Lecture 1: Fusing Statistical Survival Analysis with Topological Methods for Validation of Medical Imaging

Abstract: Synthetic scientific imagery is increasingly used to benchmark AI systems, support regulatory evaluations, and accelerate scientific discovery, but reliable methods for validating their realism remain limited. We introduce a new validation framework that combines persistent homology from computational topology with survival analysis from statistics, by treating the persistence of topological features as survival times. This enables us to harness the statistical formalism for estimating, comparing, and hypothesis testing of the associated topological survival functions, uncovering latent structural differences across all resolution scales, quantifying uncertainty, and providing interpretable insight into where and how synthetic and real images diverge. We illustrate the idea in application to synthetic digital mammography, demonstrating the  capabilities to reveal subtler multi-resolution patterns that are missed by existing validation methods. This is a joint work with Asim Dey, Jaidev Goel, Ansu Chatterjee, and Vincent Poor.

Lecture 2: Leveraging Morse theory for diffusion on non-Euclidean objects

Abstract: Topology is, by its nature and design, suited to structures that are nonlinear, multiscale, and nonstationary. However, within machine learning and data sciences, use of topology remains largely confined to topological data analysis. In this talk we discuss how tools from low-dimensional topology such as Morse theory which have remained almost exclusively contained within the domain of pure mathematics can help us to enhance generative AI such as graph diffusion models. This introductory talk is based on the joint work with Jennifer Rozenblit and Yuzhou Chen.

Lecture 1: 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.

Lecture 2: Topology in Motion: Advanced Applications to Echocardiography and Topological Mode Decomposition

Abstract: Building upon the mathematical foundations of Topological Flow Data Analysis (TFDA), this presentation explores two advanced applications bridging computational topology, clinical medicine, and operator theory. First, I will present a new TFDA theory for left ventricle (LV) blood flow. This theory successfully extracts topological features of cardiac flow structures, allowing us to identify well-organized topological vortex structures that characterize healthy blood flow patterns, as well as inefficient patterns in diseased hearts. Second, I will introduce Topological Mode Decomposition (TMD) for pattern time series. I will show that the transition matrix of topological states serves as an Ulam-type approximation of the Koopman operator, allowing us to extract structural cycles from chaotic flows. This demonstrates the profound intersection of topology, symbolic dynamics, and operator theory.

Lecture 1: Learning from the Cosmic Web: Topological Data Analysis for Cosmological Inference

Abstract: A central challenge in modern cosmology is to extract the underlying physics from the increasingly large and complex datasets provided by large-scale structure surveys. The cosmic web contains rich information about cosmological parameters and the physics of structure formation, much of which is not captured by conventional two-point statistics. Topological data analysis (TDA), and in particular persistent homology, provides a powerful framework for characterizing the multiscale structure and connectivity of complex datasets through interpretable topological summaries. These statistics are stable against perturbations and sensitive to non-local, higher-order features of the cosmic web.

In this talk, I will discuss how persistent homology can be used to extract cosmological information from the large-scale structure of the Universe. I will present recent results on constraining cosmological parameters and primordial non-Gaussianity using topological summaries, including approaches that combine persistent homology with machine learning for parameter inference. I will conclude with prospects for topology-based inference as a complementary probe of the physics encoded in the cosmic web.

Lecture 2: The Topology of Complex Systems: From Phase Transitions to Emergent Phenomena

Abstract: Many physical systems exhibit collective behavior and emergent phenomena that are difficult to characterize using conventional statistical observables. Phase transitions, in particular, are associated with qualitative changes in the structure and organization of a system, often involving correlations and collective features that extend across multiple length scales. Topological data analysis (TDA), and in particular persistent homology, offers a complementary approach to identifying such changes by quantifying the multiscale topology of complex datasets.

In this talk, I will discuss how persistent homology can be used to detect and characterize phase transitions in statistical systems. By tracking the appearance and persistence of topological features across different scales, one can construct interpretable summaries that capture critical phenomena associated with phase transitions. I will discuss how these topological signatures can help distinguish phases and identify critical temperature and critical exponents, illustrating the potential of TDA as a tool for studying complex physical systems. If time permits, I will also discuss possible connections to neuroscience, where topological methods offer a promising way to characterize collective organization and emergent structure in neural data.

Title: Persistent Tor-algebras: Topological realizations and stability 

Abstract: This talk introduces persistent Tor-algebras, which capture more refined algebraic information than standard persistent homology. Built from the face rings of simplicial complexes in a filtration, these invariants connect the theory of persistence modules with combinatorial commutative algebra and toric topology. We explain their topological interpretation through moment-angle complexes, a class of polyhedral products whose cohomology realizes the corresponding Tor-algebras. We then present several stability theorems for persistent Tor-algebras. This talk is based on joint work with A. Bahri, I. Limonchenko, T. Panov, and D. Stanley.

Title: A Mathematical Introduction to Data Assimilation and Connections to Topological Data Analysis

Abstract: Data assimilation is a framework for estimating the state of a dynamical system by combining a mathematical model with observations. It plays an essential role in applications such as weather forecasting, where the dynamics are high-dimensional and chaotic, and observations are incomplete and noisy. This talk provides a mathematical introduction to data assimilation, with an emphasis on ensemble Kalman filtering. I will present recent results from my work on the long-time accuracy of ensemble filters and numerical evidence relating the required ensemble size to the unstable directions of the dynamics. I will also introduce work related to topological data analysis and discuss potential connections between topological methods and data assimilation.

Lecture 1: Topological data analysis based machine learning

Abstract: A central challenge in artificial intelligence (AI)-driven molecular science lies in efficiently representing molecular data and developing learning architectures that capture intrinsic structure-function relationships. In this talk, I will introduce advanced mathematics-based molecular representations built on topological data analysis (TDA). Molecular structures and interactions are encoded using high-order topological representations, including Rips complexes, Alpha complexes, Neighborhood complexes, Dowker complexes, Hom-complexes, and Tor-algebras. I will explain how these representations capture multiscale and high-order structural information that conventional descriptors miss, and how they serve as input features for machine learning models. The talk will cover the mathematical foundations of these representations, practical construction algorithms, and their integration into standard ML pipelines. Examples will be drawn from molecular datasets in chemistry, biology, and materials science, demonstrating how TDA-based features uncover complex structural-functional relationships and improve predictive performance.

Lecture 2: Mathematical AI: from topological data analysis to topological deep learning

Abstract: Building on the first talk, this talk moves from topological data analysis (TDA) toward topological deep learning (TDL). Essentially, TDA is a featurization paradigm, while TDL is an end-to-end deep learning paradigm that generalizes geometric deep learning from graphs to higher-order domains such as simplicial and cellular complexes, integrating topological priors directly into differentiable architectures. I will explain the general design of TDL architectures, focusing in particular on simplicial message passing and simplicial attention modules. I will then briefly discuss sheaf-based models, including sheaf normal model, sheaf neural networks (SNNs) and SPD-sheaf networks. I will present applications across chemistry, biology, and materials science, and conclude with a summary for Mathematical AI.

Title: Supervised Learning and Classification Tasks on Hypergraphs via Categorical Concepts

Abstract: Topological Deep Learning extends graph neural networks to higher-order structures such as hypergraphs. I will first briefly explain two tasks in machine learning: supervised learning and hypergraph classification. I will then introduce two categorical concepts, symmetric simplicial sets and a categorical extension of the Weisfeiler–Lehman test, and show how they apply to supervised learning and classification tasks, respectively.

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.

Title: Choosing Filtrations for Fourier Spectra in Exact Multi-parameter Persistent Homology: Frequency-Amplitude Persistence Images and Curved Filtration Learning

Abstract: Exact multi-parameter persistent homology (EMPH) expresses the Fourier spectrum of a signal as a persistence diagram in closed form, and what the diagram captures depends on how the filtration is chosen in the multi-parameter space. With an appropriate choice, the diagram coordinates of each Fourier component are exactly its frequency and its amplitude, so the spectrum can be viewed as a set of points in the frequency-amplitude plane rather than as a function of frequency. In the persistence image built on this plane, amplitude serves both as a weight and as a position, and the image connects directly to the conventional Gaussian-smoothed spectrum. In impact-echo testing of concrete, where boundary reflections, structural modes, and noise overlap within one spectrum, this representation made defect classification more robust than reading the spectrum along frequency alone. This filtration is chosen so that the diagram reads directly as frequency and amplitude, but a filtration can also be chosen for the task itself. Regarding a filtration as good when it classifies well, we define filtrations along curves in the multi-parameter space and learn the curve jointly with the classifier, so that the way each Fourier mode enters the diagram is adapted to the task.

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.

Title: ReLU neural networks: Symmetries, fibers, and topology

Abstract: In this talk, we will study the topology of parameter fibers of univariate ReLU neural networks. We will fix a feedforward ReLU neural network architecture and associate with it a geometric object called a neuromanifold: the space of functions representable by this architecture. The fiber over a target function F consists of all parameter choices that realize F. In general, ReLU networks exhibit a rich structure of parameter symmetries, which can extend beyond neuron permutations and positive rescalings. We will show how this structure is reflected in the topology of the fibers of the parameter map.

Title: Probing the Multiscale Geometry of Learned Representations

Abstract: Can a single intrinsic dimension adequately describe the geometry of learned representations? In this talk, we explore this question using power-weighted minimum spanning trees (MSTs), which probe geometric structure through different edge-length moments. We establish a mathematical connection between MST edge-length distributions and their scaling behavior, and develop finite-sample calibrated diagnostics to assess consistency with single-dimensional scaling. Applications to neural networks and large language models reveal departures from simple geometric reference models and highlight the influence of class separation on representation geometry.

Title:Persistent Homology of Sentence Representations in LLMs and Its Application to Layer Pruning

Abstract: We study how large language models organize sentences at their output interface using persistent homology. Representing each sentence by its lm_head input, we track the H0 persistent homology of the resulting point cloud: as a distance threshold grows, sentences the model represents as close merge first, while distant ones stay apart until large scales. The merge hierarchy thus offers a multi-scale picture of how the model organizes sentences. To ask how this picture changes when a layer is removed, we needed a quantity we could compare across ablations, so we summarized the hierarchy with a compact scalar, the Cluster Persistence Index (CPI), defined as the area under the normalized Betti-0 curve, which captures how widely sentences are separated. We then measure ΔCPI, the change in CPI when a layer is skipped. Across LLaMA-2-7B, Mistral-7B, Qwen2.5-7B, and Phi-4, skipping a few early and late layers changes CPI substantially, while skipping most middle layers barely changes it. As an application, pruning about 20% of layers by ΔCPI gives lower WikiText-2 perplexity than ShortGPT on three of the four models. We also discuss what this approach misses, such as the drop on GSM8K.