Intro
- ‘Algebra about strucure and equalities. Analysis about inequalities.’
- Research themes: Properties of various algebraic objects, their relationships.
- Characterization of research effort: See linear algebra survey, complexity theory survey.
Techniques
- Algebra is about correct reasoning: symbol manipulation according to some rules.
- It is about abstraction by means of techniques such as change of variables.
- Look for structure in expressions, understand and if necessary, abstract away sub-expressions with new variables: very important! (5)
- Use properties of algebraic objects well, keep a list of such properties handy.
Common proof techniques
- Induction. Direct inference.
- Contradiction:
- Diagonalization- Order all elements, make a new element which differs from every other element.