ece598: topological data analysis

Good working knowledge of linear algebra (Math 257, 416…) and basic ideas of analysis (Math 424, 432, 444, 447…) should suffice. If in doubt, contact the instructor.

Lectures 4-5:20 pm MW in ECEB3015.

Foundations of Topology: 4 weeks

• Point-Set Topology and Continuous Spaces
  Topics :: Definition of a topological space. Open and closed sets, bases, and neighborhoods. Continuous functions between topological spaces. Key topological properties: compactness and connectedness.
  Context :: Understanding why standard metric spaces in engineering (like $\mathbb{R}^n$) are special cases of topological spaces, and how sensor data inherently live in these spaces.
• Homotopy and Topological Equivalence
  Topics :: Homotopy between continuous maps. Homotopy equivalence and deformation retracts. Introduction to the fundamental group $\pi_1(X, x_0)$.
  Context :: Formalizing the “rubber-sheet geometry” concept. Understanding which structural features of a design or dataset are invariant under continuous deformations (e.g., coordinate transformations or stretching).
• Simplicial Complexes and Proximity Graphs
  Topics :: Abstract and geometric simplicial complexes. Constructing discrete spaces from point clouds. The Vietoris-Rips complex $VR(P, r)$, Čech complex $\check{C}(P, r)$, and Alpha complexes. The Nerve Theorem.
  Context :: Turning a finite set of discrete sensor readings into a continuous topological space suitable for analysis.
• Homology Theory
  Topics :: Simplicial chains $C_p$, cycles $Z_p$, and boundaries $B_p$. The boundary operator $\partial_p$. Defining the homology group as a quotient space: $H_p = Z_p / B_p$. Computing Betti numbers $\beta_k$.
  Context :: Algorithmic counting of connected components, holes, and voids in structural models, CAD designs, or sensor networks.
• Basics of Morse Theory
  Topics :: Manifolds. Smooth functions and non-degenerate critical points. Gradient flows and cell decompositions. Morse inequalities.
  Context ::Applications in dynamical systems and control theory. Cell attachments and obstacles.

Persistent Homology and Stability: 3 weeks

• Filtrations and Persistent Homology
  Topics :: Parameterized spaces and the filtration sequence $\emptyset = K_0 \subseteq K_1 \subseteq \dots \subseteq K_n = K$. The fundamental lemma of persistent homology and the algebraic birth and death of homology classes.
  Context :: Multi-scale analysis. Observing how features evolve as the resolution parameter (e.g., sensor range) increases.
• Persistence Diagrams and Barcodes
  Topics :: Visual representations of topological persistence. Algebraic representation of birth-death pairs $(b, d)$. The space of persistence diagrams and the Bottleneck Distance $W_\infty(D_1, D_2)$.
  Context :: Quantifying the difference between the topological signatures of two distinct physical states or systems.
• The Stability Theorem
  Topics :: Bounding topological noise. The proof that for tame, continuous functions $f, g: X \to \mathbb{R}$, the bottleneck distance is bounded by the supremum norm:
  $$W_\infty(D(f), D(g)) \leq |f – g|_\infty$$
  Context :: Theoretical guarantees that bounded sensor noise or small measurement errors will not drastically alter the extracted topological features.

Stochastic Topology & Gaussian Random Fields: 4 weeks

• Topology of Random Point Processes
  Topics :: Poisson Point Processes (PPP) in $\mathbb{R}^d$. Expected topology of random geometric graphs. Subcritical, critical, and supercritical phase transitions of Betti numbers $\mathbb{E}[\beta_k]$ as $n \to \infty$ (the thermodynamic limit).
  Context :: Determining the expected connectivity and coverage holes in a randomly deployed wireless sensor network.
• Gaussian Random Fields (GRFs) and Excursion Sets
  Topics :: Modeling spatial uncertainty. Using Rice’s Formula to compute the expected Euler Characteristic of the Excursion sets.
  Context :: Modeling surface roughness in manufacturing, material stress distributions, or fluid turbulence fields.
• Morse Theory for Random Surfaces
  Topics :: Linking critical points of GRFs to topological birth/death events via the Morse Index. Predictability of topological changes in random scalar fields.
  Context :: Identifying critical stress points or fluid stagnation points strictly from the topological behavior of the scalar field.

Statistical Inference and Domain Applications: 4 weeks

• Statistical Inference in TDA
  Topics :: Noise: light and heavy tailed. Extreme outliers. The DTM function $d_{\mu, m}(x)$ and its 1-Lipschitz continuity properties.
  Context :: Filtering out spurious acoustic emission hits or defective sensor readings before calculating the system’s topology.
• Signal Processing and Time Series
  Topics : Takens’ Delay Embedding Theorem. Reconstructing phase spaces from 1D scalar measurements and applying persistent homology to detect periodicity.
  Persistence for trajectories of Brownian motions.
  Context :: Detecting regime shifts or impending failures in rotating machinery based on 1D vibrational time-series data. Financial time-series.
• Material Science & Porous Media
  Topics :: Characterizing molecular geometries, crystal structures, and 3D-printed metal porosity. Inverse problem mapping from barcodes to physical properties.
  Context :: Quantifying the quality of additive manufacturing builds by comparing the persistence diagram of the internal void space to a baseline standard.
• Final Project Presentations

• Munkres, J. R. (2000). Topology (2nd Edition). Prentice Hall.
• Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
• Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society.
• Adler, R. J., & Taylor, J. E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics.

Relevant papers will be added.

Homework assignments (4, at 12.5% each) and a final project (50%). Final project will focus on working on a dataset, finding and interpreting its topological invariants.