# What Every Mathematics Major Should Know

## Core mathematics

- Linear algebra including:
- axiomatic characterization of determinant.
- orthogonal diagonalization of symmetric real matrices.
- factorization of an invertible real matrix as the product
of an orthogonal matrix with a positive-definite
symmetric matrix
- Jordan canonical form

- 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".
- Euler's product for the zeta function.
- The classification of the completions of the rational field.

## Algebra

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

## Geometry

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

## Analysis

- Volumes of finite dimensional balls and spheres.
- The transformation rule for multiple integrals.
- Ordinary linear differential equations.
- Exterior calculus
- Stokes's theorem in Euclidean space.
- Poincare's lemma in Euclidean space.

- Elementary facts about solutions of the Cauchy-Riemann
equations.
- Fourier analysis on
**R**^{n}, **Z**^{n},
**T**^{n}, and (**Z**/m**Z**)^{n}:
invertibility of Fourier transform in these cases for
Schwartzian functions with explicit formulae for Gaussian
densities.

## Probability & Statistics

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