What is algebraic geometry?

these are all examples of algebraic curves they're curves that are cut out by polinomial equations consider this curve over here it's defined by the algebraic equation y^2 = XB - x moving everything to one side we get that y^2 - XB + x = 0 algebraic geometry begins as the studies of zeros of algebraic equations like this but at the same time it has a reputation of being incredibly abs ract but why because algebraic geometry is not just about the Zer of algebraic equations it's really about the bridge between algebra and geometry this is best explained in an example consider the unit circle x^2 + y^2 equals 1 one of the philosophies in math in general is that to understand an object you should study functions on that object for example consider a function that takes a point on the unit circle and returns just the y-coordinate so so it would map the point 1/ < tk2 comma 1/ > 2 to 1/ < tk2 it mapped the point 1/ < tk2 comma - 1/ < tk2 to minus 1/ < tk2 and so on a slightly more interesting example is the function that sends the pair X comma y to the real number x + 2 y this function would map this point to 3 over < tk2 it would map this point to -1 < tk2 now let's do something interesting consider the function G of XY defined as follows the part X2 + y^2 - 1 always evaluates to zero for points XY on the unit circle so G and F are actually the same as functions on the unit circle G of 1/ < tk2 comma 1 < tk2 is 3 overk 2 and G of this point is min-1 / < tk2 notice that these are the same outputs produced by our function f ofx y from earlier the same will be true for these two functions they are the same as functions on the unit circle more generally any function of this form will be the same as f ofx y let's phrase this in a more sophisticated language the set of pols in the variables X and Y with real coordinates is denoted r braet x comma y this is more than just a set it's closed under addition if you add two pols with real coefficients you get a polinomial with real coefficients it's also closed under multiplication an algebraic object that behaves in this way is called a ring now look at the set R of all polinomial functions defined on the unit circle we think of two pols as defining the same function if they agree on All Points XY on the unit circle then they are equal as members of R we can add and multiply functions in r therefore R is a ring some terminology this ring R is called the coordinate ring of the circle we view it as an algebraic manifestation of the circle as we will see geometric properties of the circle are encoded in the algebraic properties of its coordinate ring a small as side for those who know some ring Theory you'll realize what we're doing here is working with the quotient ring of R braet XY modulo the deal generated by x^2 + y^- 1 now if those words don't make any sense don't worry about it we won't need it for the rest of the video look at this curve y - x * y + x = 0 versus this Parabola x + y^2 equals z the first curve is a union of two curves and the second one isn't we say that this curve is reducible and this curve is irreducible let R be the ring of functions of the first curve and let s be the Ring of functions of the second curve the algebra of these two rings as we will see is quite different the first string R is kind of weird consider the function f ofx y = y + x in R it's not identically zero for example F of 1 1 is equal 2 and says is clearly non zero similarly the function G of x y = y - x is not zero on the curve either however their product is zero on all points of the curve this type of behavior might feel strange the product of two things is zero but both of the factors are non zero this never happens in the ring R it turns out and this is not immediately obvious that if the product of two pols is zero on the parabola then at least one of the two factors must be zero on the parabola this algebraic phenomenon is detecting that the first curve has two pieces it is reducing ible while the second curve has only one piece it's irreducible the geometry of whether the curve is irreducible or not is appearing in the algebra here's a more complex example the curve y^2 = x^2 * x + 1 this curve intersects itself it has a node at the origin can we detect this using algebra first let's bring all the terms to one side using the formula for a difference of squares let's write this as follows as before notice that these two functions are both nonzero on the curve but the product of these two functions is zero on the curve if you zoom into the origin this looks like two lines that are crossing each other so near the origin this curve quote unquote looks reducible now algebraically near the origin theun of x + 1 is approximately equal to one so this top equation is approximately equal to this bottom equation and the bottom equation if you'll remember was precisely the equation of the two lines crossing each other from before but there's a problem up until now we've only been looking at polinomial functions but theun of X+1 isn't a polinomial what ring does theun of X+1 live in well the secret is to use a tool from calculus we can write the tailor series of theun of x + 1 as follows previously we looked at the polom ring R of XY now we're going to look at a power Series ring R dou bracket XY this is defined as a set of all formal power Series in X comma y with r coefficients we will think of the function the square root of one as belonging to the power Series ring R braet XY or more precisely its tailor expansion belongs to this power Series ring so what we've noticed is that there are formal power series which are individually nonzero on the curve but the product of them is zero on the curve this detects the fact that if you zoom into the origin the curve has two lines that look like they're Crossing themselves but who cares about the algebra geometry dictionary that we've been talking about well the benefit of this is that algebra helps us nail down geometric intuition into rigorous math this is why if you look at most textbook in algebraic geometry most proofs use ring Theory quite heavily furthermore if I give you a shape that's carved out by these three pooms and I ask you what are the singular points is it irreducible I mean you can't visualize it and tell me the answer but you can compute its coordinate ring and tell me its properties so far we've seen two geometric properties of curves that are captured by their coordinate rings irreducibility and a curve having a node the takeaway of this is that we can translate geometric phenomenon into algebraic properties of rings but can we reverse this procedure what if you take any old ring like Z the Ring of integers is there a geometric object that corresponds to it can we view Z as the Ring of functions on some geometric object it turns out that there is a geometric object here and it is called spec Z this is an example of something called a scheme it is a geometric realization of the Ring Z this is a very Advanced topic and I won't rigorously describe in this video what a scheme is but I can draw a picture of spec Z spec Z is drawn as a line and each point on this line corresponds to a prime number and there's one point corresponding to zero well how do you view integers as functions on the space well if I give you an integer like say 10 and I ask you to evaluate it at a point on the space like two tell me the remainder of 10 after division by two so the function 10 Returns the value zero at two it Returns the value 1 at three the value 0 at 5 3 at 7 and so on and at this mysterious point zero it returns itself this seems really bizarre why on Earth would we do this it suffices to say that it turns out to be incredibly useful for instance several key results in modern number Theory make heavy use of algebraic geometry and the notion of a scheme the proof of formas theorem uses some very non-trivial ring theory that is motivated by geometric considerations an important ingredient of the proof required showing that certain rings are quote unquote complete intersections now what that is is besides the point but it's an algebraic property that's motivated by geometry so that's all for the math portion of this video I'll end off by listing some resources that you can use to learn algebraic geometry in some more detail a really good starting point is the book a guide to plain algebraic curves written by Keith Kendig it's written in a really elementary style and has lots of really captivating diagrams throughout if you look at the table of contents it starts off with a lot of examples that require only Elementary High School level algebra to understand but by the end it actually gets to some pretty deep theorems in algebraic geometry if you'd like to learn the subject in a bit more detail using more ring Theory and abstract algebra a great book is ideals varieties and algorithms the great thing about this book is that it doesn't assume any knowledge of abstract algebra and teaches everything from the ground up pretty rigorously if you want to learn about schemes and spec a really good book is algebraic geometry and arithmetic curves by Ching Liu as you can see it is very tur and it is a graduate level textbook but it has everything in it and it is written very very beautifully if this is your cup of tea by all means take a look that's all for now thanks for taking the time to learn some math with me and I'll see you next video

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Length: 11min 50sec (710 seconds)

Published: Tue Oct 17 2023