Topics In Topology

Science Honors Program: Topics in Topology (Fall 2019)

Time: Saturday from 10:00-12:30 pm

Location: Pupin 313

Course Description: This course will give an introduction to the field of topology. Roughly speaking, topology is the study of shape. To a topologist, a square and a circle have the same shape since lengths and angles do not affect shape. In this course, we will study properties that can describe and distinguish different shapes (Why does a donut have a different shape than a beach ball?). Using properties such as the Euler characteristic, homotopy groups, and homology groups, we can prove things like the fundamental theorem of algebra, Nash's equilibrium theorem, "there is a location on the earth where the wind is not blowing", and more! Other possible topics include: the classification of surfaces, curve complexes, orientations, manifolds, the Poincare-Hopf theorem, knots and braids, the Borsuk–Ulam theorem. We will also see applications of topology to questions in data science, biology, and sociology via topological data analysis. No special mathematical background is required.


Course Schedule

The material that will be covered is subject to change as the course progresses.

  • September 21, 2019:
    • Introduction, polyhedral complexes, graphs
    • Sections 1.1, 1.2, 1.3, 2.1
  • September 28, 2019:
    • Continuity, surfaces, connect sums, Euler characteristic, legal colorings of maps
    • Sections 2.2, 2.3
  • October 5, 2019:
    • More on the Euler characteristic, planarity of graphs, orientations, the classification of surfaces theorem
    • Sections 2.3, 2.4, 2.5
  • October 12, 2019:
    • Homotopy classes of loops, curve graphs, nodal surfaces
    • Sections 2.6, 2.7
  • October 19, 2019:
    • Group theory, the fundamental group
    • Sections 3.1, 3.2
  • October 26, 2019:
    • Review of the fundamental group, the fundamental group of the circle, elementary complex analysis
    • Sections: 3.3, 3.4
  • November 2, 2019:
    • The fundamental theorem of algebra, Brouwer's fixed point theorem, Nash's equilibrium theorem
    • Sections: 3.4, 3.5
  • November 9, 2019:
    • Borsuk-Ulam theorem, the Ham Sandwich theorem, introduction to simplicial complexes and manifolds
    • Sections: 3.6, 4.1
  • November 16, 2019:
    • 3-manifolds, handlebody decompositions, Dehn surgery, knot complements, knot diagrams, connect sums of knots, knot genus
    • Sections: 4.1, 4.2, 4.3
  • November 23, 2019:
    • Prime decomposition of knots, crossing numbers, unknotting numbers, the Jones/Kauffman polynomial
    • Sections: 4.3
  • November 30, 2019: No class, Thanksgiving weekend
  • December 7, 2019:
    • Word groups, chain complexes/homology, homology of a polygonal complex
    • Sections: 5.1, 5.2
  • December 14, 2019:
    • Persistent homology, applications of persistent homology to brain artery trees, stable homotopy theory
    • Slides for brain artery trees, no note on other material.


Course Notes

The final version of the class notes and slides from the brain artery trees presentation are below.