This is a snapshot about operator theory and one of its fundamental tools: the singular value decomposition (SVD). The SVD breaks up linear transformations into simpler mappings, thus unveiling their  geometric properties. This tool has become important in many areas of applied mathematics for its ability to organize information. We discuss the SVD in the concrete situation of linear transformations of the plane (such as rotations, reflections, etc.).

This Snapshot is about the generalization of “>” from ordinary numbers to so-called fields. At the end, I will touch on some ideas in recent research.

Leonhard Euler (1707–1783) – one of the greatest mathematicians of the eighteenth century and of all times – often corresponded with a friend of his, Christian Goldbach (1690–1764), an amateur and polymath who lived and worked in Russia, just like Euler himself. In a letter written in June 1742, Goldbach made a conjecture – that is, an educated guess – on prime numbers:

“Es scheinet wenigstens, dass eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey.”