An Introduction to Applied Topology
This talk is an introduction to applied and computational topology. The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3x3 pixel patches from optical images, which can be well-modeled by a Klein bottle, and the configuration space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce topological tools (such as persistent homology) for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets.
Discrete Cubical Homology of Quasimonophobic Graphs
Discrete cubical homology, introduced by Barceló and collaborators in 2014, provides a homology theory associated with discrete homotopy theory for graphs and related discrete structures. In this talk, I will give an accessible introduction to discrete cubical homology, emphasizing both the geometric intuition underlying the theory and the computational challenges that arise even in explicit calculations for small graphs. I will also discuss connections with discrete homotopy theory and present recent work on monophobic cube neighborhoods in graphs. Finally, I will describe recent results on quasimonophobic graphs, degree spectral sequences, and computations of the second discrete cubical homology group for several important families of graphs.
Topology-Aware Medical Image Segmentation: Preserving Anatomical Structure in Learning-Based Methods
Medical image segmentation methods often produce anatomically implausible results, such as disconnected regions or incorrect topology. Topological approaches address this issue by incorporating global shape and connectivity constraints into segmentation models. This talk introduces key ideas such as persistent homology, Euler characteristic–based constraints, and topology-preserving formulations, and discusses how they can be integrated with modern deep learning frameworks. Applications in medical imaging domains such as brain, vessel, and lesion segmentation are briefly highlighted, along with current challenges and future directions in topology-aware learning.
Sheaf Cohomology in Network Coding
In this expository talk, we will explain what network codings are and how they can be reframed in terms of sheaf using "Network code sheaf" as mention by Ghrist and Hiraoka. Sheaves are gadgets that capture local and global information of a topological sace. This could be continuous functions, differential forms etc. Sheaf cohomology measures the obstruction to glue together local information to get a cohorent global picture. Using the machinery of sheaves and sheaf cohomology, many important information e.g. max flow-min cut theorem, robustness, node failure can be interreted in terms of sheaf cohomology. These connections will be explored in this talk.
From Simplicial Sets to the definition of \infty-Categories
This talk is an exposition of infinity categories. Given the simplex category ∆ and its elementary coface and codegeneracy maps, a simplicial set is a contravariant functor X : ∆^op → Set. Unpacking this yields a sequence of sets X0, X1, X2,... equipped with face and degeneracy maps satisfying the simplicial identities. A morphism of simplicial sets consists of maps Xn → Yn compatible with the face and degeneracy maps. We then introduce the nerve functor N : Cat → sSet and prove it is fully faithful, and characterize its essential image by the unique inner horn-filling property. Relaxing uniqueness, requiring any filler for each inner horn, leads naturally to the definition of an ∞-category (quasicategory). We conclude with definition of ∞-groupoid and the homotopy hypothesis.
Multiparameter Persistent Homology for Hypergraphs
This talk introduces multiparameter persistent embedded homology for hypergraphs. We begin by motivating the need for hypergraph models in topological data analysis and discuss the importance of extending persistence theory to hypergraphs in the multiparameter setting. We then introduce embedded homology for hypergraphs and develop a categorical framework for hypergraph filtrations by defining filtration functors and associated persistent modules. Following this, we define the interleaving distance for these persistent modules and present stability results for the resulting multiparameter persistence framework.
Homology Theories of hypergraph
We introduce three homology theories for hypergraphs, namely Γhomology, □-homology, and ×-homology, and show that they are pairwise nonisomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that □-homology is invariant under this homotopy. Based on excision, we also identify a distinctive structural behavior exhibited by Γ-homology that further di erentiates it from □-homology.
From Persistence Diagrams to Deep Learning: A TDA Pipeline
The symposium presentation will outline a practical TDA pipeline beginning with data representation and simplicial complex construction, followed by the computation of persistence diagrams (PDs) as compact summaries of multi-scale topological information. The talk will further discuss methods for transforming these summaries into machine-learning-compatible representations, including persistence landscapes (PLs) and persistence images (PIs), which enable seamless integration with classical learning algorithms and modern deep neural networks. Particular emphasis will be placed on the stability, interpretability, and computational advantages of these representations in high-dimensional and noisy data settings.
A Review of Topological Tools in Machine Learning
During our research, we explored how ideas from topology can enrich machine learning. While machine learning models excel at pattern recognition, they often miss the bigger picture of how data is connected. Topology offers tools to capture this global structure in a stable way.
In this talk, I will review existing work on vectorized topological descriptors, differentiable topological layers and applications in medical imaging. I will highlight the challenges discussed in the literature, the results these studies provide, and how engaging with this body of work has shaped my perspective on building more interpretable and geometry‑aware learning systems that better respect the shape of data.
Euler Characteristics Tools for Topological Data Analysis
This talk reviews the paper “Euler Characteristic Tools for Topological Data Analysis.” Topological Data Analysis (TDA) provides methods for extracting geometric information from complex datasets, but persistence diagrams, while powerful, can be computationally expensive and difficult to integrate with machine learning. The reviewed work introduces Euler characteristic profiles and Hybrid Transforms as efficient alternatives for topological feature extraction. These tools generate compact, vector‑valued descriptors that are naturally suited for machine learning and avoid the computational burden of persistence diagrams. The talk highlights how these methods provide stable, computationally efficient, and machine learning‑friendly representations for modern TDA.