A Compilation of Various Examples
At some point, I decided to try learning one example from mathematics,
per day. Hilbert or Feynman or some other famous person 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. Some examples, I
think every mathematician should know while others are more for
specialists. Some really shouldn't be called examples as much as Fields
medal worthy theorems. I'm sure there are mistakes in these notes. Email
me if you spot them. Needless to say, these are not exhaustive. Also, I
occasionally mention some colleagues by name in these notes.
I have also made many notes. I've included some of the ones made on TeX
or GoodNotes but not the ones from the markdown editor Zettlr. Email me if
you're interested in the Zettlr notes.
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
- Note on 1st Chern class of CP^n and
- 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
- Note on classical
field theory, based on my reading of notes by Charles Torre.
- Notes from a talk on light rays and blackholes 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
- Short proof that
the harmonic series diverges.
- Some things I learned from the
YouTube channel 3blue1brown.