Notes



This page gives a selection of notes. Some are more polished than others.

General

A 1 page technical statement of the motivating problem of my PhD thesis

Linear Algebra

Bilinear forms

Finite geometry

The normal rational curves in PG(2,q) are ovals (via Chevalley groups)

The projective space PG(3,q)

The elliptic quadric in PG(3,2) is an ovoid

Ovoids in PG(n,q) via lattice theory (with examples)

The symplectic polar space W(3,q)

The Hermitian polar space H(3,q^2)

Translation of Section 1 of Jacques Tits, "Translations a ovoides", 1962.

Representation Theory

sl_2 representation theory

Categorification of the polynomial representation of the Weyl Algebra Based on Alistair Savage's course on categorification.

Twisted Chevalley Groups

The Witt, Virasoro and N=2 superconformal algebras.

A little table of algebras and pictures

What are the Kazhdan-Lusztig conjectures?

Analysis

Stokes' Theorem

Riemann Integration

singular homology of R^n

de Rham cohomology of R^n

de Rham cohomology of S^n

The Plancherel Formula for SL_2(R)

Algebraic Geometry

Two examples of affine schemes and the Proj construction

Intersection Cohomology (part 1)

Intersection Cohomology (part 2)

Homological Algebra

"Homological algebra is a tool used to prove nonconstructive existence theorems in algebra (and in algebraic topology). It also provides obstructions to carrying out various kinds of constructions; when the obstructions are zero, the construction is possible. Finally, it is detailed enough so that actual calculations may be performed in important cases." - from the Introduction of Weibel, 'An Introduction to Homological Algebra'.

Derived functors: Given a left exact functor F, the right derived functors of F measure "how far" F is from being exact.

Complex Analysis

"The solution of a large number of problems can be reduced, in the last analysis, to the evaluation of definite integrals; thus mathematicians have been much occupied with this task... However, among many results obtained, a number were initially discovered by the aid of a type of induction based on the passage from real to imaginary. Often passage of this kind led directly to remarkable results. Nevertheless this part of the theory, as has been observed by Laplace, is subject to various difficulties... After having reflected on this subject and brought together various results mentioned above, I hope to establish the passage from the real to the imaginary based on a direct and rigorous analysis; my researches have thus led me to the method which is the object of this memoir..."
A. L. Cauchy, 1827

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