## Algebra

Definitions and examples of group (finite and infinite, commutative and non-commutative), cyclic groups, subgroups, homomorphisms, quotients

Definitions and examples of rings and fields

Basic facts about finite dimensional vector spaces, matrices, determinants, and ranks of linear transformations

Integers and their basic properties

Polynomials with real or complex coefficients in 1 variable.

## Analysis

Basic facts about real and complex numbers

convergence of sequences and series of real and complex numbers

continuity

differentiability and Riemann integration of real valued functions defined on an interval (finite or infinite)

elementary functions (polynomial functions, rational functions, exponential and log, trigonometric functions).

## Geometry/Topology

Elementary geometric properties of common shapes and figures in 2 and 3 dimensional Euclidean spaces (example: triangles, circles, discs, spheres, etc)

Plane analytic geometry (= coordinate geometry) and trigonometry

Definition and basic properties of metric spaces, examples of subset Euclidean spaces (of any dimension), connectedness, compactness

Convergence in metric spaces, continuity of functions between metric spaces.

## General

Pigeon-hole principle (box principle)

induction

elementary properties of divisibility

elementary combinatorics (permutations and combinations, binomial coefficients)

elementary reasoning with graphs.