Abstracts

Cell differentiation can be conceptualized as movement on Waddington’s epigenetic landscape, yet reconstructing this landscape from high-dimensional single-cell data remains challenging. Here, we propose landscape analysis, a novel framework for single-cell RNA-seq data that reconstructs an RNA landscape, which is Waddington’s landscape-like structure, and performs downstream dynamical analysis utilizing this landscape. Single-cell RNA-seq measures transcript levels for approximately 20,000 genes per cell, producing a high-dimensional expression matrix. By applying RNA velocity, we convert these static profiles into vectors that predict each cell’s future transcriptional trajectory. We then perform Hodge decomposition on this velocity field to extract the potential that forms the gradient component. The resulting potential surface defines the RNA landscape’s height. Finally, by geometrically or statistically analyzing the potential, we derive biologically meaningful insights such as single-cell trajectories, time-resolved differential expression dynamics, and gene functions in cell differentiation. We applied landscape analysis to time-series scRNA-seq data of the PGCLC induction system, identifying differentiation pathways and candidate genes driving induction.

In an era where data utilization is essential across all industries, the importance of mathematics for industry has grown dramatically. However, we face the harsh reality that the value of our work is not properly recognized, and our services are often underpriced.

In particular, we mathematicians who operate in the "midstream"—translating business challenges into mathematical models—face a serious dilemma: our technical skills are appreciated, but they fail to translate into tangible business outcomes.

In this presentation, I will report on my company's real-world experience in confronting this "midstream dilemma." Based on this, I will present to you the fundamental, inherent challenges that we have uncovered within the business model of applying mathematics to industry.

Acute liver failure (ALF) is one of the most critical hepatic conditions, characterized by rapid progression and a high risk of multi-organ failure and death. Liver transplantation remains the only curative treatment, yet predicting which patients will require it is still a major clinical challenge due to the significant heterogeneity in disease progression. In our first study, we analyzed time-series clinical data from 320 patients with acute liver injury and applied machine learning techniques to identify key prognostic indicators. We found that prothrombin time (PT) serves as a central biomarker for tracking individual disease trajectories. By stratifying patients into six distinct PT dynamic patterns, we were able to quantify the severity of ALF and predict its progression from admission data. Furthermore, we demonstrated the feasibility of modeling future PT dynamics using mathematical approaches, offering a personalized framework for understanding and anticipating ALF progression. 

While liver transplantation offers a potential cure for end-stage liver disease, outcome prediction remains a critical issue, particularly in living donor liver transplantation (LDLT), which has gained prominence due to shorter wait times and better graft quality. In our second study, we retrospectively analyzed data from 748 LDLT recipients and developed a machine learning model to predict early graft loss (within 180 days postoperatively) with higher accuracy than conventional models. We stratified patients into five groups based on risk and further identified a specific intermediate-risk group (G2) with characteristics similar to those who experienced early graft loss (G1), but with different survival outcomes. Using data available within the first 30 days post- transplant, we constructed a hierarchical model capable of distinguishing these populations, facilitating earlier clinical intervention such as consideration of retransplantation or alternative donor strategies. 

Together, these studies address the continuum of liver disease—from acute liver injury to post-transplant outcomes—through the lens of time-resolved, individualized prediction. By leveraging machine learning and mathematical modeling, we present a framework that supports more precise and proactive clinical decision-making across the full trajectory of severe liver disease.

Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence diagrams and their interpretations. In this paper, we define a class of distances on graphs, called path-representable distances, and investigate structural relationships between their induced persistent homologies. In particular, we identify a nontrivial injection between the 1-dimensional barcodes induced by two commonly used graph distances: the unweighted and weighted shortest-path distances. We formally establish sufficient conditions under which such embeddings arise, focusing on a subclass we call cost-dominated distances. The injection property is shown to hold in 0- and 1-dimensions, while we provide counterexamples for higher-dimensional cases. To make these relationships measurable, we introduce the total persistence difference (TPD), a new topological measure that quantifies changes between filtrations induced by cost-dominated distances on a fixed graph. We prove a stability result for TPD when the distance functions admit a partial order and apply the method to the SNAP EU Research Institution E-Mail dataset. TPD captures both periodic patterns and global trends in the data, and shows stronger alignment with classical graph statistics compared to previously proposed PH-based measures.

Persistent homology is a widely used tool in topological data analysis, yet standard filtration methods often fail to capture anisotropic structures inherent in real-world data. We propose Ellipse Cloud, a preprocessing-based approach that enhances anisotropic features in persistent homology. Our method constructs a Vietoris–Rips (VR) filtration using ellipse tangency times instead of pairwise Euclidean distances, extending the standard VR filtration to an anisotropic setting. This formulation allows anisotropy to be incorporated into persistent homology while remaining compatible with standard computational frameworks. A key computational challenge in this framework involves determining critical time points at which expanding ellipses first interact, which we address through an efficient numerical algorithm.

To evaluate the effectiveness of our approach, we apply it to a toy problem involving a highly noisy two-dimensional point cloud with multiple ring structures. While standard persistent homology struggles to capture the underlying rings due to excessive noise, our anisotropic filtration successfully identifies optimal 1-cycles that preserve the original structures to a greater extent. More generally, our proposed preprocessing technique tends to increase the lifetime of significant persistence pairs, lowering birth values and raising death values compared to standard VR filtrations. These results suggest that incorporating anisotropic filtrations can provide more informative topological summaries of geometrical structures in data. Potential applications include sensor coverage problems, where sensors often exhibit directional sensitivity rather than isotropic coverage.

In the talk, we will also introduce the Python library `EllPHi` for anisotropic persistent homology analysis. `EllPHi` provides the fast and accurate ellipse-tangency solver. The related source codes are available in [GitHub (https://github.com/t-uda/ellphi)](https://github.com/t-uda/ellphi).

Topological Data Analysis (TDA) generally refers to utilizing topological features from data. A central topic in TDA is persistent homology, which observes data at various resolutions and summarizes topological features that persistently appear. TDA has been proven valuable in enhancing machine learning applications.  This work explores how TDA can enhance machine learning workflows, focusing on two areas: feature extraction and model evaluation. Persistent homology, while rich in structural information, is often challenging to integrate directly into statistical and machine learning frameworks. To address this, various featurization techniques map persistence-based information into Euclidean or functional spaces, enabling its incorporation into neural networks and other learning algorithms. I will examine different approaches that efficiently transform topological summaries into differentiable layers and leverage geometric representations for visualization and dimensionality reduction. In addition to feature extraction, TDA has recently been applied to evaluate data quality and model performance. By quantifying the topological structure of generated or transformed data, TDA-based methods provide robust evaluation metrics that improve the reliability of model assessment. I will present how this is done in particular in generative modeling scenarios.

In this talk, we introduce a new method to estimate Stiefel–Whitney classes—topological invariants that detect features like orientability and embedding obstructions—directly from point cloud data. We first extend classical vector bundle theory to persistent vector bundles in the setting of topological data analysis. By applying cohomology operations to persistent cohomology, we compute these classes in a persistent setting. We demonstrate the method with applications in image analysis, molecular conformation, and high-dimensional data.

In this talk, we analyze ensemble forecasting trajectories from a geometric point of view. We focus on the oriented turning angles to cluster and distinguish different weather scenarios. Ensemble forecasting is a method used in weather prediction, which consists of running multiple forecast simulations, each with slightly varied initial conditions or model parameters, to capture the inherent uncertainty in weather forecasting. Collectively, these outputs map the range and likelihood of future states, and it is crucial to identify and label them to take the proper preventive actions in each situation. We quantify the shape of the trajectory with the Frenet frame, which is a coordinate system attached to a moving point along a curve. In two dimensions, the curvature at each point of the curve defines the frame; in three dimensions, the torsion is additionally included; and analogous quantities extend to higher dimensions. As a first step in this research, we emphasize the oriented turning angle (the cumulative signed change of direction) as a feature for grouping ensemble forecasting data. We apply this to heavy‑rain datasets and show that the turning angle helps distinguish different meteorological scenarios. We think that studying these techniques further will improve interpretation of ensemble information and uncertainty assessment.