By: A Fellow Math Enthusiast 

Target Audience: Data Scientists, Systems Engineers 

Introduction: Data Has a Shape 

Standard statistics (mean, variance) treats data like a cloud of dust. But data often has structure: loops, voids, and tunnels. Topological Data Analysis (TDA) is the math of figuring out the “shape” of your data, which is incredibly useful for detecting anomalies in sensor networks or classifying complex molecules. 

The Syllabus Map 

Your Bachelor’s Subject  The TDA Implementation 
Linear Algebra  Gaussian Elimination (Smith Normal Form) 
Discrete Math  Simplicial Complexes (Connecting the Dots) 
Set Theory  Filtration (Growing the Shapes) 
Abstract Algebra  Homology Groups (Counting the Holes) 
Metric Spaces  Distance Functions (Measuring Similarity) 

The Breakdown 

  • Discrete Math (Simplicial Complexes) 
    1. The Component: Building the shape. 
    2. The Math: You treat data points as vertices. If two points are close, you draw an edge. If three are close, you fill in a triangle (a face). This structure is called a Simplicial Complex. It is essentially graph theory on steroids. 
  • Abstract Algebra (Homology) 
    1. The Component: Finding the features. 
    2. The Math: How do you define a “hole” mathematically? You use Homology Groups (H_0, H_1, H_2). 
    3. H_0: Connected components (clusters). 
    4. H_1: Loops (cycles). 
    5. H_2: Voids (bubbles). 
    6. You are essentially counting independent features in vector spaces. 
  • Linear Algebra (The Calculation) 
    1. The Component: Computing “Persistence.” 
    2. The Math: Surprisingly, TDA is mostly just Matrix Reduction (Gaussian Elimination). You build a massive “Boundary Matrix” representing the connections, and you reduce it to find which holes “persist” over a long range of scales and which ones are just noise.