What are the best ideas/theorems/formulas in mathematics? Any list is a matter of taste and personal preferences. Due to the vastness of mathematics, it is quite challenging to build a list, because one tends to focus on things which one uses most. Here is a first shot written in 31. December 2007:

## Formulas

- Completion of squares and quadratic formula by Brahmagupta. Babylonians and Chinese mathematicians
- Euler's formula exp(i x) = cos(x) + i sin(x) as a key to geometry and trigonometry.
- Euler's formula V-E+F=2 for polyhedra is a prototype of an index theorem.
- Taylors formula. A prototype of an approximation result.
- The quadratic reciprocity law.

## Theorems

- Pythagoras theorem. Used in so many geometric proofs or computations.
- The fundamental theorem of calculus and generalizations to Stokes and Ito's formula in stochastic calculus
- Zorn's lemma in the form of Tychonov or Banach's theorem.
- The fundamental equations of calculus of variations.
- The Chinese remainder theorem in elementary number theory.
- Banach's fixed point theorem. A constructive method. Prototype of many more fixed point theorems.
- Brower's fixed point theorem as archetype for other nonconstructive fixed point and index arguments.
- Gauss-Bonnet theorem in differential geometry.
- Birkhoff's ergodic theorem with other limit theorems as special cases. Like the law of large numbers.

## Algorithms

- Newtons method to solve equations up to KAM theorem. More general Gradient methods or averaging methods.
- Gaussian elimination. It's fundamental to solve linear equations.
- Fourier approximation. Basic differential equations and probability theory. Starts harmonic analysis.
- Euclid's algorithm. Fundamental to do number theoretical computations.
- LLL algorithm is prevalent in many number theoretical and cryptological applications.
- Continued fraction expansion. For example to solve Diophantine equations or to classify numbers.
- Simplex method in convex optimization.
- Bubble sort as an example of a sorting.

## Concepts

- Exponential maps in differential and Lie geometry. Logarithms both in the reals and indices in number theory.
- Indirect proofs like the irrationality of sqrt(2) and Euclids proof of the infinity of prime numbers.
- Baire category in topology. Countless of elegant existence theorems like nonalgebraic numbers, Liouville numbers.
- Diagonal arguments prototyped by Cantor, computability by Turing or decidability questions by Goedel.
- Topological spaces generalizing metric spaces. Prototype of an axiomatically clean setup.
- Galois theory as a prototype to bridge different areas of mathematics and to settle so many quests.
- Algebraic topology to use algebra to solve problems in topology.
- Group theory to classify geometry: Klein's Erlanger program.
- Scaling and renormalization arguments as an other symmetry.
- The methodology of statistics to extremize in a space of mathematical models.
- Invariants like index theorems. Archetype: Gauss-Bonnet. Leading to invariants.
- Graph theory to solve combinatorial problems. Relations with linear algebra and spectral theory.
- Generating functions. A fundamental tool in combinatorics, number theory, probability theory.
- Matrix theory. Started with determinants and theory of linear equations. Motivates most of the rest of linear algebra.
- Differential equations, up to partial differential equations and stochastic differential equations.
- Non-commutative geometry. Extends measure theory, topology and geometry to larger setup.