John Baez
Apr14-04, 05:55 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nAlso available at http://math.ucr.edu/home/baez/week205.html\n\nApril 11, 2004\nThis Week\'s Finds in Mathematical Physics - Week 205\nJohn Baez\n\nThis week I\'d like to say more about number theory, but first -\nhere\'s the most fun book on astronomy I\'ve ever seen:\n\n1) James B. Kaler, The Hundred Greatest Stars, Copernicus Books\n(Springer Verlag), New York, 2002.\n\nIt\'s just what the title says: a compilation of the author\'s 100\nfavorite stars, each with a picture and a one-page description of what\nmakes that star interesting. They\'re incredibly diverse, from the\nmammoth Eta Carinae to tiny brown dwarf Gliese 229B. You\'ll see soft\ngamma repeaters, yellow hypergiants, pulsars, Mira-type variables,\nbarium stars, symbiotic stars, and more. There\'s also an introduction\nthat explains the concepts needed to enjoy all these different kinds of\nstars, like the Hertzsprung-Russell diagram and a bit of nuclear physics.\nWith the help of this, the whole book forms a wonderful taxonomy of\nstellar astrophysics. I suggest keeping it by your bed and reading one\nstar a night - though you may wind up staying up late and devouring the\nwhole book.\n\nOn to number theory....\n\nThere\'s a widespread impression that number theory is about *numbers*,\nbut I\'d like to correct this, or at least supplement it. A large part\nof number theory - and by the far the coolest part, in my opinion - is\nabout a strange sort of *geometry*. I don\'t understand it very well,\nbut that won\'t prevent me from taking a crack at trying to explain it....\n\nThe basic idea is to push the analogy between integers and polynomials\nas far as it will go. They\'re similar because you can add, subtract and\nmultiply them, and these operations satisfy the usual rules we all\nlearned in high school:\n\nx + y = y + x (x + y) + z = x + (y + z) x + 0 = x x + (-x) = 0\n\nxy = yx (xy)z = x(yz) x1 = x\n\nx(y + z) = xy + xz\n\nAnything satisfying these rules is called a "commutative ring".\nThere are also a lot of deeper similarities between integers and\npolynomials, which I\'ll talk about later. But, there\'s a big difference!\nPolynomials are functions on the line, whereas the integers aren\'t\nfunctions on some space - at least, not in any instantly obvious way.\n\nThe fact that polynomials are functions on a space is what lets us\ngraph them. This lets us think about them using *geometry* - and also\nthink about geometry using *them*. This was the idea behind Descartes\'\n"analytic geometry", and it was immensely fruitful.\n\nSo, it would be cool if we could also think about the integers using\ngeometry. And it turns out we can, but only if we stretch our concept\nof geometry far enough!\n\nIf we do this, we\'ll see some cool things. First of all, we\'ll see\nthat algebra is just like geometry, only backwards.\n\nWhat do I mean by this? Well, whenever you have a map T: X -> Y going\nfrom the space X to the space Y, you can use it to take functions on Y\nand turn them into functions on X. Since this goes backwards, it\'s\ncalled "pulling back along T". Here\'s how it goes: if f is a function\non Y, we get a function T*(f) on X given by:\n\nT*(f)(x) = f(T(x))\n\nMoreover, functions on a space form a commutative ring, since you can\nadd and multiply them pointwise, and pulling back is a "homomorphism",\nmeaning that it preserves all the structure of a commutative ring:\n\nT*(f + g) = T*(f) + T*(g) T*(0) = 0\nT*(fg) = T*(f) T*(g) T*(1) = 1\n\nConversely, any sufficiently nice homomorphism from functions on Y to\nfunctions on X will come from some map T: X -> Y this way! Here I\'m\nsummarizing a whole bunch of different theorems, each of which goes\nalong with its own precise definition of "space", "map", and "nice".\n\nSome of these theorems are technical, but the basic idea is simple:\nwe can translate back and forth between the study of commutative rings\n(algebra) and the study of spaces (geometry) and by thinking of\ncommutative rings as consisting of functions on spaces. We get a little\ndictionary for translating between geometry and algebra, like this:\n\nGEOMETRY ALGEBRA\nspaces commutative rings\nmaps homomorphisms\n\nBut be careful: this translation turns maps into homomorphisms going\nbackwards: it\'s "contravariant". This is really important in two ways.\nFirst, suppose we have a point x in a space X. This gives a map\n\ni: {x} -> X\n\nThis, in turn, gives a homomorphism i* sending functions on X to\nfunctions on {x}. Functions on a one-point space are like numbers,\nso i* acts like "evaluation at x". Moreover, i* will tend to be\nonto: that\'s the backwards analogue of the fact that i is one-to-one!\n\nSecond, suppose we have a map from a space E onto the space X:\n\np: E -> X.\n\nIf you know some topology, think of E as a "covering space" of X. Then\nwe get a homomorphism p* from functions on X to functions on E. Moreover\np* will tend to be one-to-one: that\'s the backwards version of the fact\nthat p was onto!\n\nWe can use these examples to figure out the analogue of a "point" or a\n"covering space" in the world of commutative rings! And the resulting\nideas turn out to be crucial to modern number theory.\n\nIn "week199" I explained the analogue of a "point" for commutative rings:\nit\'s a "prime ideal". So, now I want to explain the analogue of a\n"covering space". This will expand our dictionary so that it relates\nGalois groups to fundamental groups of topological spaces... and so on.\n\nBut, we won\'t get too far if we don\'t remember why a "prime ideal" is\nlike a "point"! So, I guess I\'d better review some of "week199" before\ncharging ahead into the beautiful wilderness.\n\nWhat\'s special about the ring of functions on a space consisting of\njust one point? Take real- or complex-valued functions, for example.\nHow do these differ from the functions on a space with lots of points?\n\nThe answer is pretty simple: on a space with just one point, a function\nthat vanishes anywhere vanishes everywhere! So, the only function\nthat fails to have a multiplicative inverse is 0. For bigger spaces,\nthis isn\'t true.\n\nA commutative ring where only 0 fails to have a multiplicative inverse\nis called a "field". So, the algebraic analogue of a one-point space is\na field.\n\nThis means that the algebraic analogue of a map from a one-point\nspace into some other space:\n\ni: {x} -> X\n\nshould be a homomorphism from a commutative ring R to a field k:\n\nf: R -> k\n\nOur translation dictionary now looks like this:\n\nGEOMETRY ALGEBRA\nspaces commutative rings\nmaps homomorphisms\none-point spaces fields\nmaps from one-point spaces homomorphisms to fields\n\nIt\'s worth noting some subtleties here. In the geometry we learned in\nhigh school, once we see one point, we\'ve seen \'em all: all one-point\nspaces are isomorphic. But not all fields are isomorphic! So, if\nwe\'re trying to think of algebra as geometry, it\'s a funny sort of\ngeometry where points come in different flavors!\n\nMoreover, there are homomorphisms between different fields. These\nact like "flavor changing" maps - maps from a point of one flavor\nto a point of some other flavor.\n\nIf we have a homomorphism f: R -> k and a homomorphism from k to some\nother field k\', we can compose them to get a homomorphism f\': R -> k\'.\nSo, we\'re doing some funny sort of geometry where if we have a point\nmapped into our space, we can convert it into a point of some other\nflavor, using a "flavor changing" map.\n\nNow let\'s take this strange sort of geometry really seriously, and\nfigure out how to actually turn a commutative ring into a space!\nFirst I\'ll describe what people usually do. Eventually I\'ll describe\nwhat perhaps they really should do - but maybe you can guess before\nI even tell you.\n\nPeople usually cook up a space called the "spectrum" of the commutative\nring R, or Spec(R) for short. What are the points of Spec(R)?\nThey\'re not just all possible homomorphisms from R to all possible\nfields. Instead, we count two such homomorphisms as the same\npoint of Spec(R) if they\'re related by a "flavor changing process".\nIn other words, f\': R -> k\' gives the same point as f: R -> k if you\ncan get f\' by composing f with a homomorphism from k to k\'.\n\nThis is a bit ungainly, but luckily there\'s a quick and easy way to tell\nwhen f: R -> k and f\': R -> k\' are related by such a flavor changing\nprocess, or a sequence of such processes. You just see if they have\nthe same kernel! The "kernel" of f: R -> k is the subset of R consisting\nof elements r with\n\nf(r) = 0\n\nThe kernel of a homomorphism to a field is a "prime ideal", and two\nhomomorphisms are related by a sequence of flavor changing processes\niff they have the same kernel. Furthermore, every prime ideal is\nthe kernel of a homomorphism to some field. So, we can save time by\ndefining Spec(R) to be the set of prime ideals in R.\n\nFor completeness I should remind you what a prime ideal is! An "ideal"\nin a ring R is a set closed under addition and closed under multiplication\nby anything in R. It\'s "prime" if it\'s not all of R, and whenever the\nproduct of two elements of R lies in the ideal, at least one of them\nlies in the ideal.\n\nSo, we have something like this:\n\nGEOMETRY ALGEBRA\nspaces commutative rings\nmaps homomorphisms\none-point spaces fields\nmaps from one-point spaces homomorphisms to fields\npoints of a space prime ideals of a commutative ring\n\nNow let\'s use these ideas to study "branched covering spaces" and their\nanalogues in algebra. This week I\'ll talk about two examples. The first\nis very geometrical, and it should be familiar to anyone who has studied\na little complex analysis. The second is more algebraic, and it\'s\nimportant in number theory. But, the cool part is that they fit into\nthe same formalism!\n\nIf you don\'t know what a branched covering space is, don\'t worry:\nwe\'ll start with the very simplest example. We\'ll look at this map from\nthe complex plane to itself:\n\np: C -> C\n\np(z) = z^2\n\nExcept for zero, every complex number has two square roots, so this map\nis two-to-one and onto away from the origin. In fact, away from the\norigin you can visualize this thing locally as two sheets of paper\nsitting above one. But these two sheets have a global complication: if\nyou start on the top sheet and hike once around the origin, you wind up\non the bottom sheet - and vice versa! In topology we call this sort of\nthing a "double cover". When we include the point z = 0 things get\neven more complicated, since the two sheets meet there. So we have\nsomething trickier: a "branched cover". In general, a branched cover\nis like a covering space except that the different "sheets" can merge\ntogether at certain points, called "branch points".\n\nNow let\'s think about this algebraically. To keep from getting confused,\nlet\'s write\n\nz^2 = w\n\nso that p is a map from the "z plane" down to the "w plane", sending each\npoint z to the point z^2 = w. The ring of polynomial functions on the z\nplane is called C[z]; the ring of polynomial functions on the w plane is\ncalled C[w]. We can pull functions from the w plane back up to the z\nplane:\n\np*: C[w] -> C[z]\n\nand p* works in the obvious way, taking any function f(w) to the function\nf(z^2).\n\nJust as p is onto, p* is one-to-one! So, we can think of C[w] as\nsitting inside C[z], consisting of those polynomials in z that only\ndepend on z^2: the even functions. We say C[w] is a "subring" of C[z],\nor equivalently, that C[z] is an "extension" of C[w].\n\nIn this example we can get the bigger ring from the smaller one by\nthrowing in solutions of some polynomial equations, so we call it an\n"algebraic extension". We\'ve already seen some algebraic extensions,\nnamely algebraic number fields, where take the field of rational numbers\nand throw in some solutions of polynomial equations. Algebraic extensions\ncan be complicated, but this one is really simple: we just start with\nC[w] and throw in the solution of *one* polynomial equation, namely\n\nz^2 = w\n\nIt turns out that quite generally, algebraic extensions of commutative\nrings act a lot like branched covering spaces. I probably don\'t have\nthe technical details perfectly straight, but let\'s add this to our\ntranslation dictionary, because it\'s an important idea:\n\nGEOMETRY ALGEBRA\nspaces commutative rings\nmaps homomorphisms\none-point spaces fields\nmaps from one-point spaces homomorphisms to fields\npoints of a space prime ideals of a commutative ring\nbranched covering spaces algebraic extensions of commutative rings\n\nNow let\'s have some fun: let\'s see how our algebraic concept of "point",\nnamely "prime ideal", interacts with our branched double cover of the\ncomplex plane. There\'s something straightforward going on, but also\nsomething more subtle and interesting.\n\nThe straightforward thing is that any point up on the z plane maps to one\ndown on the w plane. We don\'t need fancy algebra to see this! But, it\'s\nworth doing algebraically. According to the fancy algebraic definition,\na "point" in the spectrum of the commutative ring C[z] is a prime ideal.\nBut as you might hope, these are the same as good old-fashioned points\nin the complex plane!\n\nIt works like this: given any point x in C, we get a homomorphism from\nC[z] to C called "evaluation at x", which sends any polynomial f to\nthe number f(x). The kernel of this is the prime ideal consisting of\nall polynomials that vanish at x. These are just the polynomials\ncontaining a factor of z - x, so we call this ideal\n\n<z - x>\n\nSo, we get some prime ideals in C[z] from points of C this way. But in\nfact there\'s a theorem that *every* prime ideal in C[z] is of this form!\nSo, we get a one-to-one correspondence\n\nSpec(C[z]) = C\n\nSimilarly,\n\nSpec(C[w]) = C\n\nNow let\'s think about our branched cover\n\np: C -> C\n\nin different ways. It starts out life as a map from the z plane down\nto the w plane. We can use this to pull back functions on the w\nplane up to the z plane:\n\np*: C[w] -> C[z]\n\nBut then, by general abstract baloney, the inverse image under p* of\nany prime ideal in C[z] is a prime ideal back in C[w]. This gives a\nmap from Spec(C[z]) to Spec(C[w]). But this is just a map from the z\nplane to the w plane! And it\'s the same map p we started with. If you\ndon\'t see why, it\'s a good exercise to check this.\n\nSo: we translated from geometry to algebra and back to geometry, and\nwe got right back where we started. Note that each time we translated,\nour description of the map p got turned around backwards.\n\nBut there\'s a subtler and more interesting thing we can do with our\nbranched cover. We can take a point down on the w plane and look at\nthe points up on the z plane that map down to it!\n\nUsually there will be two, but for the origin there\'s just one. This\nmuch is clear from thinking geometrically. But if we think algebraically,\nwe\'ll see something funny going on at the origin. We can already see\nit geometrically: the origin is where the two sheets of our branched\ncover meet, so we call it a "branch point". But the algebraic viewpoint\nsheds an interesting new light on this.\n\nWhat we\'ll do is take a prime ideal in C[w] and push it forwards via\n\np*: C[w] -> C[z]\n\nThe resulting subset won\'t be an ideal, but it will "generate" an\nideal, meaning we can take the smallest ideal containing it. This\nideal won\'t be prime, but we can "factor" it into prime ideals: there\'s\na fairly obvious way to multiply ideals, and we happen to be working\nwith rings where there\'s a unique way to factor any ideal into prime\nideals.\n\nLet\'s try it. First pick a number x that\'s not zero. It gives a prime\nideal in C[w], namely\n\n<w - x>\n\nNext push this ideal forwards via p* and let it generate an ideal in\nC[z], namely\n\n<z^2 - x>\n\nThis is not prime, but we can factor it, which in this case simply\namounts to factoring the polynomial that generates it:\n\n<z^2 - x> = <(z - sqrt(x)) (z + sqrt(x))>\n\n= <z - sqrt(x)> <z + sqrt(x)>\n\nWe get a product of two prime ideals, corresponding to two points in\nthe z plane, namely +sqrt(x) and -sqrt(x). These are the two points\nthat map down to x.\n\nIn this sort of situation, we say the prime ideal <w - x> "splits"\ninto the prime ideals <z - sqrt(x)> and <z + sqrt(x)> when we go from\nC[w] to the extension C[z]. This is just an overeducated way of saying\nthe number x has two different square roots.\n\nBut suppose x = 0. This doesn\'t have two square roots! Everything\nworks the same except we get\n\n<z^2> = <z> <z>\n\nWe say the prime ideal <w> "ramifies" when we go from C[w] to the\nextension C[z]. We still get a product of prime ideals; they just happen\nto be the same. This is a way of making sense of the funny notion that\nthe number 0 has two square roots... which just happen to be the same!\nLots of mathematicians and physicists talk about "repeated roots" when\nan equation has "two solutions that just happen to be equal". This is\njust a way of making that precise.\n\nBut all this algebraic machinery must seem like overkill if this is the\nfirst time you\'ve seen it. It pays off when we get to more algebraic\nexamples. So, let me sketch the simplest one.\n\nLet Z be the ring of integers, and let Z[i] be the ring of Gaussian\nintegers, namely numbers of the form a+bi where a and b are integers.\nZ[i] is an algebraic extension of Z, since we can get it by throwing in\na solution z of the polynomial equation\n\nz^2 = -1\n\nThis equation is quadratic, just like it was in the example we just\ndid! Now we\'re throwing in a square root of -1 instead of a square\nroot of some function on the complex plane. But if we take the analogy\nbetween geometry and algebra seriously, this extension should still give\nsome sort of "branched double cover"\n\np: Spec(Z[i]) -> Spec(Z)\n\nWhat\'s this like?\n\nIt\'s actually really interesting, but I\'ll just *sketch* how it works.\n\nThe points of Spec(Z) are prime ideals in Z. In "week199" we saw\nthat except for the prime ideal <0>, these are generated by prime\nnumbers.\n\nSimilarly, except for <0>, the prime ideals in Z[i] are generated by\n"Gaussian primes": Gaussian numbers that have no factors except\nthemselves and the "units" 1, -1, i and -i. (A "unit" in a ring is an\nelement with a multiplicative inverse; we don\'t count units as primes.)\n\nThe map p sends each Gaussian prime to a prime, and it\'s fun to work\nout how this goes... but it\'s even more fun to work backwards! Let\'s\ntake primes in the integers and see what happens when we let them\ngenerate ideals in the Gaussian integers! This is like taking points\nin the base space of a branched cover and seeing what\'s sitting up\nabove them.\n\nFor example, the prime 5 "splits". It has two prime factors in the\nGaussian integers:\n\n5 = (2 + i)(2 - i)\n\nso in Z[i] the ideal it generates is a product of two prime ideals:\n\n<5> = <2 + i> <2 - i>\n\nThis means that two different points in Spec(Z[i]) map down to the\npoint <5> in Spec(Z), namely <2 + i> and <2 - i>. So we indeed have\nsomething like a double cover!\n\nOn the other hand, the prime 2 "ramifies". It has two prime factors in\nthe Gaussian integers:\n\n2 = (1 + i)(1 - i)\n\nbut these two Gaussian primes generate the same prime ideal:\n\n<1 + i> = <1 - i>\n\nsince if we multiply 1+i by the unit -i we get 1-i. So, in the\nGaussian integers we have\n\n<2> = <1 + i> <1 + i>\n\nA repeated factor! This is just what happened to the branch point in\nour previous example: it had "two points sitting over it, which happen\nto be the same".\n\nSo far, everything seems to be working nicely. But, besides splitting\nand ramification, there\'s a third thing that happens here, which didn\'t\nhappen in our example involving the complex plane. In fact, this third\noption never happens when we\'re doing algebraic geometry over the complex\nnumbers!\n\nHere\'s how it works. Consider the prime 3. This is still prime in the\nGaussian integers! It doesn\'t split, and it doesn\'t ramify. If we\nfactorize the ideal generated by 3 in Z[i] we just get\n\n<3> = <3>\n\nIt doesn\'t do anything - it just sits there! So, we say this prime is\n"inert".\n\nThis may seem boring, but it\'s actually mysterious - and downright\nMADDENING if we take the analogy between geometry and algebra seriously.\nIt\'s weird enough to have a "branched" cover where sheets merge at\ncertain points, but at least in that case we can *see* they\'ve merged:\na prime ideal in our subring generates an ideal in the extension that\'s\nnot prime, but is a product of several prime factors, some of which\nhappen to be the same. But when a prime ideal in our subring generates\na PRIME ideal in the extension, it\'s as if our "cover" has just ONE\nsheet over this point in the base space! And if this happens for a\nquadratic extension - as it just did - something seems to have gone\nhorribly wrong with the nice idea that "quadratic extensions are like\nbranched double covers".\n\nLuckily, this puzzle has a nice resolution. We shouldn\'t have\ndecategorified! When we started discussing "points" for a commutative\nring, we saw they form a category in a nice way: there are points of\ndifferent "flavors", with "flavor-changing operations" going between\nthem. Then we freaked out and turned this category into a set by\ndecreeing that two point are the same whenever there\'s a morphism\nbetween them. If we hadn\'t done this, we\'d have seen more clearly how\n"inert" primes fit into a nice pattern along with "split" and "ramified"\nones.\n\nI\'ll probably talk about this more sometime, and also look more carefully\nat what happens to all the different primes when we go to the Gaussian\nintegers - to show you that we are, indeed, doing number theory!\n\nBut for now, I just want to make a few comments about this idea of\npoints of different "flavors".\n\nIn fact Grothendieck proposed an even more general idea of this sort in\nhis second approach to "schemes", which is simpler but much less widely\ndiscussed than his first approach. Basically, he said that given a\ncommutative ring R, we should not only consider points that are\nhomomorphisms from R to any *field*, but also to any *commutative ring*.\nFor each commutative ring k we get a set consisting of all "k-points",\nof R, namely homomorphisms\n\nf: R -> k\n\nAnd, for each homomorphism g: k -> k\' we get a "flavor changing\noperation" that sends k-points to k\'-points. So, we get a functor\nfrom CommRing to Set! He called such a functor a "scheme". We can\nget schemes from commutative rings as just described - these are called\n"affine schemes" - but there are also others, for example those coming\nfrom projective varieties.\n\nAnyway, here are some places to read more about number theory... mostly\nwith an emphasis on the geometric viewpoint and the issue of "splitting,\nramification and inertia".\n\nFor a really quick and friendly no-nonsense introduction, try this:\n\n2) Harold M. Stark, Galois Theory, Algebraic Number Theory, and Zeta\nFunction, in From Number Theory to Physics, eds. M. Waldschmit et al,\nSpringer, Berlin, 1992, pp. 313-393.\n\nTo dig a lot deeper, try this book by Neukirch:\n\n3) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher,\nSpringer, Berlin, 1986.\n\nI already mentioned it, but it\'s worth mentioning again, because it\'s\npretty elementary, and very clear on the analogy between "function\nfields" (fields of functions on Riemann surfaces) and "number fields"\n(algebraic number fields).\n\nThis book by Borevich and Shafarevich doesn\'t make the analogy to\ngeometry explicit:\n\n4) Z. I. Borevich and I. R. Shafarevich, Number Theory, trans.\nNewcomb Greenleaf, Academic Press, New York, 1966.\n\nHowever, it has a nice concept of a "theory of divisors" for a\ncommutative ring - and if you know a bit about divisors from algebraic\ngeometry, you\'ll see that this is *secretly* very geometrical! They\nshow how to classify algebraic extensions of commutative rings using\na theory of divisors, and show how to get a theory of divisors using\n"valuations". This manages to accomplish a lot of what other texts do\nusing "adeles", without actually mentioning adeles. I find this\ninstructive.\n\nThis book goes much further in the geometric direction, but still\nwithout introducing schemes:\n\n5) Dino Lorenzini, An Invitation to Arithmetic Geometry,\nAmerican Mathematical Society, Providence, Rhode Island, 1996.\n\nIt\'s really great - very pedagogical! It develops number fields and\nfunction fields in parallel. You\'ll need to be pretty comfy with\ncommutative algebra to work all the way through it, though.\n\nIf you want to learn about schemes - not the kind I just talked about,\njust the usual sort, which still includes cool "spaces" like Spec(Z) -\ntry these:\n\n6) V. I. Danilov, V. V. Shokurov, and I. Shafarevich, Algebraic Curves,\nAlgebraic Manifolds and Schemes, Springer, Berlin, 1998.\n\n7) David Eisenbud and Joe Harris, The Geometry of Schemes, Springer,\nBerlin, 2000.\n\nSchemes have a reputation for being scary, but both these books try hard\nto make them less so, including lots of actual *pictures* of things like\nSpec(Z[i]) sitting over Spec(Z).\n\nTo wrap things up, I just want to mention two papers on subjects I\'m\nfond of....\n\nIn "week172" I discussed Tarski\'s "high school algebra problem".\nThis asks whether every identity involving 1, +, x, and exponentials\nthat holds in the positive natural numbers follows from the eleven\nwe learned in high school:\n\nx + y = y + x (x + y) + z = x + (y + z)\n\nxy = yx (xy)z = x(yz)\n\n1x = x\n\nx^1 = x 1^x = 1\n\nx(y + z) = xy + xz\n\nx^(y + z) = x^y x^z (xy)^z = x^z y^z x^(yz) = (x^y)^z\n\nThe rules of this game allow only purely equational reasoning - not\nstuff like mathematical induction. The reason is that this is secretly\na problem about "universal algebra" or "algebraic theories", as\nexplained in "week200".\n\nIt turns out the answer is NO! In fact there are infinitely many\nmore independent identities! Here is the first one, due to Wilkies:\n\n[(x + 1)^x + (x^2 + x + 1)^x]^y [(x^3 + 1)^y + (x^4 + x^2 + 1)^y]^x =\n[(x + 1)^y + (x^2 + x + 1)^y]^x [(x^3 + 1)^x + (x^4 + x^2 + 1)^x]^y\n\nI just found a paper, apparently written after "week172", which gives\na very detailed account of this problem:\n\n8) Stanley Burris and Karen Yeats, The saga of the high school\nidentities, available at\nhttp://www.thoralf.uwaterloo.ca/htdocs/MYWORKS/preprints.html\n\nIt includes some new results, like the smallest known algebraic\ngadget satisfying all the high school identities but not Wilkies\'\nidentity - but also more interesting things that are a bit harder to\ndescribe.\n\nAlso, here\'s a cool paper relating some of Ramanujan\'s work to\nstring theory:\n\n9) Antun Milas, Ramanujan\'s "Lost Notebook" and the Virasoro Algebra,\navailable as math.QA/0309201.\n\nA *lot* of Ramanujan\'s weird identities turn out to be related to concepts\nfrom string theory, suggesting that he was born about a century too soon\nto be fully appreciated... but this paper tackles an identity of his that\nnobody had managed to explain using string theory before.\n\n----------------------------------------------------------------------\n\nQuote of the week (which applies very nicely to number theory though\nit\'s about a landscape in southern Utah):\n\n"One discovery opens another, and then another. Everything in this\ncountry is nested like Russian dolls. Even a solid artifact in front\nof me drew back into other levels. Schemes within schemes." - Craig\nChilds, Soul of Nowhere\n\n-----------------------------------------------------------------------\nPrevious issues of "This Week\'s Finds" and other expository articles on\nmathematics and physics, as well as some of my research papers, can be\nobtained at\n\nhttp://math.ucr.edu/home/baez/\n\nFor a table of contents of all the issues of This Week\'s Finds, try\n\nhttp://math.ucr.edu/home/baez/twf.html\n\nA simple jumping-off point to the old issues is available at\n\nhttp://math.ucr.edu/home/baez/twfshort.html\n\nIf you just want the latest issue, go to\n\nhttp://math.ucr.edu/home/baez/this.week.html\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Also available at http://math.ucr.edu/home/baez/week205.html
April 11, 2004
This Week's Finds in Mathematical Physics - Week 205
John Baez
This week I'd like to say more about number theory, but first -
here's the most fun book on astronomy I've ever seen:
1) James B. Kaler, The Hundred Greatest Stars, Copernicus Books
(Springer Verlag), New York, 2002.
It's just what the title says: a compilation of the author's 100
favorite stars, each with a picture and a one-page description of what
makes that star interesting. They're incredibly diverse, from the
mammoth \Eta Carinae to tiny brown dwarf Gliese 229B. You'll see soft
\gamma repeaters, yellow hypergiants, pulsars, Mira-type variables,
barium stars, symbiotic stars, and more. There's also an introduction
that explains the concepts needed to enjoy all these different kinds of
stars, like the Hertzsprung-Russell diagram and a bit of nuclear physics.
With the help of this, the whole book forms a wonderful taxonomy of
stellar astrophysics. I suggest keeping it by your bed and reading one
star a night - though you may wind up staying up late and devouring the
whole book.
On to number theory....
There's a widespread impression that number theory is about *numbers*,
but I'd like to correct this, or at least supplement it. A large part
of number theory - and by the far the coolest part, in my opinion - is
about a strange sort of *geometry*. I don't understand it very well,
but that won't prevent me from taking a crack at trying to explain it....
The basic idea is to push the analogy between integers and polynomials
as far as it will go. They're similar because you can add, subtract and
multiply them, and these operations satisfy the usual rules we all
learned in high school:
x + y = y + x (x + y) + z = x + (y + z) x +[/itex] = x x + (-x) =
xy = yx (xy)z = x(yz) x1 = x
x(y + z) = xy + xz
Anything satisfying these rules is called a "commutative ring".
There are also a lot of deeper similarities between integers and
polynomials, which I'll talk about later. But, there's a big difference!
Polynomials are functions on the line, whereas the integers aren't
functions on some space - at least, not in any instantly obvious way.
The fact that polynomials are functions on a space is what lets us
graph them. This lets us think about them using *geometry* - and also
think about geometry using *them*. This was the idea behind Descartes'
"analytic geometry", and it was immensely fruitful.
So, it would be cool if we could also think about the integers using
geometry. And it turns out we can, but only if we stretch our concept
of geometry far enough!
If we do this, we'll see some cool things. First of all, we'll see
that algebra is just like geometry, only backwards.
What do I mean by this? Well, whenever you have a map T: X -> Y going
from the space X to the space Y, you can use it to take functions on Y
and turn them into functions on X. Since this goes backwards, it's
called "pulling back along T". Here's how it goes: if f is a function
on Y, we get a function T*(f) on X given by:
T*(f)(x) = f(T(x))
Moreover, functions on a space form a commutative ring, since you can
add and multiply them pointwise, and pulling back is a "homomorphism",
meaning that it preserves all the structure of a commutative ring:
T*(f + g) = T*(f) + T*(g) T*(0) =
T*(fg) = T*(f) T*(g) T*(1) = 1
Conversely, any sufficiently nice homomorphism from functions on Y to
functions on X will come from some map T: X -> Y this way! Here I'm
summarizing a whole bunch of different theorems, each of which goes
along with its own precise definition of "space", "map", and "nice".
Some of these theorems are technical, but the basic idea is simple:
we can translate back and forth between the study of commutative rings
(algebra) and the study of spaces (geometry) and by thinking of
commutative rings as consisting of functions on spaces. We get a little
dictionary for translating between geometry and algebra, like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
But be careful: this translation turns maps into homomorphisms going
backwards: it's "contravariant". This is really important in two ways.
First, suppose we have a point x in a space X. This gives a map
i: {x} -> X
This, in turn, gives a homomorphism i* sending functions on X to
functions on {x}. Functions on a one-point space are like numbers,
so i* acts like "evaluation at x". Moreover, i* will tend to be
onto: that's the backwards analogue of the fact that i is one-to-one!
Second, suppose we have a map from a space E onto the space X:
p: E -> X.
If you know some topology, think of E as a "covering space" of X. Then
we get a homomorphism p* from functions on X to functions on E. Moreover
p* will tend to be one-to-one: that's the backwards version of the fact
that p was onto!
We can use these examples to figure out the analogue of a "point" or a
"covering space" in the world of commutative rings! And the resulting
ideas turn out to be crucial to modern number theory.
In "week199" I explained the analogue of a "point" for commutative rings:
it's a "prime ideal". So, now I want to explain the analogue of a
"covering space". This will expand our dictionary so that it relates
Galois groups to fundamental groups of topological spaces... and so on.
But, we won't get too far if we don't remember why a "prime ideal" is
like a "point"! So, I guess I'd better review some of "week199" before
charging ahead into the beautiful wilderness.
What's special about the ring of functions on a space consisting of
just one point? Take real- or complex-valued functions, for example.
How do these differ from the functions on a space with lots of points?
The answer is pretty simple: on a space with just one point, a function
that vanishes anywhere vanishes everywhere! So, the only function
that fails to have a multiplicative inverse is . For bigger spaces,
this isn't true.
A commutative ring where only fails to have a multiplicative inverse
is called a "field". So, the algebraic analogue of a one-point space is
a field.
This means that the algebraic analogue of a map from a one-point
space into some other space:
i: {x} -> X
should be a homomorphism from a commutative ring R to a field k:
f: R -> k
Our translation dictionary now looks like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
It's worth noting some subtleties here. In the geometry we learned in
high school, once we see one point, we've seen 'em all: all one-point
spaces are isomorphic. But not all fields are isomorphic! So, if
we're trying to think of algebra as geometry, it's a funny sort of
geometry where points come in different flavors!
Moreover, there are homomorphisms between different fields. These
act like "flavor changing" maps - maps from a point of one flavor
to a point of some other flavor.
If we have a homomorphism f: R -> k and a homomorphism from k to some
other field k', we can compose them to get a homomorphism f': R -> k'.
So, we're doing some funny sort of geometry where if we have a point
mapped into our space, we can convert it into a point of some other
flavor, using a "flavor changing" map.
Now let's take this strange sort of geometry really seriously, and
figure out how to actually turn a commutative ring into a space!
First I'll describe what people usually do. Eventually I'll describe
what perhaps they really should do - but maybe you can guess before
I even tell you.
People usually cook up a space called the "spectrum" of the commutative
ring R, or Spec(R) for short. What are the points of Spec(R)?
They're not just all possible homomorphisms from R to all possible
fields. Instead, we count two such homomorphisms as the same
point of Spec(R) if they're related by a "flavor changing process".
In other words, f': R -> k' gives the same point as f: R -> k if you
can get f' by composing f with a homomorphism from k to k'.
This is a bit ungainly, but luckily there's a quick and easy way to tell
when f: R -> k and f': R -> k' are related by such a flavor changing
process, or a sequence of such processes. You just see if they have
the same kernel! The "kernel" of f: R -> k is the subset of R consisting
of elements r with
f(r) =
The kernel of a homomorphism to a field is a "prime ideal", and two
homomorphisms are related by a sequence of flavor changing processes
iff they have the same kernel. Furthermore, every prime ideal is
the kernel of a homomorphism to some field. So, we can save time by
defining Spec(R) to be the set of prime ideals in R.
For completeness I should remind you what a prime ideal is! An "ideal"
in a ring R is a set closed under addition and closed under multiplication
by anything in R. It's "prime" if it's not all of R, and whenever the
product of two elements of R lies in the ideal, at least one of them
lies in the ideal.
So, we have something like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
points of a space prime ideals of a commutative ring
Now let's use these ideas to study "branched covering spaces" and their
analogues in algebra. This week I'll talk about two examples. The first
is very geometrical, and it should be familiar to anyone who has studied
a little complex analysis. The second is more algebraic, and it's
important in number theory. But, the cool part is that they fit into
the same formalism!
If you don't know what a branched covering space is, don't worry:
we'll start with the very simplest example. We'll look at this map from
the complex plane to itself:
p: C -> C
p(z) = z^2
Except for zero, every complex number has two square roots, so this map
is two-to-one and onto away from the origin. In fact, away from the
origin you can visualize this thing locally as two sheets of paper
sitting above one. But these two sheets have a global complication: if
you start on the top sheet and hike once around the origin, you wind up
on the bottom sheet - and vice versa! In topology we call this sort of
thing a "double cover". When we include the point z = things get
even more complicated, since the two sheets meet there. So we have
something trickier: a "branched cover". In general, a branched cover
is like a covering space except that the different "sheets" can merge
together at certain points, called "branch points".
Now let's think about this algebraically. To keep from getting confused,
let's write
z^2 = w
so that p is a map from the "z plane" down to the "w plane", sending each
point z to the point z^2 = w. The ring of polynomial functions on the z
plane is called C[z]; the ring of polynomial functions on the w plane is
called C[w]. We can pull functions from the w plane back up to the z
plane:
p*: C[w] -> C[z]
and p* works in the obvious way, taking any function f(w) to the function
f(z^2).
Just as p is onto, p* is one-to-one! So, we can think of C[w] as
sitting inside C[z], consisting of those polynomials in z that only
depend on z^2: the even functions. We say C[w] is a "subring" of C[z],
or equivalently, that C[z] is an "extension" of C[w].
In this example we can get the bigger ring from the smaller one by
throwing in solutions of some polynomial equations, so we call it an
"algebraic extension". We've already seen some algebraic extensions,
namely algebraic number fields, where take the field of rational numbers
and throw in some solutions of polynomial equations. Algebraic extensions
can be complicated, but this one is really simple: we just start with
C[w] and throw in the solution of *one* polynomial equation, namely
z^2 = w
It turns out that quite generally, algebraic extensions of commutative
rings act a lot like branched covering spaces. I probably don't have
the technical details perfectly straight, but let's add this to our
translation dictionary, because it's an important idea:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
points of a space prime ideals of a commutative ring
branched covering spaces algebraic extensions of commutative rings
Now let's have some fun: let's see how our algebraic concept of "point",
namely "prime ideal", interacts with our branched double cover of the
complex plane. There's something straightforward going on, but also
something more subtle and interesting.
The straightforward thing is that any point up on the z plane maps to one
down on the w plane. We don't need fancy algebra to see this! But, it's
worth doing algebraically. According to the fancy algebraic definition,
a "point" in the spectrum of the commutative ring C[z] is a prime ideal.
But as you might hope, these are the same as good old-fashioned points
in the complex plane!
It works like this: given any point x in C, we get a homomorphism from
C[z] to C called "evaluation at x", which sends any polynomial f to
the number f(x). The kernel of this is the prime ideal consisting of
all polynomials that vanish at x. These are just the polynomials
containing a factor of z - x, so we call this ideal
<z - x>
So, we get some prime ideals in C[z] from points of C this way. But in
fact there's a theorem that *every* prime ideal in C[z] is of this form!
So, we get a one-to-one correspondence
Spec(C[z]) = C
Similarly,
Spec(C[w]) = C
Now let's think about our branched cover
p: C -> C
in different ways. It starts out life as a map from the z plane down
to the w plane. We can use this to pull back functions on the w
plane up to the z plane:
p*: C[w] -> C[z]
But then, by general abstract baloney, the inverse image under p* of
any prime ideal in C[z] is a prime ideal back in C[w]. This gives a
map from Spec(C[z]) to Spec(C[w]). But this is just a map from the z
plane to the w plane! And it's the same map p we started with. If you
don't see why, it's a good exercise to check this.
So: we translated from geometry to algebra and back to geometry, and
we got right back where we started. Note that each time we translated,
our description of the map p got turned around backwards.
But there's a subtler and more interesting thing we can do with our
branched cover. We can take a point down on the w plane and look at
the points up on the z plane that map down to it!
Usually there will be two, but for the origin there's just one. This
much is clear from thinking geometrically. But if we think algebraically,
we'll see something funny going on at the origin. We can already see
it geometrically: the origin is where the two sheets of our branched
cover meet, so we call it a "branch point". But the algebraic viewpoint
sheds an interesting new light on this.
What we'll do is take a prime ideal in C[w] and push it forwards via
p*: C[w] -> C[z]
The resulting subset won't be an ideal, but it will "generate" an
ideal, meaning we can take the smallest ideal containing it. This
ideal won't be prime, but we can "factor" it into prime ideals: there's
a fairly obvious way to multiply ideals, and we happen to be working
with rings where there's a unique way to factor any ideal into prime
ideals.
Let's try it. First pick a number x that's not zero. It gives a prime
ideal in C[w], namely
<w - x>
Next push this ideal forwards via p* and let it generate an ideal in
C[z], namely
<z^2 - x>
This is not prime, but we can factor it, which in this case simply
amounts to factoring the polynomial that generates it:
<z^2 - x> = <(z - \sqrt(x)) (z + \sqrt(x))>
= <z - \sqrt(x)> <z + \sqrt(x)>
We get a product of two prime ideals, corresponding to two points in
the z plane, namely +\sqrt(x) and -\sqrt(x). These are the two points
that map down to x.
In this sort of situation, we say the prime ideal <w - x> "splits"
into the prime ideals <z - \sqrt(x)> and <z + \sqrt(x)> when we go from
C[w] to the extension C[z]. This is just an overeducated way of saying
the number x has two different square roots.
But suppose x = . This doesn't have two square roots! Everything
works the same except we get
<z^2> = <z> <z>
We say the prime ideal <w> "ramifies" when we go from C[w] to the
extension C[z]. We still get a product of prime ideals; they just happen
to be the same. This is a way of making sense of the funny notion that
the number has two square roots... which just happen to be the same!
Lots of mathematicians and physicists talk about "repeated roots" when
an equation has "two solutions that just happen to be equal". This is
just a way of making that precise.
But all this algebraic machinery must seem like overkill if this is the
first time you've seen it. It pays off when we get to more algebraic
examples. So, let me sketch the simplest one.
Let Z be the ring of integers, and let Z[i] be the ring of Gaussian
integers, namely numbers of the form a+bi where a and b are integers.
Z[i] is an algebraic extension of Z, since we can get it by throwing in
a solution z of the polynomial equation
z^2 = -1
This equation is quadratic, just like it was in the example we just
did! Now we're throwing in a square root of -1 instead of a square
root of some function on the complex plane. But if we take the analogy
between geometry and algebra seriously, this extension should still give
some sort of "branched double cover"
p: Spec(Z[i]) -> Spec(Z)
What's this like?
It's actually really interesting, but I'll just *sketch* how it works.
The points of Spec(Z) are prime ideals in Z. In "week199" we saw
that except for the prime ideal <0>, these are generated by prime
numbers.
Similarly, except for <0>, the prime ideals in Z[i] are generated by
"Gaussian primes": Gaussian numbers that have no factors except
themselves and the "units" 1, -1, i and -i. (A "unit" in a ring is an
element with a multiplicative inverse; we don't count units as primes.)
The map p sends each Gaussian prime to a prime, and it's fun to work
out how this goes... but it's even more fun to work backwards! Let's
take primes in the integers and see what happens when we let them
generate ideals in the Gaussian integers! This is like taking points
in the base space of a branched cover and seeing what's sitting up
above them.
For example, the prime 5 "splits". It has two prime factors in the
Gaussian integers:
5 = (2 + i)(2 - i)
so in Z[i] the ideal it generates is a product of two prime ideals:
<5> = <2 + i> <2 - i>
This means that two different points in Spec(Z[i]) map down to the
point <5> in Spec(Z), namely <2 + i> and <2 - i>. So we indeed have
something like a double cover!
On the other hand, the prime 2 "ramifies". It has two prime factors in
the Gaussian integers:
2 = (1 + i)(1 - i)
but these two Gaussian primes generate the same prime ideal:
<1 + i> = <1 - i>
since if we multiply 1+i by the unit -i we get 1-i. So, in the
Gaussian integers we have
<2> = <1 + i> <1 + i>
A repeated factor! This is just what happened to the branch point in
our previous example: it had "two points sitting over it, which happen
to be the same".
So far, everything seems to be working nicely. But, besides splitting
and ramification, there's a third thing that happens here, which didn't
happen in our example involving the complex plane. In fact, this third
option never happens when we're doing algebraic geometry over the complex
numbers!
Here's how it works. Consider the prime 3. This is still prime in the
Gaussian integers! It doesn't split, and it doesn't ramify. If we
factorize the ideal generated by 3 in Z[i] we just get
<3> = <3>
It doesn't do anything - it just sits there! So, we say this prime is
"inert".
This may seem boring, but it's actually mysterious - and downright
MADDENING if we take the analogy between geometry and algebra seriously.
It's weird enough to have a "branched" cover where sheets merge at
certain points, but at least in that case we can *see* they've merged:
a prime ideal in our subring generates an ideal in the extension that's
not prime, but is a product of several prime factors, some of which
happen to be the same. But when a prime ideal in our subring generates
a PRIME ideal in the extension, it's as if our "cover" has just ONE
sheet over this point in the base space! And if this happens for a
quadratic extension - as it just did - something seems to have gone
horribly wrong with the nice idea that "quadratic extensions are like
branched double covers".
Luckily, this puzzle has a nice resolution. We shouldn't have
decategorified! When we started discussing "points" for a commutative
ring, we saw they form a category in a nice way: there are points of
different "flavors", with "flavor-changing operations" going between
them. Then we freaked out and turned this category into a set by
decreeing that two point are the same whenever there's a morphism
between them. If we hadn't done this, we'd have seen more clearly how
"inert" primes fit into a nice pattern along with "split" and "ramified"
ones.
I'll probably talk about this more sometime, and also look more carefully
at what happens to all the different primes when we go to the Gaussian
integers - to show you that we are, indeed, doing number theory!
But for now, I just want to make a few comments about this idea of
points of different "flavors".
In fact Grothendieck proposed an even more general idea of this sort in
his second approach to "schemes", which is simpler but much less widely
discussed than his first approach. Basically, he said that given a
commutative ring R, we should not only consider points that are
homomorphisms from R to any *field*, but also to any *commutative ring*.
For each commutative ring k we get a set consisting of all "k-points",
of R, namely homomorphisms
f: R -> k
And, for each homomorphism g: k -> k' we get a "flavor changing
operation" that sends k-points to k'-points. So, we get a functor
from CommRing to Set! He called such a functor a "scheme". We can
get schemes from commutative rings as just described - these are called
"affine schemes" - but there are also others, for example those coming
from projective varieties.
Anyway, here are some places to read more about number theory... mostly
with an emphasis on the geometric viewpoint and the issue of "splitting,
ramification and inertia".
For a really quick and friendly no-nonsense introduction, try this:
2) Harold M. Stark, Galois Theory, Algebraic Number Theory, and \Zeta
Function, in From Number Theory to Physics, eds. M. Waldschmit et al,
Springer, Berlin, 1992, pp. 313-393.
To dig a lot deeper, try this book by Neukirch:
3) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher,
Springer, Berlin, 1986.
I already mentioned it, but it's worth mentioning again, because it's
pretty elementary, and very clear on the analogy between "function
fields" (fields of functions on Riemann surfaces) and "number fields"
(algebraic number fields).
This book by Borevich and Shafarevich doesn't make the analogy to
geometry explicit:
4) Z. I. Borevich and I. R. Shafarevich, Number Theory, trans.
Newcomb Greenleaf, Academic Press, New York, 1966.
However, it has a nice concept of a "theory of divisors" for a
commutative ring - and if you know a bit about divisors from algebraic
geometry, you'll see that this is *secretly* very geometrical! They
show how to classify algebraic extensions of commutative rings using
a theory of divisors, and show how to get a theory of divisors using
"valuations". This manages to accomplish a lot of what other texts do
using "adeles", without actually mentioning adeles. I find this
instructive.
This book goes much further in the geometric direction, but still
without introducing schemes:
5) Dino Lorenzini, An Invitation to Arithmetic Geometry,
American Mathematical Society, Providence, Rhode Island, 1996.
It's really great - very pedagogical! It develops number fields and
function fields in parallel. You'll need to be pretty comfy with
commutative algebra to work all the way through it, though.
If you want to learn about schemes - not the kind I just talked about,
just the usual sort, which still includes cool "spaces" like Spec(Z) -
try these:
6) V. I. Danilov, V. V. Shokurov, and I. Shafarevich, Algebraic Curves,
Algebraic Manifolds and Schemes, Springer, Berlin, 1998.
7) David Eisenbud and Joe Harris, The Geometry of Schemes, Springer,
Berlin, 2000.
Schemes have a reputation for being scary, but both these books try hard
to make them less so, including lots of actual *pictures* of things like
Spec(Z[i]) sitting over Spec(Z).
To wrap things up, I just want to mention two papers on subjects I'm
fond of....
In "week172" I discussed Tarski's "high school algebra problem".
This asks whether every identity involving 1, +, x, and exponentials
that holds in the positive natural numbers follows from the eleven
we learned in high school:
x + y = y + x (x + y) + z = x + (y + z)
xy = yx (xy)z = x(yz)
1x = x
x^1 = x 1^x = 1
x(y + z) = xy + xz
x^(y + z) = x^y x^z (xy)^z = x^z y^z x^(yz) = (x^y)^z
The rules of this game allow only purely equational reasoning - not
stuff like mathematical induction. The reason is that this is secretly
a problem about "universal algebra" or "algebraic theories", as
explained in "week200".
It turns out the answer is NO! In fact there are infinitely many
more independent identities! Here is the first one, due to Wilkies:
[(x + 1)^x + (x^2 + x + 1)^x]^y [(x^3 + 1)^y + (x^4 + x^2 + 1)^y]^x =
[itex][(x + 1)^y + (x^2 + x + 1)^y]^x [(x^3 + 1)^x + (x^4 + x^2 + 1)^x]^y
I just found a paper, apparently written after "week172", which gives
a very detailed account of this problem:
8) Stanley Burris and Karen Yeats, The saga of the high school
identities, available at
http://www.thoralf.uwaterloo.ca/htdocs/MYWORKS/preprints.html
It includes some new results, like the smallest known algebraic
gadget satisfying all the high school identities but not Wilkies'
identity - but also more interesting things that are a bit harder to
describe.
Also, here's a cool paper relating some of Ramanujan's work to
string theory:
9) Antun Milas, Ramanujan's "Lost Notebook" and the Virasoro Algebra,
available as math.QA/0309201.
A *lot* of Ramanujan's weird identities turn out to be related to concepts
from string theory, suggesting that he was born about a century too soon
to be fully appreciated... but this paper tackles an identity of his that
nobody had managed to explain using string theory before.
----------------------------------------------------------------------
Quote of the week (which applies very nicely to number theory though
it's about a landscape in southern Utah):
"One discovery opens another, and then another. Everything in this
country is nested like Russian dolls. Even a solid artifact in front
of me drew back into other levels. Schemes within schemes." - Craig
Childs, Soul of Nowhere
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
April 11, 2004
This Week's Finds in Mathematical Physics - Week 205
John Baez
This week I'd like to say more about number theory, but first -
here's the most fun book on astronomy I've ever seen:
1) James B. Kaler, The Hundred Greatest Stars, Copernicus Books
(Springer Verlag), New York, 2002.
It's just what the title says: a compilation of the author's 100
favorite stars, each with a picture and a one-page description of what
makes that star interesting. They're incredibly diverse, from the
mammoth \Eta Carinae to tiny brown dwarf Gliese 229B. You'll see soft
\gamma repeaters, yellow hypergiants, pulsars, Mira-type variables,
barium stars, symbiotic stars, and more. There's also an introduction
that explains the concepts needed to enjoy all these different kinds of
stars, like the Hertzsprung-Russell diagram and a bit of nuclear physics.
With the help of this, the whole book forms a wonderful taxonomy of
stellar astrophysics. I suggest keeping it by your bed and reading one
star a night - though you may wind up staying up late and devouring the
whole book.
On to number theory....
There's a widespread impression that number theory is about *numbers*,
but I'd like to correct this, or at least supplement it. A large part
of number theory - and by the far the coolest part, in my opinion - is
about a strange sort of *geometry*. I don't understand it very well,
but that won't prevent me from taking a crack at trying to explain it....
The basic idea is to push the analogy between integers and polynomials
as far as it will go. They're similar because you can add, subtract and
multiply them, and these operations satisfy the usual rules we all
learned in high school:
x + y = y + x (x + y) + z = x + (y + z) x +[/itex] = x x + (-x) =
xy = yx (xy)z = x(yz) x1 = x
x(y + z) = xy + xz
Anything satisfying these rules is called a "commutative ring".
There are also a lot of deeper similarities between integers and
polynomials, which I'll talk about later. But, there's a big difference!
Polynomials are functions on the line, whereas the integers aren't
functions on some space - at least, not in any instantly obvious way.
The fact that polynomials are functions on a space is what lets us
graph them. This lets us think about them using *geometry* - and also
think about geometry using *them*. This was the idea behind Descartes'
"analytic geometry", and it was immensely fruitful.
So, it would be cool if we could also think about the integers using
geometry. And it turns out we can, but only if we stretch our concept
of geometry far enough!
If we do this, we'll see some cool things. First of all, we'll see
that algebra is just like geometry, only backwards.
What do I mean by this? Well, whenever you have a map T: X -> Y going
from the space X to the space Y, you can use it to take functions on Y
and turn them into functions on X. Since this goes backwards, it's
called "pulling back along T". Here's how it goes: if f is a function
on Y, we get a function T*(f) on X given by:
T*(f)(x) = f(T(x))
Moreover, functions on a space form a commutative ring, since you can
add and multiply them pointwise, and pulling back is a "homomorphism",
meaning that it preserves all the structure of a commutative ring:
T*(f + g) = T*(f) + T*(g) T*(0) =
T*(fg) = T*(f) T*(g) T*(1) = 1
Conversely, any sufficiently nice homomorphism from functions on Y to
functions on X will come from some map T: X -> Y this way! Here I'm
summarizing a whole bunch of different theorems, each of which goes
along with its own precise definition of "space", "map", and "nice".
Some of these theorems are technical, but the basic idea is simple:
we can translate back and forth between the study of commutative rings
(algebra) and the study of spaces (geometry) and by thinking of
commutative rings as consisting of functions on spaces. We get a little
dictionary for translating between geometry and algebra, like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
But be careful: this translation turns maps into homomorphisms going
backwards: it's "contravariant". This is really important in two ways.
First, suppose we have a point x in a space X. This gives a map
i: {x} -> X
This, in turn, gives a homomorphism i* sending functions on X to
functions on {x}. Functions on a one-point space are like numbers,
so i* acts like "evaluation at x". Moreover, i* will tend to be
onto: that's the backwards analogue of the fact that i is one-to-one!
Second, suppose we have a map from a space E onto the space X:
p: E -> X.
If you know some topology, think of E as a "covering space" of X. Then
we get a homomorphism p* from functions on X to functions on E. Moreover
p* will tend to be one-to-one: that's the backwards version of the fact
that p was onto!
We can use these examples to figure out the analogue of a "point" or a
"covering space" in the world of commutative rings! And the resulting
ideas turn out to be crucial to modern number theory.
In "week199" I explained the analogue of a "point" for commutative rings:
it's a "prime ideal". So, now I want to explain the analogue of a
"covering space". This will expand our dictionary so that it relates
Galois groups to fundamental groups of topological spaces... and so on.
But, we won't get too far if we don't remember why a "prime ideal" is
like a "point"! So, I guess I'd better review some of "week199" before
charging ahead into the beautiful wilderness.
What's special about the ring of functions on a space consisting of
just one point? Take real- or complex-valued functions, for example.
How do these differ from the functions on a space with lots of points?
The answer is pretty simple: on a space with just one point, a function
that vanishes anywhere vanishes everywhere! So, the only function
that fails to have a multiplicative inverse is . For bigger spaces,
this isn't true.
A commutative ring where only fails to have a multiplicative inverse
is called a "field". So, the algebraic analogue of a one-point space is
a field.
This means that the algebraic analogue of a map from a one-point
space into some other space:
i: {x} -> X
should be a homomorphism from a commutative ring R to a field k:
f: R -> k
Our translation dictionary now looks like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
It's worth noting some subtleties here. In the geometry we learned in
high school, once we see one point, we've seen 'em all: all one-point
spaces are isomorphic. But not all fields are isomorphic! So, if
we're trying to think of algebra as geometry, it's a funny sort of
geometry where points come in different flavors!
Moreover, there are homomorphisms between different fields. These
act like "flavor changing" maps - maps from a point of one flavor
to a point of some other flavor.
If we have a homomorphism f: R -> k and a homomorphism from k to some
other field k', we can compose them to get a homomorphism f': R -> k'.
So, we're doing some funny sort of geometry where if we have a point
mapped into our space, we can convert it into a point of some other
flavor, using a "flavor changing" map.
Now let's take this strange sort of geometry really seriously, and
figure out how to actually turn a commutative ring into a space!
First I'll describe what people usually do. Eventually I'll describe
what perhaps they really should do - but maybe you can guess before
I even tell you.
People usually cook up a space called the "spectrum" of the commutative
ring R, or Spec(R) for short. What are the points of Spec(R)?
They're not just all possible homomorphisms from R to all possible
fields. Instead, we count two such homomorphisms as the same
point of Spec(R) if they're related by a "flavor changing process".
In other words, f': R -> k' gives the same point as f: R -> k if you
can get f' by composing f with a homomorphism from k to k'.
This is a bit ungainly, but luckily there's a quick and easy way to tell
when f: R -> k and f': R -> k' are related by such a flavor changing
process, or a sequence of such processes. You just see if they have
the same kernel! The "kernel" of f: R -> k is the subset of R consisting
of elements r with
f(r) =
The kernel of a homomorphism to a field is a "prime ideal", and two
homomorphisms are related by a sequence of flavor changing processes
iff they have the same kernel. Furthermore, every prime ideal is
the kernel of a homomorphism to some field. So, we can save time by
defining Spec(R) to be the set of prime ideals in R.
For completeness I should remind you what a prime ideal is! An "ideal"
in a ring R is a set closed under addition and closed under multiplication
by anything in R. It's "prime" if it's not all of R, and whenever the
product of two elements of R lies in the ideal, at least one of them
lies in the ideal.
So, we have something like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
points of a space prime ideals of a commutative ring
Now let's use these ideas to study "branched covering spaces" and their
analogues in algebra. This week I'll talk about two examples. The first
is very geometrical, and it should be familiar to anyone who has studied
a little complex analysis. The second is more algebraic, and it's
important in number theory. But, the cool part is that they fit into
the same formalism!
If you don't know what a branched covering space is, don't worry:
we'll start with the very simplest example. We'll look at this map from
the complex plane to itself:
p: C -> C
p(z) = z^2
Except for zero, every complex number has two square roots, so this map
is two-to-one and onto away from the origin. In fact, away from the
origin you can visualize this thing locally as two sheets of paper
sitting above one. But these two sheets have a global complication: if
you start on the top sheet and hike once around the origin, you wind up
on the bottom sheet - and vice versa! In topology we call this sort of
thing a "double cover". When we include the point z = things get
even more complicated, since the two sheets meet there. So we have
something trickier: a "branched cover". In general, a branched cover
is like a covering space except that the different "sheets" can merge
together at certain points, called "branch points".
Now let's think about this algebraically. To keep from getting confused,
let's write
z^2 = w
so that p is a map from the "z plane" down to the "w plane", sending each
point z to the point z^2 = w. The ring of polynomial functions on the z
plane is called C[z]; the ring of polynomial functions on the w plane is
called C[w]. We can pull functions from the w plane back up to the z
plane:
p*: C[w] -> C[z]
and p* works in the obvious way, taking any function f(w) to the function
f(z^2).
Just as p is onto, p* is one-to-one! So, we can think of C[w] as
sitting inside C[z], consisting of those polynomials in z that only
depend on z^2: the even functions. We say C[w] is a "subring" of C[z],
or equivalently, that C[z] is an "extension" of C[w].
In this example we can get the bigger ring from the smaller one by
throwing in solutions of some polynomial equations, so we call it an
"algebraic extension". We've already seen some algebraic extensions,
namely algebraic number fields, where take the field of rational numbers
and throw in some solutions of polynomial equations. Algebraic extensions
can be complicated, but this one is really simple: we just start with
C[w] and throw in the solution of *one* polynomial equation, namely
z^2 = w
It turns out that quite generally, algebraic extensions of commutative
rings act a lot like branched covering spaces. I probably don't have
the technical details perfectly straight, but let's add this to our
translation dictionary, because it's an important idea:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
points of a space prime ideals of a commutative ring
branched covering spaces algebraic extensions of commutative rings
Now let's have some fun: let's see how our algebraic concept of "point",
namely "prime ideal", interacts with our branched double cover of the
complex plane. There's something straightforward going on, but also
something more subtle and interesting.
The straightforward thing is that any point up on the z plane maps to one
down on the w plane. We don't need fancy algebra to see this! But, it's
worth doing algebraically. According to the fancy algebraic definition,
a "point" in the spectrum of the commutative ring C[z] is a prime ideal.
But as you might hope, these are the same as good old-fashioned points
in the complex plane!
It works like this: given any point x in C, we get a homomorphism from
C[z] to C called "evaluation at x", which sends any polynomial f to
the number f(x). The kernel of this is the prime ideal consisting of
all polynomials that vanish at x. These are just the polynomials
containing a factor of z - x, so we call this ideal
<z - x>
So, we get some prime ideals in C[z] from points of C this way. But in
fact there's a theorem that *every* prime ideal in C[z] is of this form!
So, we get a one-to-one correspondence
Spec(C[z]) = C
Similarly,
Spec(C[w]) = C
Now let's think about our branched cover
p: C -> C
in different ways. It starts out life as a map from the z plane down
to the w plane. We can use this to pull back functions on the w
plane up to the z plane:
p*: C[w] -> C[z]
But then, by general abstract baloney, the inverse image under p* of
any prime ideal in C[z] is a prime ideal back in C[w]. This gives a
map from Spec(C[z]) to Spec(C[w]). But this is just a map from the z
plane to the w plane! And it's the same map p we started with. If you
don't see why, it's a good exercise to check this.
So: we translated from geometry to algebra and back to geometry, and
we got right back where we started. Note that each time we translated,
our description of the map p got turned around backwards.
But there's a subtler and more interesting thing we can do with our
branched cover. We can take a point down on the w plane and look at
the points up on the z plane that map down to it!
Usually there will be two, but for the origin there's just one. This
much is clear from thinking geometrically. But if we think algebraically,
we'll see something funny going on at the origin. We can already see
it geometrically: the origin is where the two sheets of our branched
cover meet, so we call it a "branch point". But the algebraic viewpoint
sheds an interesting new light on this.
What we'll do is take a prime ideal in C[w] and push it forwards via
p*: C[w] -> C[z]
The resulting subset won't be an ideal, but it will "generate" an
ideal, meaning we can take the smallest ideal containing it. This
ideal won't be prime, but we can "factor" it into prime ideals: there's
a fairly obvious way to multiply ideals, and we happen to be working
with rings where there's a unique way to factor any ideal into prime
ideals.
Let's try it. First pick a number x that's not zero. It gives a prime
ideal in C[w], namely
<w - x>
Next push this ideal forwards via p* and let it generate an ideal in
C[z], namely
<z^2 - x>
This is not prime, but we can factor it, which in this case simply
amounts to factoring the polynomial that generates it:
<z^2 - x> = <(z - \sqrt(x)) (z + \sqrt(x))>
= <z - \sqrt(x)> <z + \sqrt(x)>
We get a product of two prime ideals, corresponding to two points in
the z plane, namely +\sqrt(x) and -\sqrt(x). These are the two points
that map down to x.
In this sort of situation, we say the prime ideal <w - x> "splits"
into the prime ideals <z - \sqrt(x)> and <z + \sqrt(x)> when we go from
C[w] to the extension C[z]. This is just an overeducated way of saying
the number x has two different square roots.
But suppose x = . This doesn't have two square roots! Everything
works the same except we get
<z^2> = <z> <z>
We say the prime ideal <w> "ramifies" when we go from C[w] to the
extension C[z]. We still get a product of prime ideals; they just happen
to be the same. This is a way of making sense of the funny notion that
the number has two square roots... which just happen to be the same!
Lots of mathematicians and physicists talk about "repeated roots" when
an equation has "two solutions that just happen to be equal". This is
just a way of making that precise.
But all this algebraic machinery must seem like overkill if this is the
first time you've seen it. It pays off when we get to more algebraic
examples. So, let me sketch the simplest one.
Let Z be the ring of integers, and let Z[i] be the ring of Gaussian
integers, namely numbers of the form a+bi where a and b are integers.
Z[i] is an algebraic extension of Z, since we can get it by throwing in
a solution z of the polynomial equation
z^2 = -1
This equation is quadratic, just like it was in the example we just
did! Now we're throwing in a square root of -1 instead of a square
root of some function on the complex plane. But if we take the analogy
between geometry and algebra seriously, this extension should still give
some sort of "branched double cover"
p: Spec(Z[i]) -> Spec(Z)
What's this like?
It's actually really interesting, but I'll just *sketch* how it works.
The points of Spec(Z) are prime ideals in Z. In "week199" we saw
that except for the prime ideal <0>, these are generated by prime
numbers.
Similarly, except for <0>, the prime ideals in Z[i] are generated by
"Gaussian primes": Gaussian numbers that have no factors except
themselves and the "units" 1, -1, i and -i. (A "unit" in a ring is an
element with a multiplicative inverse; we don't count units as primes.)
The map p sends each Gaussian prime to a prime, and it's fun to work
out how this goes... but it's even more fun to work backwards! Let's
take primes in the integers and see what happens when we let them
generate ideals in the Gaussian integers! This is like taking points
in the base space of a branched cover and seeing what's sitting up
above them.
For example, the prime 5 "splits". It has two prime factors in the
Gaussian integers:
5 = (2 + i)(2 - i)
so in Z[i] the ideal it generates is a product of two prime ideals:
<5> = <2 + i> <2 - i>
This means that two different points in Spec(Z[i]) map down to the
point <5> in Spec(Z), namely <2 + i> and <2 - i>. So we indeed have
something like a double cover!
On the other hand, the prime 2 "ramifies". It has two prime factors in
the Gaussian integers:
2 = (1 + i)(1 - i)
but these two Gaussian primes generate the same prime ideal:
<1 + i> = <1 - i>
since if we multiply 1+i by the unit -i we get 1-i. So, in the
Gaussian integers we have
<2> = <1 + i> <1 + i>
A repeated factor! This is just what happened to the branch point in
our previous example: it had "two points sitting over it, which happen
to be the same".
So far, everything seems to be working nicely. But, besides splitting
and ramification, there's a third thing that happens here, which didn't
happen in our example involving the complex plane. In fact, this third
option never happens when we're doing algebraic geometry over the complex
numbers!
Here's how it works. Consider the prime 3. This is still prime in the
Gaussian integers! It doesn't split, and it doesn't ramify. If we
factorize the ideal generated by 3 in Z[i] we just get
<3> = <3>
It doesn't do anything - it just sits there! So, we say this prime is
"inert".
This may seem boring, but it's actually mysterious - and downright
MADDENING if we take the analogy between geometry and algebra seriously.
It's weird enough to have a "branched" cover where sheets merge at
certain points, but at least in that case we can *see* they've merged:
a prime ideal in our subring generates an ideal in the extension that's
not prime, but is a product of several prime factors, some of which
happen to be the same. But when a prime ideal in our subring generates
a PRIME ideal in the extension, it's as if our "cover" has just ONE
sheet over this point in the base space! And if this happens for a
quadratic extension - as it just did - something seems to have gone
horribly wrong with the nice idea that "quadratic extensions are like
branched double covers".
Luckily, this puzzle has a nice resolution. We shouldn't have
decategorified! When we started discussing "points" for a commutative
ring, we saw they form a category in a nice way: there are points of
different "flavors", with "flavor-changing operations" going between
them. Then we freaked out and turned this category into a set by
decreeing that two point are the same whenever there's a morphism
between them. If we hadn't done this, we'd have seen more clearly how
"inert" primes fit into a nice pattern along with "split" and "ramified"
ones.
I'll probably talk about this more sometime, and also look more carefully
at what happens to all the different primes when we go to the Gaussian
integers - to show you that we are, indeed, doing number theory!
But for now, I just want to make a few comments about this idea of
points of different "flavors".
In fact Grothendieck proposed an even more general idea of this sort in
his second approach to "schemes", which is simpler but much less widely
discussed than his first approach. Basically, he said that given a
commutative ring R, we should not only consider points that are
homomorphisms from R to any *field*, but also to any *commutative ring*.
For each commutative ring k we get a set consisting of all "k-points",
of R, namely homomorphisms
f: R -> k
And, for each homomorphism g: k -> k' we get a "flavor changing
operation" that sends k-points to k'-points. So, we get a functor
from CommRing to Set! He called such a functor a "scheme". We can
get schemes from commutative rings as just described - these are called
"affine schemes" - but there are also others, for example those coming
from projective varieties.
Anyway, here are some places to read more about number theory... mostly
with an emphasis on the geometric viewpoint and the issue of "splitting,
ramification and inertia".
For a really quick and friendly no-nonsense introduction, try this:
2) Harold M. Stark, Galois Theory, Algebraic Number Theory, and \Zeta
Function, in From Number Theory to Physics, eds. M. Waldschmit et al,
Springer, Berlin, 1992, pp. 313-393.
To dig a lot deeper, try this book by Neukirch:
3) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher,
Springer, Berlin, 1986.
I already mentioned it, but it's worth mentioning again, because it's
pretty elementary, and very clear on the analogy between "function
fields" (fields of functions on Riemann surfaces) and "number fields"
(algebraic number fields).
This book by Borevich and Shafarevich doesn't make the analogy to
geometry explicit:
4) Z. I. Borevich and I. R. Shafarevich, Number Theory, trans.
Newcomb Greenleaf, Academic Press, New York, 1966.
However, it has a nice concept of a "theory of divisors" for a
commutative ring - and if you know a bit about divisors from algebraic
geometry, you'll see that this is *secretly* very geometrical! They
show how to classify algebraic extensions of commutative rings using
a theory of divisors, and show how to get a theory of divisors using
"valuations". This manages to accomplish a lot of what other texts do
using "adeles", without actually mentioning adeles. I find this
instructive.
This book goes much further in the geometric direction, but still
without introducing schemes:
5) Dino Lorenzini, An Invitation to Arithmetic Geometry,
American Mathematical Society, Providence, Rhode Island, 1996.
It's really great - very pedagogical! It develops number fields and
function fields in parallel. You'll need to be pretty comfy with
commutative algebra to work all the way through it, though.
If you want to learn about schemes - not the kind I just talked about,
just the usual sort, which still includes cool "spaces" like Spec(Z) -
try these:
6) V. I. Danilov, V. V. Shokurov, and I. Shafarevich, Algebraic Curves,
Algebraic Manifolds and Schemes, Springer, Berlin, 1998.
7) David Eisenbud and Joe Harris, The Geometry of Schemes, Springer,
Berlin, 2000.
Schemes have a reputation for being scary, but both these books try hard
to make them less so, including lots of actual *pictures* of things like
Spec(Z[i]) sitting over Spec(Z).
To wrap things up, I just want to mention two papers on subjects I'm
fond of....
In "week172" I discussed Tarski's "high school algebra problem".
This asks whether every identity involving 1, +, x, and exponentials
that holds in the positive natural numbers follows from the eleven
we learned in high school:
x + y = y + x (x + y) + z = x + (y + z)
xy = yx (xy)z = x(yz)
1x = x
x^1 = x 1^x = 1
x(y + z) = xy + xz
x^(y + z) = x^y x^z (xy)^z = x^z y^z x^(yz) = (x^y)^z
The rules of this game allow only purely equational reasoning - not
stuff like mathematical induction. The reason is that this is secretly
a problem about "universal algebra" or "algebraic theories", as
explained in "week200".
It turns out the answer is NO! In fact there are infinitely many
more independent identities! Here is the first one, due to Wilkies:
[(x + 1)^x + (x^2 + x + 1)^x]^y [(x^3 + 1)^y + (x^4 + x^2 + 1)^y]^x =
[itex][(x + 1)^y + (x^2 + x + 1)^y]^x [(x^3 + 1)^x + (x^4 + x^2 + 1)^x]^y
I just found a paper, apparently written after "week172", which gives
a very detailed account of this problem:
8) Stanley Burris and Karen Yeats, The saga of the high school
identities, available at
http://www.thoralf.uwaterloo.ca/htdocs/MYWORKS/preprints.html
It includes some new results, like the smallest known algebraic
gadget satisfying all the high school identities but not Wilkies'
identity - but also more interesting things that are a bit harder to
describe.
Also, here's a cool paper relating some of Ramanujan's work to
string theory:
9) Antun Milas, Ramanujan's "Lost Notebook" and the Virasoro Algebra,
available as math.QA/0309201.
A *lot* of Ramanujan's weird identities turn out to be related to concepts
from string theory, suggesting that he was born about a century too soon
to be fully appreciated... but this paper tackles an identity of his that
nobody had managed to explain using string theory before.
----------------------------------------------------------------------
Quote of the week (which applies very nicely to number theory though
it's about a landscape in southern Utah):
"One discovery opens another, and then another. Everything in this
country is nested like Russian dolls. Even a solid artifact in front
of me drew back into other levels. Schemes within schemes." - Craig
Childs, Soul of Nowhere
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
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For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html