into a finite set of discrete topological states. Simple application examples will be provided to demonstrate this core idea and set the stage for advanced data-driven analysis.

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Title: Near-diagonal asymptotics and the failure of strong universality in random Cech persistence

Abstract: The strong universality conjecture of Bobrowski and Skraba asserts that, under a prescribed data-dependent additive centering, the empirical laws of log-log transformed persistence ratios of random geometric complexes share a universal limit across sampling models, dimensions, filtrations, and homological degrees. They conjectured a left-skewed Gumbel limit and used it as a null law for testing topological significance. We show that, when the uncentered empirical laws and the centerings have deterministic limits, the conjecture forces a second-order quantity of each model's limiting law, determined by ratios close to one and unchanged by additive shifts, to agree across models and dimensions. To compute this quantity for degree-one Cech persistence, we bypass the global persistence pairing with a local proxy based on edges and their first incident triangles in the equivalent alpha filtration. In dimensions two and three, a Palm-Mecke calculation evaluates the proxy intensity exactly for stationary Poisson processes, and two-sided topological bounds show that it agrees with the true normalized persistence intensity through second order.

After transfer to finite samples, the quantity equals π^2/32 and 4/π^2-1/4 for the limiting laws of independent uniform samples from the unit square and the unit cube, respectively. The centered empirical laws converge weakly in probability to distinct deterministic limits, and their mean laws, the expectations of these random laws, also converge weakly to these limits. Hence the conjecture fails even for mean laws. Neither limiting law belongs to the left-skewed Gumbel location-scale family.

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Title: Topology-Aware Representation Alignment for Semi-Supervised Vision-Language Learning

Abstract: Vision-language models have shown strong performance, but they often generalize poorly to specialized domains. While semi-supervised vision-language learning mitigates this limitation by leveraging a small set of labeled image-text pairs together with abundant unlabeled images, existing methods remain fundamentally pairwise and fail to model the global structure of multimodal representation manifolds. Most existing topology-based alignment methods rely on persistence diagram matching, which neither guarantees geometric alignment nor uses the image-text pairing information central to vision-language learning. We propose Topology-Aware Multimodal Representation Alignment (ToMA), a framework that uses persistent homology to decompose pairwise relations into $H_0$-death and $H_1$-birth relations and aligns them across modalities through available cross-modal correspondences. This lightweight decomposition captures both connectivity and cycle-closing structure without constructing 2-simplices. Experiments show clear improvements on remote sensing, while fashion results reveal strong gains in classification and particularly strong retrieval improvements for difficult queries. Additional analysis shows that ToMA provides more consistent gains than alternative topology-based objectives and that lightweight $H_1$-birth edges provide useful higher-order structural signals.

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