There is no particular order to the items below. Caveat lector: A lot of these notes reflect my initial attempts at understanding a particular topic, much of the writing is quite bad, riddled with errors and incomplete.
The Poincare-Birkhoff-Witt theorem [.pdf]
Dixmier's version of Schur's lemma [.pdf] [.html]
What I would have learnt in freshman year linear algebra, had I been paying attention.
Categories and functors [.pdf]
Tensor products [.pdf]
A basic primer on tensor products over commutative rings with unity. Also, if you are afraid of tensor products then take a look at Prof. Gowers page on losing your fear of them here.
Algebraic number theory problems and solutions [.html]
Some homework sets and my solutions to them for an introductory algebraic number theory class I took in Fall 2006 as a graduate student.
The Law of Quadratic Reciprocity [.pdf]
The ring of real polynomials on the unit circle [.pdf]
This is a full workup of the ring R[X,Y]/<X2+Y2-1> down to its class group computation. All the arguments extend naturally if we replace X2+Y2-1 by f=aX2+bY2+C as long as f is irreducible in R[X,Y]. (Here R stands for the real numbers).
The Geometry of Numbers [.pdf]
This is a short expository write-up on Minkowski's convex body theorem and its applications to algebraic number theory.
The Geometry of Numbers (talk) [.pdf]
Slides to a talk I gave on Minkowski's convex body theorem at Colorado College in 2005.
The Orbit-Stabilizer Theorem [.pdf]
Completing the square [.pdf]
An geometric construction that shows how to obtain the quadratic formula.
Solving the cubic [.pdf]
A follow up to geometrically completing the square. The write up here shows (algebraically) that the general cubic can be solved in terms of radicals.
Wilson's Theorem [.html]
(p-1)! = -1 mod p.
The First Mayr-Meyer Ideal over the Rationals [.pdf]
A little project I did as an undergrad.
Complex analysis notes [.html]
Notes from an introductory complex analysis course I took in 2004 as an undergraduate. The notes were all prepared by the professor teaching the course: Travis Kowalski.
Cesaro summability of Fourier series [.pdf]
Continuity of a function is usually not strong enough to say anything conclusively about the convergence of its Fourier series. On the other hand, continuity does suffice to establish what is called Cesaro summability of the series.
A little glimpse of graph theory [.html]
Some recreation with amicable rectangles [.html]
Last updated: 15 April, 2008