Terry Tao, in a post on Google Buzz, has given an overview of mathematics in the form of multiple "punch-lines" of the requested variety.
Here are three examples from Tao's post:
Algebra is the mathematics of the "equals" sign, of identity, and of the "main term"; analysis is the mathematics of the "less than" sign, of magnitude, and of the "error term".
Algebra prizes structure, symmetry, and exact formulae; analysis prizes smoothness, stability, and estimates.
Most of geometry would not be classified as either algebra or analysis, but simply as geometry.
Definitely Tao's aphorisms are thought-provoking and inspiring ... but are they useful ? Don't ask me! :)
Partly inspired by Tao's essay, here is a one-sentence definition of quantum mechanics (as optimized for systems engineers) …
- Quantum mechanics is the algebraic geometry of $n$-particle Hamiltonian flows and Lindbladian compressions as pulled-back onto the natural $r$-indexed stratification of $r$'th secant varieties of $n$-factor Segre varieties whose $r\to\infty$ limit is … $n$-particle Hilbert space.
… and it turns out to be very useful (and great fun) to rewrite standard quantum physics texts like Charles Slichter's Principles of Magnetic Resonance based upon this one sentence definition.
Joseph Landsberg's recent Bull. AMS review "Geometry and the complexity of matrix multiplication" (2008)—which 2008), which has been praised by in multiple MathOverflow posters—provides a readable posts, provides an overview of the remarkably broad utilityof rank-indexed —despite their unwieldy name—of stratifications of secant varieties of Segre varieties (which extends far beyond quantum physics).
