![]() ![]() They’ve been working away for centuries, and by now their mastery of polynomial equations is truly staggering. Thus, by restricting the breadth of their investigations, algebraic geometers can dig deeper. Why does algebraic geometry restrict itself to polynomials? Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra. The equation is, and it’s called “cubic” because we’re multiplying at most three variables at once. ![]() It’s famous because it is the variety with the most nodes (those pointy things) that is described by a cubic equation. After all, physics problems involve plenty of functions that aren’t polynomials. As a graduate student, this seemed like a terrible limitation. In its classic form, this subject considers only polynomial equations-equations that describe not just curves, but also higher-dimensional shapes called “varieties.” So is fine, and so is, but an equation with sines or cosines, or other functions, is out of bounds-unless we can figure out how to convert it into an equation with just polynomials. How could any mathematician not fall in love with algebraic geometry? Here’s why. And one thing that did not matter to me, I believed, was algebraic geometry. I already knew that there was too much mathematics to ever learn it all, so I tried to focus on what mattered to me. So I went to graduate school-to a math department, but motivated by physics. And then, surely, these laws would give a clue to the deeper puzzle: why the universe is governed by mathematical laws in the first place. I decided that there must be some deep mystery here, that we might someday understand, but only after we understood what the laws of physics actually are: not the pretty good approximate laws we know now, but the actual correct laws.Īs a youthful optimist I felt sure such laws must exist, and that we could know them. I failed utterly, though I managed to get my first publishable paper out of the wreckage. I studied mathematical logic and tried to prove that any universe containing a being like us, able to understand the laws of that universe, must have some special properties. As far as I can tell, that hypothesis raises more questions than it answers. As he put it, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”ĭespite Wigner’s quasi-religious language, I didn’t think that God was an explanation. I was fascinated by Eugene Wigner’s question about the “unreasonable effectiveness” of mathematics in describing the universe. I learned quantum mechanics and general relativity, studying the necessary branches of math as I went. In college I wound up majoring in math, in part because I was no good at experiments. ![]() Science is the magic that actually works.Īnd so I learned to love math, but in a certain special way: as the key to physics. The mysterious symbols seemed like magic spells. But later, when I realized that by fiddling around with equations I could learn about the universe, I was hooked. I found long division insufferably boring, and refused to do my math homework, with its endless repetitive drills. My parents were a bit worried, because they knew physicists needed mathematics, and I didn’t seem very good at that. While I couldn’t understand it, I knew right away that I wanted to. ![]() When I was eight, he gave me a copy of the college physics textbook he wrote. Whenever my uncle came to town, he’d open his suitcase, pull out things like magnets or holograms, and use them to explain physics to me. My uncle Albert Baez, father of the famous folk singer Joan Baez, worked for UNESCO, helping developing countries with physics education. The great mathematician Leonhard Euler dreamt this up in 1745.Īs a kid I liked physics better than math. It’s not obvious that you can describe this using a polynomial equation, but you can. You get a curve with three sharp corners called a “deltoid”, shown in red above. For example, roll a circle inside a circle three times as big. We can describe many interesting curves with just polynomials. ![]()
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