Lecture Notes: Math for Everyone - Geometry and Topology

Introduction

• Speaker: Steve Trle, University of San Francisco
• Area of study: Geometry, Topology, Computer Graphics
• Event: Evening Math Talk
• Incentive: Pizza in the hallway post-talk

Historical Context of Geometry

• Traditional Geometry: Ancient Greece, Pythagoras, Archimedes
• Evolution: 2000 years of Euclidean geometry
• Foundation: Plane geometry used in architecture, art, science
• Transition: 1800s, Gauss's study of surfaces
• Discovery: Bolyai's hyperbolic plane, 200-year anniversary

Modern Geometry: 1823-2023

• Impact on Math and Science
  • Riemann's generalization of geometry
  • Einstein's relativity
  • Contemporary influence in mathematics

Key Concepts in Geometry

• Curvature: Central to modern geometry
  • Euclidean Geometry
    • Circumference of a circle: 2πr
    • Area growth: Quadratic
  • Non-Euclidean Geometry
    • Positive curvature: Sphere, slower growth
    • Negative curvature: Saddle, faster growth
    • Curvature quantified by deviation from Euclidean formulas

Vision and Geometry

• Vision as a geometric sense: Light as rays
• Perception of size and distance
  • Euclidean space: Linear perception
  • Positively curved space: Objects appear larger
  • Negatively curved space: Objects appear smaller

Visualizing Non-Euclidean Spaces

• Positively Curved Space

  • Example: Sphere, North Pole to South Pole vision
  • Phenomenon: Whole vision engulfed by an object
  • Application: Apollo mission, seeing more than half of Earth

• Negatively Curved Space

  • Example: Saddle, space expansion
  • Phenomenon: Objects shrink, vast space
  • Application: Space growth rate faster

Mixed Curvature Spaces

• Stacking spaces with different curvatures
  • H2 x R: Mixed negative and zero curvatures
  • Applications: Visualization of mixed geometries

Advanced Geometries

• Non-traditional spaces (e.g., Nil geometry, SL2 R, Sol geometry)
• Applications: Geometry in unique mathematical spaces

Application of Geometry in Topology

• Understanding surfaces and curves

• Concept: Unrolling surfaces

  • Use of tiling (e.g., octagons in hyperbolic plane)
  • Uniformization theorem

• 3D Spaces

  • Unrolling complex spaces into geometric forms
  • Thurstin's conjecture: Geometrization of all 3D spaces
  • Perelman's proof

Geometry in Data Science and Machine Learning

• Use in graph theory

  • Graphs as data structures
  • Embedding graphs into geometric spaces

• Challenges

  • Different graphs prefer different curvatures
  • Finding suitable embedding spaces

• Real-world applications

  • Legal cases graph
  • Structuring data for efficient analysis

Conclusion

• Geometry's evolution from ancient Greece to modern applications
• Diverse applications in science, topology, and data science
• Encouragement to explore further through papers and discussions

Q&A Session

• Discussion on dimensional space requirements for embedding graphs
• Similarities with symmetric spaces for specific groups
• Inviting further questions and interactions post-talk