The amazing story of Geometry

The problem of figuration of space is quite an ancient matter. Euclid's Elements first proposed a set of rules Mathematic(ian)s should use to infer "geometrical" properties of a suitable conglomerate of objects (points, lines and polygons in the plane). It's worth to say that formally speaking Euclid's Elements are inconsistent (one of the first propositions [1] in Euclid's book gives a recipe to build an equilateral triangle having prescribed side, but nothing prevents the two circles needed in this construction from having empty intersection); nonetheless his work is worth to be considered the first piece of abstract Mathematics: a set of objects (points and lines) and a bunch of inference rules (the relation of incidence, some formal properties of such relation, mimicing the "natural" behaviour of forms in physical space) is all one needs to do "Geometry".

Jump forward to the IX century and you'll meet (Abū Jaʿfar Muhammad ibn Mūsā) al-Khwārizmī, commonly accepted by mathematical storiography as the father of Algebra. Astronomer, philosopher, mathematician, he combined the work of Diophantus and Brahmagupta to obtain the first set of rules to handle with quadratic equations. He classified "all the different types" of degree-two equation and proposed both an algebraic and a geometrical way of solution: every algebraically solvable problem which lacks a geometrically evident solution, has to be considered incomplete: no "algorithm" (the european word is nothing but an adaption of al-Khwārizmī's name) can stand alone to justify a solving procedure.

Rene' Descartes has to be considered the father of analytical geometry. He was the first to state the greatest abstract principle of modern Geometry: a geometric problem can be translated in an algebraic one via coordinates. A precise knowledge of the rules governing the algebraic equations of a subset of the plane totally equals a precise knowledge of the space itself. Numbers are points on a line; a conic is the closed curve arising as the set of zeroes of a degree-two polynomial in x and y; any polynomial of degree n defines a precise curve on the plane: this is made by two simpler (smaller) curves if and only if the polynomial can be factored as a product.

If only Kant knew more Geometry! Space is not a "pure form" of epistemic intuition; it is determined by a more atavic concept, the primeval idea of a coordinate system. Husserl understood Geometry as the obvious science of space, and Algebra as the natural mathematization of time: the idea of an interplay between the two dimensions is quite more ancient than Einstein's theory of relativity.

William Rowan Hamilton sits just in the middle of the story. He wanted to find a plausible algebraic model in which to write down Maxwell's equations (involving pseudo-vectorial objects, elements of structures now called Clifford-Hamilton alegbras). Its tentative quaternionic geometry fails to be the right idea: the natural metric on the space of quaternions fails to model the natural metric on space-time, because of a signature problem (the "physical" space admits isotropic vectors, pointing the direction where light beams propagate: no such vector exists in the quaternionic 4-space). So despite their actractiveness, pseudovectors seem not to be the right object to describe our universe. Nonetheless, a new formalism to understand the classical motion of bodies, analytical mechanics, stemmed from the work of Hamilton: he used to project the equations of motion of a body on the cotangent bundle of its configuration space, and worked out the formalism in which Quantum Mechanics was written a century after.

At the same time, the Italian school in Algebraic Geometry (working in many italian universities like Padua, Rome, Cremona, Florence, Naples, ...) is pursuing a much more subtle task. Italian school has been deeply influenced by the work of B. Riemann and Emmy Noether: Noether's life and genius could be one of the best "manifesto" of a tentative book about women in Science: from Abstract Algebra to celestial and "mundane" Mechanics, there's no field of pure Mathematics (and engineering, Physics, hydrodynamics...) she didn't touched and radically changed. Anyway she was a woman in the XIX century, so Noether's genius was completely stonewalled by the academics: "Is this a university or a public restroom?". The profoundness of the ideas Noether's work uncovered can be stated in this way: we can associate to any algebraic curve (defined to be the zero-set of a suitable polynomial) a ring, named the ring of coordinates of the curve. This ring encodes geometric properties translating them into algebraic ones (this is what I like to call the "figuration/representation paradigm" in Mathematics): in the end a suitable family of subsets of the ring (its prime ideals) mimic the behaviour of points in a topological space. This topological space is the curve we started from. Primes are points, she wrote: the set of prime ideals turns out to be a topological space homeomorphic to the curve we started from!

This situation rapidly shed a light on a deep problem: we know how to pass from a curve to its coordinate ring, and we also know that geometric properties of the curve can be translated into algebraic properties of the ring. Can this correspondence be reversed? Can we hope that to any algebraic object can be associated a geometric one, in such a way that its coordinate ring is the ring we started from? Can we extend the "dictionary" between Algebra and Geometry in order to draw algebraic problems, and solve equations to attack geometric ones? This is the cornerstone in Modern Mathematics, this point of view permeates any kind of ideas in Algebraic and Differential Geometry, Algebraic Number Theory, General, Algebraic and Differential Topology, Homological Algebra, Homotopy Theory...

This is in a few words what A. Grothendieck did. He proved that this dictionary often really exists, he taught us how to read it, he propoesd further generalizations of this basic idea: "Algebra is like Geometry, done backwards". Representing a space, in such a way to solve equations, in such a way to coordinatize the Raum witnessing our geometric quest, in such a way to shed a light on the "logical" problem of the highly non-boolean behaviour of physical entities...