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. 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.
Notes on Geometry and Physics
- Note on symplectic
geometry and it's relationship to classical mechanics.
- Note on
classical and quantum mechanics from a rather mathematical viewpoint.
- 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 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
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