# What Every Mathematics Major Should Know

## Core mathematics

1. Linear algebra including:
1. axiomatic characterization of determinant.
2. orthogonal diagonalization of symmetric matrices.
3. factorization of an invertible matrix as the product of an orthogonal matrix with a positive-definite symmetric matrix
4. Jordan canonical form
2. Topological spaces: notions of homeomorphism, compactness, and connectedness. Generalization of the extreme and intermediate value theorems. Metric spaces: equivalence of compactness with "complete and totally bounded".
3. Euler's product for the zeta function.
4. The classification of the completions of the rational field.

## Algebra

1. General knowledge of groups, rings, fields.
2. Structure of finitely generated abelian groups.
3. Principal ideal domains: examples, relation to long division, uniqueness of factorization.
4. Field extensions of finite degree: construction, examples, Galois theory in characteristic zero.
5. Structure of finite fields.

## Geometry

1. The classical transformation groups.
2. The concept of Riemannian manifold: geodesic distance and curvature.
3. The classification of compact oriented 2-manifolds by genus.
4. Examples: projective spaces, projective and affine hypersurfaces, tori, ...

## Analysis

1. Volumes of finite dimensional balls and spheres.
2. The transformation rule for multiple integrals.
3. Ordinary linear differential equations.
4. Exterior calculus
1. Stokes's theorem in Euclidean space.
2. Poincare's lemma in Euclidean space.
5. Elementary facts about solutions of the Cauchy-Riemann equations.
6. Fourier analysis on Rn, Zn, Tn, and (Z/mZ)n: invertibility of Fourier transform in these cases for Schwartzian functions with explicit formulae for Gaussian densities.

## Probability & Statistics

1. The concept of (abstract) discrete probability space.
2. The notion of random variable as function on a probability space.
3. The unit interval as example of a "continuous probability space".
4. The normal distribution.
5. The central limit theorem.