It is me again to bother you. Since my last post I started to look at some seemingly "serious" mathematics for background study. Accidentally I went into a university bookshop and came across some "differential geometry", "algebraic geometry", "geometric analysis", etc. I am a writer, and my math skill remains at pre-calculus level, but I am (doubly) confident that the subject I learned long ago called "analytic geometry" contains quite a handful of geometric figures and coordinate systems. I was genuinely surprised that those "advanced geometry" textbooks I randomly picked-up contain very few figures, and radically different from what I expected if they have any. Although I work with literature and languages, the descriptions in math textbooks sound to me just like Alienese. I was hoping that the geometric figures might be enlightening somehow, and I was so so wrong.
Well, this is my question: where are the geometric figures in those "advanced geometry" textbooks?
PS The same applies to what I heard about topology. I learned it from wikipedia that mathematicians are like changing coffee cups to donuts in topology, so I guess I might see loads of "cups and donuts", or similar stuff, in topology textbook. I was rather disappointed when I open a book called "Introduction to General Topology", all that I read is about some set things.
PS2 When I was roaming around the web, I learned that a prestigious mathematician called Shinichi Mochizuki proves "abc conjecture". The abc conjecture I read from wikipedia sounds quite "algebra" to me, but on Shinichi Mochizuki's homepage he called himself "inter-universal geometer". I take that "inter-universal" is like something very powerful. But how come a geometer solves an algebra problem?
Best Answer
Analytic geometry, invented by Descartes and Fermat, was the beginning of algebraic geometry and of differential geometry.
Their contemporary Desargues's work, the mathematization of the discovery by Renaissance painters of perspective, was also an important source for the branch of algebraic geometry called projective geometry but had less influence until it was rediscovered by Poncelet (in Russian captivity as a prisoner of war, after Napoleon 's defeat).
Analytic geometry is still indispensable if you want to understand these branches of mathematics.
And drawings are capital in the study and understanding of modern algebraic geometry: I have rarely heard a talk at a conference on algebraic geometry which was not accompanied by them, with some mathematicians exercising real artistic talent and making heavy use of colored chalk, or more recently of beautiful computer animations.
For various reasons this is not always reflected in books: the most egregious example being Grothendieck and Dieudonné's Eléments de Géométrie Algébrique, which laid out the foundations of modern Algebraic Geometry six decades ago and does not contain a single drawing.
Note however that the excellent textbooks by Fulton, Hartshorne, Qing Liu, Perrin, Reid, Shafarevich, ... contain lots of magnificent illustrations.
Finally, I find it regrettable that many young learners of algebraic geometry are not exposed to computations in analytic geometry: many questions on this site are from obviously excellent and intelligent students who unfortunately have been exposed to much abstract concepts but lack practice in down-to-earth calculations, so that they have difficulty in, say, proving that a quadric in three space is birational to a plane.
One of the difficulties of modern algebraic geometry, and the cause of its reputation as a hard subject, is the necessity to learn to use quite abstract tools (cohomology of sheaves, commutative algebra,...) but also simultaneously to be able to understand beautiful classical results like Pascal's mystic hexagram or Cayley-Bacharach's bound on the dimension of the complete linear system associated to a divisor on a curve.