Working over , a representation of a finite group is a homomorphism , where is a -linear space. Hence acts on via its image in the general linear group. I’ll often write or for this action, rather than . Nonetheless it can be instructive to write out some of these ideas and arguments with the full notation.

From the perspective of category theory, a group can be described as a category with one object, and such that every arrow is an isomorphism, the arrows thus representing the elements of . Thus, we can also say that a representation is a functor , and where is the image of ‘s single object and the arrows in map to automorphisms of . With this description, it’s clear that a mapping of representations should be a natural transformation of such functors. Hence, given two representations and , a linear mapping is a morphism of representations if the following diagram commutes for all :

Surpressing the symbols and , we can write , suggesting the name “-linear” for a mapping of this sort.

Now we’ll develop these notions from a different angle, which can lead to many elegant arguments and definitions. Let be a finite group, and consider the free -vector space on the elements of . Such a space also parametrizes the set of (set-theoretic) functions : simply take any element and define for any

We can give this the structure of a (non-commutative) -algebra by identifying each element with , where , and defining a ring multiplication by setting . Multiplication for other elements is defined by forcing the distributive law. If is abelian, then this is a commutative -algebra. This ring is called .

Note that nearly always contains zerodivisors. Consider the cyclic group of order with generator . Then it is easy to see using the definitions above that is isomorphic to , where is the polynomial ring. Since is not a prime ideal, contains zerodivisors. Since any non-trivial finite group contains cyclic subgroups, the associated ring contains as a subring for some , and thus contains zerodivisors.

Let be a -linear space, and let be a representation. Then can act on from the left via the action

Note that

so that this is a well-defined action. Hence induces a -module structure on .

On the other hand, let be a -module. Clearly is a -vector space, because acts via . For each , define a mapping by for all . Because has a module action on , and lies in the centre of , we have

so is linear and is clearly a homomorphism, that is, a representation of . These two preceding correspondences are inverse (up to isomorphism).

Let and be two -modules. Then a -linear map is, in particular, -linear and -linear. Thus a morphism of -modules induces a map of representations, and a mapping of representations extends via linearity to a -linear map. This establishes an equivalence between the category of -modules and that of -representations of .

This equivalence gives us practically two ways to define anything. For example, a subrepresentation can be defined as subspace such that is stable under the action of . Equivalently, a subrepresentation is a -submodule of the associated -module, since a subrepresentation must be stable under the action of both and . In the next post I’ll describe some basic results on subrepresentations.

]]>In this post, we’re going to need a couple of facts from high-school algebra. The first is a technique for solving quadratic equations called “completing the square”. This refers to the fact that

You can verify this for yourself by expanding the brackets using the FOIL method on the right-hand side. The second algebra fact is that

commonly referred to as the “difference of two squares”, and is another fact you can check for yourself using FOIL.

Recall that a conic in is a curve given by an equation that is homogeneous of degree 2. That is, an equation of the kind

(1)

where the ‘s are any number — we only exclude the possibility that all six ‘s are zero. This covers every possible equation you can make with 3 letters multiplied together in pairs, once all term have been moved to the left hand side (there’s no reason whatever to think of and as separate equations when all we want to study is the geometry of the solutions).

In the first article, I talked about the affine plane , and how curves there are equivalent if one can be transformed into the other by a substitution

where , which we referred to as non-degeneracy. These substitutions can equivalently be thought of as the transformation of one figure into another equivalent figure, or as simply a change of the coordinate system used to describe ; they are two sides of the same coin, and you can choose your preferred interpretation. The same thing is true in , but using the letters , , and rather than and . This suggests that some conics might be equivalent to others. Can we classify the equivalent types of conics in ? And what relation will they have to Apollonius’ classification of the Euclidean/affine conic sections?

I claim that up to equivalence, there are really only five conics in , despite the infinitely many equations of the type that are possible.

One way to show this is to employ a technique called the Lagrange diagonalization algorithm. It uses coordinate transformations like the kind above to remove the “mixed” terms , , and from the picture. I’ll now explain the algorithm and give an example. This is definitely the most formula-heavy part of the whole series, so feel free to just skim-read it and pick up again at the conclusion, although the techniques used are nothing more than the two facts from basic algebra outlined in the intro.

- Look at the left hand side of your equation, and look for a squared term like , , or . When you find one, look for a mixed term containing the squared letter. So for example, if you find , look to see if there is a term such as or . If there is no such mixed term, nothing to do here, move on to the next squared term, such as . If there is, then “complete the square” (as in the introduction) on the squared and mixed term you have found. For example, if the terms are and , rewrite them like this
Then just make the non-degenerate transformation.

This means that

Notice how we have essentially “killed off” a mixed term using the coordinate transformation. With a little re-arranging of the two substitutions above, this also implies that , so if there is another mixed term, say , we can swap the for . This means there are no more s and s in the equation, only and . We can then repeat the process outlined about on the resulting and term. Once this is done, look for more squared terms you can use to eliminate mixed terms with further transformations—these may now be terms like and so on.

- But what if there are no squared terms at some stage in the algorithm? What if we are stuck with only mixed terms? In that case, we can take a mixed term like and make the non-degenerate transformation
so that by the “difference of two squares” formula above. Now we have some squared terms to work with, but we may have created some more mixed terms in the process. So go back to step 1. and use the newly created squared terms to eliminate more mixed terms.

- Repeat steps 1 and 2 until there are no more mixed terms left. The resulting equation is not uniquely defined: it depends on the choices you made along the way about what order to do things in. You and I could each perform the algorithm and get different results. We’ll deal with that after.Let’s do a fully worked example of this part of the algorithm to make sure that it is clear. Our conic is defined by an equation, say, . There are no squared terms, so let’s pick and make the transformation in step 2: , and . Thus the equation becomes
Remember: this equation defines an equivalent projective curve to the original equation, because all we did was make a valid non-degenerate change of coordinates. Now we can use the term to kill off the term. Just write . Then we can make the change of coordinates , , so the equation becomes:

Finally, we can use to eliminate the term. I can write

and then make the transformation and . So the equation becomes

Now there are no more mixed terms, so we are done.

- If we relabel all our symbols so that the s are hidden, we are left with an equation that looks like
where , , and are some combination of the ‘s in the original equation, and may or may not be zero. The final step is to make the non-dengerate transformation , and then again for and . Thus we have the following situation. If is positive, then . If is negative, then . If , there’s nothing to do. The same is true analogously for and , and and . After making

*these*changes, we are left with a new equation of the form , where , and are either , , or , and the number of s, s, and s appearing in this form*is*uniquely defined. You and I could do the algorithm, or indeed any algorithm using valid coordinate changes, to get the equation to a form with only s, s and s as coefficients, but we’d always end up with the same number — the only possible difference is the names of the variables, but we can always swap these around anyway. This is a result called (for some mysterious reason) “Sylvester’s law of inertia”^{1}, but it’s a bit more advanced so I won’t prove^{2}it here.

Therefore, if we swap the names of the variables if need be, and perhaps replace an equation with its negative (since and clearly have the same solutions), then every conic in is equivalent to one of the following five conics:

That’s it for the algebra. Now what about the geometry?

In the previous article, I wrote that we could obtain an affine curve from a projective curve by simply writing , and then studying the points in corresponding to . These are just the points of the projective curve where , and is commonly called an “affine piece” of the curve. The “shortcut” method for doing this calculation is just to rewrite the equation, but replacing with , and replace by and by . Geometrically, we are intersecting the curve with a copy of , and studying the intersection. Hence we will do that for each of the five conics to see what these affine pieces of the curve look like, and then briefly examine the points at infinity with respect to the affine pieces.

First, . Wherever , its points are equivalent to . This implies that on these points. However, could be anything. So the affine part of this curve contains all points — it’s just a line running along the -axis in our coordinate system. Is this a conic section? Well, actually yes: it’s a particularly ugly conic section obtained by intersecting a cone with a plane that just glances the edge of the cone:

The point also satisfies the equation. This point does not lies in the affine piece where , but is the *point at infinity* where the opposite ends of the affine line meet. Projective lines are like circles — they loop back on themselves.

Now for . Repeating the process, we have obtain the affine curve . But that’s a difference of two squares, so this can also be written as . This equation is true whenever is zero, or is zero. Therefore this curve is comprised of two straight lines, described by and , which intersect at the point . It’s a cross shape, and is also a degenerate conic section:

The points where , are and . These are the points the two lines disappear off to in the distance.

Then we have . In the projective plane, because does not refer to a projective point, this equation holds only at . However, this point does lie in the affine plane where , so when we take it to the affine piece, we find that also defines a single point, sitting at . It corresponds to another degenerate conic section—when the plane only cuts at the point of the cone.

Now we come to the two most interesting cases. Let’s rewrite the curve equation to and study the affine part . The affine curve we obtain here is : an affine ellipse, one of Apollonius’s conic sections. On the other hand, if , then we have which has no projective solutions, and so there are no points at infinity. But that’s exactly what we would expect, because ellipses do not extend infinitely far.

But what about the other conic sections of Apollonius? If we relabel variables to , and , we have , and then write this as , we can study the part where . But that’s just the affine curve : precisely the points of a hyperbola. If we check the points at infinity by examining the curve where , we have , with solution and . These are the same points at infinity as with the cross above, and with a picture, we can see why:

The arms of the hyperbola meet with the cross at the same points at infinity.

What about the parabola, the third conic section? Returning to , we can write this using the difference of two squares formula , and then making the non-degenerate change of coordinates and . Hence we have . Then at the part of the curve where , we have , the parabola. There is a single point infinitely far from this parabola, at in coordinates.

These considerations lead us to the inescapable conclusion that the three conic sections identified by Apollonius are just three pieces of a single projective curve . The projective plane unifies our vision of all three curves, which can now be studied as a single geometric entity. It is an example of how even ideas many hundreds of years old can be given a new perspective by the abstract power of more modern mathematics.

Now we come to the final projective conic, given by the equation . This one presents an unfortunate problem, because it doesn’t seem to describe a projective curve at all. It could only be solved if , , and were all zero, but is not a point in projective space. This equation doesn’t have any geometry.

But perhaps we’re being too hasty. There is a way out, but it involves moving to an even more abstract space. The complex numbers are the number system obtained by adding square roots to negative numbers, and are a foundational part of much of modern mathematics and physics. And we can transfer everything we have done so far to the complex world. We can study equations where the numbers involved are complex numbers, form an affine plane that has points given in coordinates as pairs where and are each complex numbers, and form the complex projective plane by dividing three-dimensional complex affine space into complex lines through the origin, and has coordinates as ratios where each entry is complex. By this point, there is no way that any human can imagine what a complex projective space “looks” like^{3}. We can work with it perfectly well mathematically, but any visualization we do is via analogy, which with training and intuition can be surprisingly accurate guides to navigating this strange world.

We can study conics given by equations

where now the s are complex numbers, and even repeat steps 1-3 of the algorithm above. In step 4, however, we can even go a step further. In the complex world, there is no distinction between positive and negative numbers. Once we have arrived at the form , where , , and , are complex we can just make the change of coordinates , , and . These square roots always exist in the complex numbers, no matter what , and are.^{4} Now the curve can only possibly be in one of three forms (after possibly swapping the names of the variables):

simply depending on how many of , , and were zero.

Classical — and much of modern — algebraic geometry is really the study of curves, surfaces, and higher dimensional geometric objects living in complex projective spaces. In the complex world, *every* algebraic equation has a geometric meaning. The theory is more unified, and frankly easier, owing to a number of powerful results about complex geometry that make the correspondence between geometry and algebra as rich as can be.

Let’s take one last look at that final curve . In the complex case, this definitely has solutions, for example . In fact, it has infinitely many. And as a final nod to our friend Apollonius, we can make another non-degenerate change of coordinates , so that the equation becomes

In the complex projective plane, that fifth conic, the most degenerate and problematic of them all, is one and the same with the fourth conic, which was the most beautiful and interesting. Apollonius’s conic sections live inside it, as the complex numbers contain the ordinary “real” numbers, and so this complex curve contains all the original solutions to the fourth equation. The whole family of conic sections, studied by the ancient Greeks, and later by Islamic and Renaissance mathematicians, are still there despite the long journey they have been on, through increasingly abstract spaces, from , to , to .

]]>

Last week, we took the Euclidean plane and forgot what angles and distances were, while retaining the concept of parallel lines. This gave us a new space called the affine plane, . Our mission today is to describe an extension of the affine plane called the projective plane, much beloved by algebraic geometers.

Why do algebraic geometers like projective space? One reason is that as well as studying the geometry of equations, they like to study the geometry of *systems* of equations. Consider the pair of lines:

A pair that satisfies both equations is a point lying on both lines—the only one is . Now consider

It is obvious that these two equations have no common solutions, and therefore, there’s no geometry lying behind this system. Each equation individually describes a line parallel to the other. The system as a whole describes the intersection of two parallel lines: nowhere.

When standing in the middle of a road it seems like the parallel edges of the road do meet in the distance. What if there were a way to make this idea of parallel lines meeting at an infinitely far away point a mathematical reality?

The projective plane is a way of extending the affine plane so that parallel lines really do meet at infinity. Rather than being some “hack”, this really is where figures defined by algebraic equations “want” to live. It gives us a more complete, unified picture of algebraic geometry. We don’t lose any of the geometry of the affine situation: wherever you’re standing in projective space, everything “looks” affine from your perspective. Nonetheless, there is a broader perspective that allows us to see the infinitely far-away horizon, just like how on planet Earth everything looks flat, until we go up in a space shuttle and see the true geometry from space. So what is the projective plane?

When introducing a new concept in mathematics, it is best if it can be fully described using objects and concepts that are already well understood and defined, to guarantee the new concept fits in with the rest of mathematics. One common way to do this is via an *equivalence relation*: take an existing structure, divide it into distinct parts called *equivalence classes*, and then use the classes as single entities in a new structure, retaining all the properties of the old structure except those that were destroyed in the partitioning process. We’ll use a space we already know — the affine space — to construct the projective plane. However, once the construction is complete, the projective plane will stand as a mathematical entity on its own; rarely will we have to lift the hood to see how it was originally built.

The first step is to introduce the affine space . There’s nothing mysterious here. You’ve met the 2d affine plane . So the 3d affine space is just like ordinary 3d space but with no concept of angle or length, but parallel lines (and now, planes) are to be respected. The next step is to choose a privileged point, which we’ll called for now — any point will do for this construction.

The equivalence relation I’m going to introduce on the affine space is to say that two points and are in the same class if there is a line through and that also passes through , the privileged point, in which case we write to show they are now “equivalent”, but not equal. There’s a slight problem with this though: under this definition every point would be in the same class as , since for any there is certainly a line passing through and that also passes through . The easy fix is to “puncture” the space, removing from all further considerations.

Now every point in (with missing) is equivalent to every other point on an infinitely long line through and where used to be. These lines through are the equivalence classes of our equivalence relation. And now for the punchline: the points of the projective plane are the equivalence classes of the relation described above.

In other words, is a geometric space whose points are lines through a chosen point in . Or in fancy mathematical notation^{1}:

If you find this hard to get your head around, don’t worry: it is. We can’t even properly draw a projective curve; it is too far removed from ordinary 2d space. It isn’t even clear why this is a 2d plane at this stage.

Let’s add some coordinate to make this plane easier to navigate. If I briefly go back to , I can choose a coordinate system that has . What this means for the equivalence relation is that if and only if, supposing , then where is a non-zero number. So for example, . Since these two points lie on the same line through in , they represent the same point in .

Now we just let inherit this coordinate system: a point in is given coordinates by taking the coordinates of any point in the line that compromises it in . The coordinates for a point in are therefore not unique. However, the point *is* uniquely determined by the *ratios* between the coordinates, so we write

for any non-zero , where the colons remind us that it is ratios that count. Of course, does not refer to any point in , because we excluded it from when constructing the space. And that’s it: we no longer have to think about at all. We are just working on a strange plane, where points are uniquely identified by a ratio of 3 coordinates. Let’s do some geometry and see if we can get more of a feel for this.

Using these special coordinates I can now explain why is an extension of . I somehow have to provide a method for embedding the entirety of into without losing any information—each point in should be correspond to a unique point in . The trick here is as follows: by the rule that it is coordinate ratios that matter, any point in where is the same as . If we then just look at the first two coordinates, then is point on the affine plane. By a reverse argument, we can choose any we like, and assign a point on the affine plane by

(1)

For simplicity, we can just choose , so that . ^{2}. The important thing is that each pair is of a different ratio to , and so corresponds to a different point in . Therefore contains . In fact there are many ways to contain inside , for instance by choosing a different value for , or by sending to , or however you please, so that wherever we are in , there is a local neighbourhood that looks just like . This means that to study just some region of a figure in , we can just use the comparatively simpler methods of affine geometry. For this post we’ll stick with the simple embedding as much as possible.

Now let’s see what this means for the parallel lines and we met above. We can embed the points as we described above by , so that these affine lines now lives in . But we can go a step further, and continue the lines in by applying this same transformation to the equations themselves:

It is not hard to see that, after choosing an embedding, say by setting , then the solutions to these equations in and correspond to solutions of the corresponding affine lines. But then if we multiply through by , we obtain the following relations between projective coordinates that lines satisfy:

These new “projectivized” equations do have a solution in projective coordinates. Namely, the point . This point does not lie on our affine plane at all, because we chose our affine plane to specifically be that subset of where . Rather, it is “infinitely far away”.

The set of all points are therefore the “points at infinity” with respect to this copy of that lives inside , and there aren’t that many of them: there’s point for each number , forming an (affine) line, plus the point . Together, we call this “line at infinity”^{3}. Thus is simply with one additional line added. This is an intuitive reason that deserves to be called a 2d plane—It is much more like than the original that it was constructed from.

Finally, for this rather long and challenging post, we’ll consider solutions of equations in the projective plane. There’s a problem. Suppose I have an equation, like . A triple of coordinates such as satsifies this equation. However, , which when considered as projective coordinates is the same point as , does not. The fix for this is to consider the solutions of homogeneous equations only. These are equations in which every term is of the same degree. So for example, is homogeneous, but is not. Since we’re studying conics, which we defined to be algebraic curves of degree 2, we require every term to be of degree 2. This means that if we have the general conic equation

then *any* coordinate triple representing the same point in will satisfy the equation, since if we have two coordinates of the same point (with , of course), then

if and only if

This insistence on studying only homogeneous equations may seem like a limitation. However, we have already seen that have a way of switching seamlessly between non-homogeneous equations for plane curves in and into homogeneous equations in , and switching back is just as easy. For example the hyperbola:

and this projective curve contains a perfect copy of the original affine curve.

That covers all I’d like to say about the projective plane. But where is all this going? What are the equivalents of Apollonius’ conic sections in the projective plane? How many are there, and how do they relate to the original conic sections from the first post? Is there anything to be gained from migrating to an even weirder geometric space? Find out in the final post of the series.

]]>In the 2nd century BC, Apollonius of Perga studied and classified the so-called conic sections: geometric curves obtained by intersecting a flat 2-dimensional plane with a hollow cone. If the plane is intersected with the cone at a shallow angle, the resulting curve is an ellipse, the circle being a special case.

On the other hand, intersecting with a steep inclination gives us a hyperbola.

Lying directly between these two, when the plane is parallel to the cone, we have the parabola.

Apollonius considered these figures as living the plane as described by Euclid of Alexandria: a flat 2d world where overlapping circles and straight lines give rise to the notions of distance and angle, the angles of a triangle sum to 180 degrees, and all lines meet once except for parallel lines. It’s the plane we all remember from school.

These three families of curves are still studied by all modern students of geometry, and indeed physics, as Newton’s theory of gravitation predicts that the orbits of planets and comets all follow the path of a conic section. The three curves share many similar characteristics. In this series, I’ll explore a more modern, abstract view of conic sections, leading to a beautiful interpretation of the relationship between them.

When Descartes introduced coordinate systems to geometry, it became possible to describe these curves with algebraic equations — that is, equations that use only numbers, and some chosen letters like and , combined using only addition, subtraction, and multiplication. When an equation involves two or more letters, there are generally infinitely many ways to solve it. Thus, instead of trying to “solve for ” as we did in high school, it makes more sense to understand the *geometry* of the solutions.

If the Euclidean plane is given the usual coordinate system by choosing some central point to act as together with a pair of perpendicular coordinate axes, then every conic section can be described by giving an equation involving and . If a point satisfies the equation — that is, the equation is true when and — then that point lies on the conic section, and otherwise it does not. For example, the points of an ellipse satisfy an algebraic equation of the form:

where and are any two numbers that determine the size and shape of the ellipse^{1}

Let’s be more concrete. If the ellipse in question is the circle , then the point lies on the circle, because

On the other hand, the point does not lie on the circle, because

The family of hyperbolas are almost the same, except that they satisfy an equation of the form

Finally, the points on a parabola satisfy an equation that looks something like

Again in these two latter cases, the choice of and determine the overall shape of the curve.

The ability to express geometric ideas using algebraic equations — and vice versa, to express the relationships described by algebraic equations as geometrical figures — is the inspiration for algebraic geometry. The goal this of series is to explain a modernized^{2} version of Apollonius’ classification of conic sections: the classification of conics in the projective plane. Along the way, we’ll have to understand what is meant by the terms conic and projective plane.

The algebro-geometric definition of conic is easy: a conic is any plane curve described by equations where the coordinates are multiplied together in pairs, like , , or , but never any higher products (for example, not or ). All of Apollonius’ conic sections are conics. The equation for the parabola has appearing by itself without being multiplied by or . That’s okay too, but the equation must have at least one “double” term to be considered a conic equation. To use the correct lingo, we say that a conic is a curve of degree 2.

The projective plane is a weird alternative to the Euclidean plane; understanding precisely what it is will take the remainder of this post and the next. We’ll begin by re-considering the Euclidean plane. One of the most fundamental properties of any geometric space is what it means for two figures in the space to be essentially equivalent. Consider the following two rectangles.

Are these rectangles equivalent? From the Euclidean perspective, the answer is is yes, as one is just a rotated version of the other. Two shapes in the Euclidean plane are equivalent if one can be transformed into the other in such a way that all distances—and hence also all angles—are preserved. Another perspective is that one can be turned into the other by tilting your head. If the plane is given Descartes’ coordinate system, the mathematical equivalent of tilting your head is changing that coordinate system by moving and rotating the coordinate axes together. In mathematics, changing a figure while leaving it equivalent and changing the coordinate system used to describe it are two sides of the same coin, just as in the rectangle example above. Consequently, what it means for two figures to be equivalent can often be specified by describing the valid coordinate changes in that space. The first step toward the projective plane will be to *relax* the Euclidean criteria for equivalence of figures.

The problem is that, when studying algebraic curves, lengths and angles are not really an aspect of the algebraic structure of the equations, so saying that figures can only be considered equivalent if their lengths and angles are the same is too restrictive for algebraic geometry. Thinking back to the example of the ellipse above, we recall that ellipses of all shapes and sizes are described by an equation that looks like

for some and . So two ellipses maybe have different sizes and shapes when we draw them on paper, but because the “shape” of their algebraic equations are the same (just with different numbers and )^{3}, the algebraic geometer has no need to distinguish between them. Hence, we will consider two figures to be equivalent if one can be turned into the other by means of rotating, reflecting, or moving—these are the Euclidean transformations—but also by shearing, or uniformly stretching the entire plane. In Euclid’s world, two triangles are equivalent if each corresponding angle and side is the same—in this new space we are describing, *all *triangles are equivalent. This can be summarized by saying that our transformations of the plane must take all evenly space parallel lines to evenly spaced parallel lines.

Thankfully in these articles, we’ll never have to move our curves around the plane, so it’s even simpler for us. I can now explain what equivalence means by describing the allowable coordinate transformations. In this space, two figures described in coordinates are considered equivalent if we can turn one into the other with a transformation like so:

(1)

This is enough to rotate, reflect, shear, and stretch any figure—moving the figure would be controlled by also adding an extra constant in each equation. We’ll also add an extra condition, that , a condition that I’ll call “non-degeneracy”. This guarantees that the entire plane does not get squashed into a single line or point. Such a transformation would still leave all parallel lines parallel — they’d just all be the same line, which is no good. Hence, by change of “change of coordinates” or “transformation”, I mean anything that can be described by a substitution in form of , subject to the non-degeneracy rule.

As an example of this in action, consider the rather grisly algebraic equation

Since the equation is of degree , it’s a conic. But which of Apollonius’ conic sections is it? Well,

and so if we write

as per the prescription, we obtain

so the figure was a hyperbola all along^{4}.

We have now developed a new geometric space, more suitable for algebraic geometry, called the *affine plane*. It is similar to the Euclidean plane, in that it has a similar notion of parallel lines, but “angle” and “length” are meaningless concepts. Shapes that differ in angle and size are considered to be the same shape, just like in the Euclidean plane, a pair of shapes related by movement and rotation are considered to be the same. Let’s give these two spaces symbols. I’ll call the Euclidean plane , and the affine plane , where the meanings of and are obvious, and the represents that these are 2d spaces.

Geometers and philosophers from an earlier time may have objected to the mathematical validity of such a space because it is unintuitive. Intuitively, lengths and angles are what makes two shapes equivalent. It’s easy to see this by drawing them. But this is the nature of modern mathematics. It is abstract, often defying our everyday intuition and accurate pictorial representation. We will still use pictures to aid our intuition, but must remember there are mathematical characteristics of the affine space that cannot be fully captured by the picture.

We obtained the affine plane from the Euclidean plane by removing structure, namely, the concepts of length and angle, but retaining the concept of parallel lines. But for reasons that will be explained soon, this is still not enough. In the next post, we must explore the even stranger projective plane, in which even parallel lines “eventually meet”. If the affine plane was constructed by disfiguring the Euclidean space, we will construct the projective plane by disfiguring affine space. But until then, please add your questions and comments below.

]]>There are alternative plug-ins that do LaTeX equations, notably ones that use MathJax. If I cannot get this fixed, I’ll switch. However, the alternative plug-ins only allow basic equations, whereas I already have a post that makes heavy use of diagrams. WP-QuickLaTeX has to this point been ideal, since it supports many major LaTeX packages and allows me to customize my preamble.

]]>The whole point of the typesetting system is that it relieves you of having to do this. Your job is just to type the document, with a suitable choice of document class, and use simple commands to logically organize your document like `\chapter`

, `\includegraphics`

, and `\cite`

and so on.

You should basically never have to manually control things like line spacing, indentation, or font size. I have seen source files with dozens of new lines (`\\\\\\\\\`

) at the end of each paragraph. Now it is fair enough that you might want the final output of your document to use blank lines to delimit paragraphs instead of indentation. But the solution is still to type out your document as normal, and then *search for how you would change this in the final output*. In this situation, incidentally, the correct solution is to add

`\usepackage{parskip}`

to the preamble of your source file.

There are a few little quirks that can make the spacing in your document go a little bit crazy. The first thing is to be aware that in your source file, a single new line is essentially treated by the compiler like a space; indeed some authors write their source files with every sentence on a new line so that it is easier to edit. A double new-line is treated the same as `\par`

; ie, a new paragraph. You should be careful with this. If you are formatting an equation in a “display math” environment, it should be treated as part of the current paragraph, not as a paragraph unto itself. Therefore, don’t insert blank lines before and after the equation in the source file; it may look “right” in the source file because there is a little blank space either side of the equation in the output, but actually, you’re creating new paragraphs. In particular, you’ll have unwanted indentation in the following line.

You should also be aware that there are several ways to insert and remove whitespace in both text and math modes. `\!`

removes some whitespace, `\ `

inserts a space, `\,`

inserts a short space, and `\quad`

inserts a rather long space. Here are some use-cases:

- When you complete a sentence in LaTeX, the space following the full stop is longer than a standard space. This has the side effect that if you write something like “i.e.”, the next word will be spaced incorrectly. You an fix this by manually inserting the correct length of space with
`i.e.\`

. - I find short spaces are useful in math equations involving differential forms, or any other time you want to multiply entities whose symbols consist of more than one character. If I have a 1-form like , you’ll notice this looks a lot nicer that . That’s because I inserted a short space
`\,`

In LaTeX, we write left and right quotes with ``like this'`

or ```like this''`

, Not using the character “. It will look wrong if you do. There are also three lengths of dashes available: `-`

for a hyphen, `--`

for an en-dash and `---`

for an em-dash (and of course in math mode, `-`

inserts a minus sign. It’s worth reading up on when to use each of these kinds of dash if you’re not sure.

When formatting mathematics, it’s common to want to have brackets, braces, and such in different sizes. For example

looks terrible, but

does not. The answer is to use `\left(`

and `\right)`

to get the delimiters to resize depending on what is inside them. You can replace the `(`

and `)`

in the previous commands with any other delimiter as you please. Note that to get braces, `\{`

and `\}`

are required.

So should you just use `\left(`

and `\right)`

all the time? The answer is no. For one, it’ll make your source file harder to read. But more importantly, due to some quirk in how these things work, it can screw up the spacing. Essentially, the TeX compiler works hard to make all your text nice and flush with the margins and so on. This means subtly adding and removing whitespace on every line so that it looks just right, even from math equations. For some reason, any math enclosed in these dynamic delimiters is off-limits in this process; the TeX compiler can’t add or remove whitespace inside \left\right pairs, so there’s a good chance that any line containing these symbols will not look quite right. So, use it only when necessary. If things start looking bad, remember you can manually set the size of a delimiter with commands like `\big,`

`\bigg`

and so on.

I really hope these issues with delimiters are eventually fixed in a later version of LaTeX.

You should never, ever, type full words in math-mode because . However, words do sometimes arise inside mathematical formulas. Usually, just using `\text{ your words here }`

is enough. If your word is a mathematical operator, such as the appearing in the titie, then there are two possibilities. Either one of the standard mathematical packages like amsmath will contain that operator as an actual command, in which case you should format it like `\sin{(x)}`

; or you can define the operator yourself (again using amsmath) like this `\operatorname{sin}{(x)}`

. Note the braces around the argument in both cases; this sets the spacing correctly. It is the difference between and .

When writing any large mathematical document, there are certain symbols you will be using again and again. Your LaTeX preamble should be absolutely stuffed with ways to make these easier to type. The three most useful are `\newcommand`

, `\renewcommand`

, and `\DeclareMathOperator`

. The first two work in the same way, except that `\renewcommand`

allows you to overwrite existing commands. I mostly use this to overwrite built in commands, like usually `\P`

`inserts the paragraph symbol , but I’ve never used that, so I set it to , symbolizing a projective space. The `

`\DeclareMathOperator`

command is like `\operatorname`

except that you can reuse the operator. So for example, I can write `\DeclareMathOperator{\trdg}{tr.d} `

to create a transcendence degree operator, and this means it has the nice font and spacing that you’d expect from an operator whenever I now write `\trdg`

. Almost everybody I know has commands for the common symbols , , , and so on.

Commands can also take arguments. It’s easy to explain this with an example

\newcommand{\prtl}[2]{\frac{\partial #1}{\partial #2}}

is the command I use for partial derivatives. The `[2]`

means it expects 2 arguments, and they’re referenced by `#1`

and `#2`

.

If you’re at university and you’ve been asked to produce a LaTeX document for the first time, the chances are you’ve just been told to launch some software pre-installed on the university machines and given a brief explanation of how to start writing and which button to use to compile the document. You might not bad aware that LaTeX is simply a language, which can be edited using any software you like, and that there are many free TeX compilers out there that can produce your document for you. So don’t just accept whatever software you’ve been given: look around for software that suits you, and configure it to your tastes. You could use a high-performance programming text editor like vim, emacs, or atom (all 3 of these have plug-ins for compiling and previewing your document from inside the editor), or you could use something like TeXstudio, a LaTeX document development environment with every LaTeX-related feature you could possibly want (except perhaps a vim mode). At the moment I’m using vim with the vimtex plugin, but I am flirting with the idea of emacs.

There are different document classes for a reason. Don’t use “article” for everything. There are lots of different classes to choose from, some standard and some provided by additional packages. Writing a report? Use the report class!

Similarly, choose the right environment for the task. There are environments for lots of common task. In particular for mathematics students, don’t just type `\textbf{Theorem 2}`

to start a theorem or something. There are theorem and proof environments provided by the amsthm. I can’t go into what all of these are here, but there will almost certainly be an environment for whatever you want to produce. Just take the time to look for the required packages.

I will plug the wonderful environment `IEEEeqnarray`

provided by the package `IEEEtrantools`

. This is the equation environment to end all equation environments as far as I’m concerned. It’s just an extremely flexible equation array environment, which gives you full control over the alignment and spacing of every column, the numbering of every row, and so on. This is *the *answer for anyone who has ever tried to wrestle with the align or eqnarray enviroments, only to have their equations spread out horribly across the page. I include a link to a use manual for this enviroment at the bottom.

You won’t appreciate this until you’ve tried to type `M\"{o}bius`

`transformation`

or `B\'{e}zout's identity`

a few times. With this input encoding you can just type `Möbius`

`transformation`

.

Links related to this article.

- http://texcatalogue.ctan.org/bytopic.html look here for enviroments, classes, and packages that suit your particular needs.
- http://moser-isi.ethz.ch/docs/typeset_equations.pdf “How To Typeset Equations In LaTeX”. The document that introduced me to the
`IEEEeqnarray`

environment, and I have never looked back since. - Some LaTeX editors/plugins
- https://www.texstudio.org/ – Massive LaTeX development environment. This one is recommended if you are not experienced with programming text editors but want access to lots of convenient features.
- https://www.gnu.org/software/auctex/ – The impressive Auctex plugin for Emacs.
- https://github.com/lervag/vimtex – A vim plugin for producing LaTeX documents. Very lightweight, designed to be used in conjunction with other more specialized plugins but much better than the more widely-known LaTeX-suite, which tries to do everything but is rather clunky for it.

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**Theorem**

Let be an ideal in the Jacobson radical (ie. the intersection of all maximal ideals) in a Noetherian ring . Then

As most sources are quick to point out, this “intuitive” fact can fail if the ring is not Noetherian.

**Proof**

We first wish to prove that

The inclusion is obvious. For the reverse inclusion, let generate , and let be the set of all homogeneous polynomials in letters over of degree such that for , we have that . Generate an ideal from , and choose a finite generating subset .

Let be the maximum degree of all the ‘s, and let . Then can be represented as where is a homogeneous polynomial over of degree . Therefore

Since and each are homogeneous, we can take ‘s to be homogeneous too, and of degree , since any terms not of this degree will cancel anyway and so are redundant. It follows that no is constant since was specifically chosen to be higher degree than any . Thus

We have that and . Therefore .

Now use Nakayama’s lemma to deduce the result.

Original source: http://www.jmilne.org/math/xnotes/ca.html

Next semester I enter my final year. This week, I selected the courses I’ll be taking.

**Masters project:**Not quite sure what this will involve yet. The supervisor says he will like me to be introduced to sheaf cohomology but hasn’t decided where to take that.**Algebraic Geometry:**Natch. The syllabus changed a lot this year because there is a new lecturer taking it, but it looks great.**Manifolds, Homology and Morse Theory:**Interesting looking course with one of my favourite lecturers.**Number Theory****Higher Arithmetic:**When I mention this one people think I’m taking a class in multiplying very large numbers.**Measure Theory:**I felt I could not leave my undergraduate without learning about measure and Lebesgue integration.**Complex Dynamical Systems:**My university has a dynamical systems research group and the lecturer is very passionate about the subject, so this should be good.

A bit of a resolution is to take better advantage of office hours. I’m one of those students who maybe for reasons of vanity (fear of looking stupid, for example; totally irrational, I know) doesn’t make much use of office hours, but not taking advantage of access to professional mathematicians is a great way to hobble oneself.

Leave a comment about what you’ll be doing next year!

]]>The fact that our devices are rarely more than a few feet away has essentially eliminated the experience of actual boredom, if we choose to take that road. This has been me all too often. Check messages while going to the loo. Read reddit during class. Listen to audiobooks on headphones while doing anything boring that requires my hands.

Some see this as a victory over an uncomfortable emotion that has plagued humanity for all time, but many others are unconvinced that this counts as a victory. Manoush Zomorodi, tech journalist and author of *Bored and Brilliant*, points out that many of the greatest innovations in history were spawned of boredom; great thinkers like Einstein and Darwin made times in their days — every day — to walk around aimlessly and allow their minds to wander. And who hasn’t had the experience of gaining new insight on a problem or project while in the shower? Is it a coincidence that the shower is the one place we can’t use our phones at all? I think not.

It’s no secret that many of the apps we use on our devices have been specifically designed to absorb our attention as much as possible. It’s an uphill battle to allow ourselves the time to think, without the influence of a constant flow of information/

Recently, I’ve been trying to re-introduce a little boredom into my life. I’ve been finding this really hard at home, since my living room contains a smart TV, a Nintendo, a laptop, and usually my mobile phone — a stimulation conglomeration. But when out and about, at least, where the only real culprit is my phone, I’ve found a few things that have really helped me out.

- Keep the phone in a bag rather than a pocket, when practical. Gentlemen, this could mean getting a man-bag, or as I affectionately call mine: a “handy-bag”. Trust me, they’re great. Carrying everything in your pockets is impractical anyway. Keeping the phone in a bag adds an extra barrier between you and your instinct to immediately pick up the phone when there’s a lull in the day.
- Disable unnecessary notifications. For some reason, our phones come pre-configured to interrupt us as often as possible and notify us about the most inconsequential things. Be ruthless, and enable only the notifications you really care about.
- Use a usage-tracking app. This can be really eye-opening, showing you exactly which apps you’re using, when, and for how long, as well as how many times you’ve “just checked” your phone that day. I use one called Quality Time, which tracks my usage in a widget on the homescreen, and allows me to set periods in the day when certain apps and notifications are disabled, to allow greater focus on work or the people I’m with.
- Make your phone deliberately less fun. For instance, I swapped Firefox for Firefox Focus. Firefox Focus is a web-browser designed for privacy; it deletes all cookies and temporary files, and retains no history. This has the unintentional side-effect of making using social websites less fun because you have to reenter your password each time you visit. You could also delete some of your more addictive apps you’ve identified using the usage-tracking app.
- The most important one: find certain times in the day where you will just allow your mind to wander. I’ve found two situations where this works well for me. One is while walking or traveling anywhere. No podcasts, no games on the bus. Just enjoy the journey, and think. The other is any time spent waiting. For example, if I’m at a café and waiting for my order, I try to avoid just staring at my phone while I wait. Just look around, take in the environment. People watch. It’s quite fun actually, even if you feel a bit self-conscious at first.

I’ve added a new page of “Stuff I like“, which is just a big list of recomendations. At the moment, it’s just a few pieces of software I like. Updates to come.

Links for this post:

http://www.manoushz.com – Manoush Zomorodi’s website, with links to her various projects about staying an authentic human in today’s tech-driven world.

https://play.google.com/store/apps/details?id=com.zerodesktop.appdetox.qualitytime – The usage tracking app I use.

]]>The derivative operator on polynomials in is

on a monomial, and is defined on polynomials via linear extension. The definition extends to formal power series and rational functions in the obvious way you’d expect from knowing calculus.

Over an arbitrary field, we’re no longer defining the slope of a curve or the instantaneous rate of change, but this is still a surprisingly useful operation. It can be used to check for repeated roots if and are co-prime, or whether is separable. The notion extends to the concept of derivations — linear maps that obey the product rule from calculus — and there is a whole field of differential algebra that studies commutative rings with derivations. Moreover, derivations on the local ring of a point can also be used to define tangents in algebraic geometry, as they are in differential geometry.

One of the nice things about algebraic differentiation is that many of the basic things you might want to prove can be done using induction on the degree of the polynomial (a degree polynomial is just where is degree ) or a double induction on the degree and number of letters if you’re working over .

For example, consider the ring , where is another indeterminate with no relations. We get an algebraic version of Taylor’s theorem by

This can easily be proved by induction on the degree of . Now we can “let go to 0″. What is the algebraic interpretation of this? Well, it harks back to the notion of “infinitesimals” from early/non-standard analysis. We quotient out by , so that in we have that is now “so small” that its square is . We can now *define* the algebraic derivative to be the unique solution to the equation

I find this really pretty. And in fact, we can do one better: we can divide this equation through by (which always obtains a new polynomial, since is a factor on both sides), and then let truly “go to 0” by quotienting again by . We obtain that

Side note: I’m aware that a very similar post to this appeared on reddit’s /r/math this week. This is coincidental; I actually wrote this post before the Snake Lemma one.

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