A Compilation of Various Examples

At some point, I decided to try learning one example from mathematics, per day. Arnold, Hilbert, Feynman, or some other famous people said it is good to always have (counter)examples on hand. That way, one can use them as testing grounds when learning a new result or when looking for evidence in support of a particular claim.

The examples I seek can be constructions, lemmas, theorems, etc. I was mildly successful at first but lost steam, eventually (around the middle of 2019). Still, here are some examples of varying depth. I'm sure there are many mistakes in these notes. Email me if you spot them. Also, I occasionally mention some colleagues by name in these notes.

• Fiber Bundles
• Topology
• Lie Groups
• Differential Geometry
• Complex Geometry
• Symplectic Geometry

I have also written some notes. I've included some of the ones made on TeX or GoodNotes but not the ones from the markdown editor Zettlr of which there are hundreds. Email me if you're interested in the Zettlr notes but good luck guessing what I've written!

Notes on Geometry

• Note on the Poincaré-Hopf theorem via Morse theory.
• Note on the simplest example of the Atiyah-Singer Index Theorem I know: d+d* and the Euler characteristic of genus-g Riemann surfaces.
• Note on Atiyah's "New Invariants of 3- and 4-Dimensional Manifolds." The original paper is brilliant and outlines some relationships between Floer theory and gauge theory, topology and geometry. You really should read that first. I add some details and pictures every now and then.
• Note on Witten's "Supersymmetry and Morse Theory" paper.
• Note from a talk on the Jones Polynomial, by Witten.
• Note on Milnor fibrations and Picard-Lefschetz theory.
• Very brief note on why virtual fundamental cycles appear in symplectic topology.
• Very brief note on complex K3 surfaces.
• Note from a talk on Heegaard Floer homology.
• Note on 1st Chern class of CP^n and relative homotopy.
• Note on basic contact geometry, Boothby-Wang bundles, Liouville domains, symplectic (co)homology, and wrapped Lagrangian Floer homology.

Notes on Physics

• Note on symplectic geometry and it's relationship to classical mechanics.
• Note which compares classical and quantum mechanics from a rather mathematical viewpoint.
• Note on my very basic impression of quantum field theory.
• Note on classical field theory, based on my reading of notes by Charles Torre.
• Notes from a talk on light rays and black holes by Witten: Part 1 and Part 2.

Other Random Notes

• Notes from a first semester graduate course on real analysis I took. The examples for different modes of convergence are rather useful.
• Finite fields: notes from an abstract algebra class I taught.
  
• Irrationality of pi via Niven.
• Short proof that the harmonic series diverges.
• Some things I learned from the YouTube channel 3blue1brown.