Favorite definitions, theorems, proofs, etc.?

[honestrosewater]

After seeing infinite sets defined negatively, I liked seeing them defined as sets that are equivalent to one of their proper subsets. I always thought diagonal argument[/url] was cool.
Do you have a favorite definition, theorem, proof, bit of knowledge you found especially insightful or useful... ?

 

[matt grime]

*cough* that'd be Dedekind infinite, which is not, strictly speaking, the same as infinite.

Surely the best results in mathematics are those that allow us to talk about continuity without reference to epsilon and delta: pointset topology

 

[honestrosewater]

*cough* that'd be Dedekind infinite, which is not, strictly speaking, the same as infinite.

What, it said... eh, well, now I like it even more.

 

[matt grime]

They are equivalent if we have the axiom of choice at hand, but otherwise not necessarily so, though I know of no examples of this. However that is not at all surprising given I know next to nothing about set theory.

 

[honestrosewater]

Yeah, that's what it said.
I also find the cut theorem, I don't know, suspiciously obvious. Here's one statement from my favorite UKish logician Wilfrid Hodges: ("l=" means "semantically entails"- i.e. decision not proof)
If X is a finite set of formulae and p and q are formulae, then if (X l= p) and (X, p l= q), then (X l= q).
So p is unnecessary and cut out, but it makes me nervous and I have to remind myself of the proof.

So now I'm curious- how do you transform a doughnut into a teacup? Edit: Is it very complicated?

 

[Haelfix]

The proof of the Atiyah Singer index theorem is my personal favorite. A Beautiful, elegant, completely nontrivial result that is expressed simply in one line, and is perhaps the most powerful concept in the latter half of the twentieth century bringing together several fields of mathematics and physics.

 

[matt grime]

Donut to teacup? That's easy to visualize. Imagine the donut is made of plasticine, then just push your thumb into its surface - that creates a hollow. Surely from there you can see how to continue to make the hollow the inside of the cup and the hole the handle? Of course a formula is going to be horrendous.

 

[honestrosewater]

Donut to teacup? That's easy to visualize. Imagine the donut is made of plasticine, then just push your thumb into its surface - that creates a hollow. Surely from there you can see how to continue to make the hollow the inside of the cup and the hole the handle? Of course a formula is going to be horrendous.

Oh, okay, I wasn't sure how you were allowed to change it, but I looked it up. You just aren't allowed to tear it. Because then two points that were arbitrarily close to each other aren't.?

 

[matt grime]

Yep, deformations must be continuous, ie no ripping.

 

[mathwonk]

haelfix, are you saying the several hundred page proof of Atiyah Singer, as in R.T. Seeley et.al., is your model of elegance? or the statement? (or atiyah's 2 - line summary)

this proof is presumably "just" a generalization of hirzebruch's argument for his version of the riemann roch theorem.

if it is indeed the proof you find elegant, would you be willing to summarize the argument for us?

as i recall these proofs are slick but rather unenlightening, establishing axioms for the index then checking the axioms, a la the earlier innovative argument by Washnitzer.

And what do you think of the heat equation approach of patodi? or was that the elegant one?

You are certainly in good company selecting this theorem, as Atiyah - Singer just won the Abel prize for it, as I am sure you know.


i like lefschetz's proof there cannot be a vector field on the sphere with no zeroes as follows; choose any point where there is a non zero vector and visualize a small circle around that point where the vectors are all going roughly the same direction. Then visualize those vectors emanating from the boundary circle, qua the boundary of the complement of that disc. You will eventually be able to see that they wind twice around the boundary of the complementary disc, hence there must be, counting with multiplicities, two zeroes outside that disc.

i also like Newton's proof that every monotone function is "riemann" integrable, as found e.g. in comenetz' or apostol's calculus books, (or Newton).

i also like baby versions of atiyah - singer, like riemann - roch (the original version by riemann and roch), as well as later incarnations by washnitzer, hirzebruch, fulton. i am still trying to absorb even the statement of grothendieck's version.

there is an english translation of riemann's works coming out which should still be interesting.

 

[mathwonk]

in the spirit of axiomatic arguments for "index theorems" here is a little such proof for hirzebruch riemann roch for plane curves.

the problem is to compute the dimension L(D) of the space of meromorphic functions on the curve, having pole divisor supported in the divisor D. The difficulty is that this is an analytic and not a topological invariant, hence cannot be computed by the usual method of degeneration to easier cases.

Hence the whole idea is to replace it by a related topological invariant, and compute that instead. that is called the hirzebruch riemann roch theorem HRR.

By sheaf theory one equates the number L(D) with the dimension of a "zeroth" cohomology group, and then because a curve has dimension one, there is another more mysterious first cohomology group whose dimension is i(D). then the "index" of D, chi(D) = L(D)-i(D). it is this which we propsoe to compute topologically.

step 1): show the difference chi(D) -chi(O), where O is the zero divisor, = degree(D) = number of points in the divisor D, surely a topological invariant. this is a trivial sheaf theoretic exact sequences count.

step 2) show chi(O) is a topolopgical invariant. by more trivial exact sequences [i.e. fund thm of linear algebra: dim source = dim image + dim kernel] one shows two things:

i) chi(O) = chi(X) is a linear deformation invariant, i.e. chi(X) depends only on the degree of the plane curve X.

ii) chi(XunionY) = chi(X) + chi(Y) - number of intersection points of X,Y.


Since any plane curve can be deformed linearly to a union of two plane curves of lower degree, so we can use induction to compute chi(O). And since a conic is not only isomorphic to a line, but also deforms to a union of two lines, meeting at one point, we have chi(line) = chi(line)+chi(line)-1, so chi(line) = 1, which starts the induction.

[the proof of i) is by a sort of rouche's principle, i.e. one computes the chi by intersection or "integrating" an object defined on the whole plane, over the curve, and this operation is invariant under deforming the curve.]

now we already have that chi(D) = chi(O) + deg(D) is a toplogical invariant, and to compute it, we finish by showing that chi(O) = 1-g, where g is the topologicsal genus.

To do that we only have to establish the same properties i) and ii) which hold for chi(O), also for the arithmetic genus 1-g.

One can show, again by degeneration, that the topological genus of a plane curve satisfies g = (1/2)(d-1)(d-2) where d is the degree. [idea: degenerate to a union of d lines and note that there are exactly this number of holes in the resulting figure, e.g. a triangle has one hole, so g = 1].

Hence i) above holds, and then the formula for 1-g can be shown by high school level arithmetic to satisfy relation ii) as well. (use induction.)

This proves HRR for plane curves.

To deduce the full RRT one must identify the term i(D). This last step exceeds what is provided by Atiyah - Hirzebruch or any index theorem, and requires a "vanishing result" or a "duality" result. Namely (Roch) i(D) = L(K-D) where K is the divisor of a differential form. Hence (Riemann) i(D) = 0, if deg(D) > deg(K) = 2g-2, so in that case L(D) = deg(D) + 1-g, is itself topological.

I love this topic. but notice all the proofs above are so slick they proceed without the need of understanding anything! as a student you may like this, but as a researcher it sems to make life harder, at least for pedestrians like me.

Notice though, if you hope to prove a theorem in "all" dimensions, or some other lofty setting, you may need a proof technique that is so formal, that you do not need to process mentally all the grubby details, i.e. one that does not require you to understand everything, but allows you to free yourself up to just calculate. Ironic, eh?

 

[mathwonk]

does this help anyone? if you always wanted to know how to prove the famous riemann roch theorem, or even to know what it says, i ask you to try to read the previous post. unlikely as it may seem, this was an actual attempt to render an esoteric subject accessible to more people.

 

[fourier jr]

Surely the best results in mathematics are those that allow us to talk about continuity without reference to epsilon and delta: pointset topology

yes it's always nice when that can be done. i also like the definition of a group. it seems to have just enough axioms to be useful. you'd be giving up a lot if anyone of the axioms were thrown away. i also like the pigeonhole principle (or ramsey's theorem).

 

[mathwonk]

continuity is amazing, like proving an equation f(x) = 0 must have a solution just beacuse there is a point a such that f(a) < 0 and a point b where f(b) > 0, and f is continuos on [a,b].

the higher dimensional versions of this are even more amazing, and these use stokes theorem, so i like that, and its use in differential topology in general.

like proving brouwers fixed point theorem, or the fundamental theorem of algebra, or that no smooth tangent vector fields on the sphere can be all non zero.

this type of topological argument by deforming to a more doable case, is what lies also behind the HRR argument above, and the Atiyah Singer index theorem. It goes on and on.


I like deformation theory also in algebraic geometry, the theory of moduli, wherein one viosualizes a whole family of geometry surfaces or curves as just points on another surface. i.e. one speaks of what it emans for a sequence of surfaces to converge to another one, and then tracks what happens to various invariants of the surface, like the homology.

One of the greatest inovations was the infinitesimal deformation theory of spencer and kodaira, developed by grothendieck, wherein one can define a computable tangent space to such a family of surfaces. it is always a cohomology group.

it turns out for example then tanegnt space to the family of all curves of genus g, at the curve X, is dual to the space of quadratic differentials on X! By RRT this has dimension 3g-3, (when g > 1), so the moduli space of all curves has that dimension, as Riemann knew by other methods.

 

[mathwonk]

many people like the little arguemnt referred to as "steinitz exchange lemma" for proving the cardinality of all maximal independent collections of vectors is the same, in a finite dimenbsional vector space.

I have just found that argument in riemanns famous paper on abelian functions, for the case of a maximal collection of non bounding curves on a given surface, 14 years before steinitz' birth.

at the beginning of that same paper riemann "puts off until a future time, a treatment of [topology] in a manner entirely independent of measurement", which apparently never appeared, perhaps until hausdorff's treatment roughly 50 years later.

 

[mathwonk]

the sheaf theoretic proof of the de rham theorem is rather amazing, i.e. sheaf theory plus poincares lemma. as in spivaks diff geom book vol 1.

 

[The Rev]

You people make me feel like a prole for saying this, but I'm rather fond of the good ole Pythagorean Theorem.

$$\phi$$

The Rev

 

[mathwonk]

i also like it. and the non linear version in harold jacobs book: i.e. imagine 2 - dimensional cutouts of the head of pythagoras, erected on each side of a right triangle. Then the sum of the areas of the two heads erected on the two legs of the triangle, equals the area of the head erected on the hypotenuse.


I.e. pythagoras says that if you erect squares on each side of a right triangle, then the sum of the areas of the two squares on the two legs, equals the area of the square on the hypotenuse, but it has nothing to do with the figures being squares.

i also like euler's (generalized) formula V-E+F = 2-2g, where g is the genus of the polyhedron, in fact that may be my favorite theorem of all time.

in fact it is closely related to andre weil's sheaf theoretic proof of the de rham theorem. i.e. eulers formula says that if you decompose a polyhedron into pieces, namely the faces, that have no geometry themselves, i.e. they are convex, then the geometry of the whole polyhedron (its genus) is determined by the way the faces fit together, i.e. how many edges and so on there are, in comparison to how many faces and vertices.

this is what weil's proof of de rham says: i.e. if you cover a manifold by locally trivial, e.g. convex open sets, where the differential equation dW = 0 always implies that W = dU, for some U, then the de rham cohomology of the manifold is entirely determined by the topological cech cohomology of the open cover.

I.e. the idea is that to understand the global structure of something, anything, you decompose it into pieces such that each piece has no interesting geometry. then the global structure is determined by how those trivial pieces fit together.

 

[mathwonk]

another theorem i like is bezout and its refinements. i.e. we know a polynomial in one variable of degree d has at most d roots, and if we count complex roots and multiple roots proeprly, it has "exactly" d roots.

then bezout asks how many common solutions two equations f = 0, g = 0 of two variables have? If we count complex solutions, and solutioons at infinity, and count multiple intersections of the two curves proeprly, then they have deg(f).deg(g) common roots.


but sometimes in higer dimensions mkroe inhtersting things happen: i.e. two surfaces in three space have usually a curve of intersection, but sometimes, three surfaces do too. Say the intersection of two surfaces breaks up into a union of two separate curves. Then pass a third surface so it completely contains oine of the curves but not the other. Then it meets the other in a finite number of points.

so the interscetion of the three surfaces is one curve plus a finite number of points. What can you say then about the number of points?


This arises in a classical problem called the rpoblem of appolonius. i.e. given 5 conics in the complex plane, how many conics are tangent everywhere they meet to all 5? The equation for a conic represents a point in the projective 5 space of all conics. And the set of those conics tangent to a given conic is a hypersurface in the space of all conics. Hence 5 such hypersurfaces, one for each given conic, should meet in a finite number of points, the set of conics meeting all 5 given conics tangentially.

but notice that any double line meets everything it meets "tangentially", i.e. doubly. so there is a whole surface of extraneous solutions to our problem of finding conics tangent to 5 given ones, the surface of double lines, one for each line in the plane. so how many actual conics meet 5 given ones tangentially?

the correct solution was given by chasles in the 19th century and possibly helped give rise to schuberts calculus of intersections and the refinement of counting "excess" intersections, as develoepd in the last century by Fulton, MacPherson, and others, see Fulton's Intersection theory.

 

[mathwonk]

i also always liked the theorem that a triangle inscribed in a semi circle is a right triangle.

and i love the refinement of pythagoras, the "law of cosines".

i.e. a^2 + b^2 = c^2 - 2ab cos(t).

this shows that the squared length of a side of a triangle is expressed as a sort of square: c^2 = a^2 + b^2 - 2ab cos(t) = (A-B).(A-B) where A.B = B.A = ab cos(t), and A,B are the direction vectors rerpesenting the two sides of the triangle with lengths a,b.

this ointroduces algebra into geometry in a very powerful way. too bad no one explained it to me this way in high school, where i stupidly thought the law of cosines just a useless pain in the neck to memorize.

 

[fourier jr]

every compact metric space is the continuous image of a Cantor set
wtf
Urysohn & Alexandroff proved it. that blew my mind the first time i saw it

or how about the Stone-Weierstrass theorem, a special case of which is the fact that every real-valued continuous function on a closed interval can be uniformly approximated by polynomials

 

[mathwonk]

there is also a nice extension of stone weierstrass:

i.e. stone weierstrass characterizes uniformly dense subalgebras of the algebra of continuous functions on a compact T2 space, as constant - containing subalgebras that separate points, and moreover the points of the space are in one - one correspondence with the maximal ideas of the algebra.

this leads to a notion of "compacitifications" of non compact T2 spaces as corresponding to such subalgebras of the algebra of continuous functions. for example the subalgebra of those functions ahving limits at infinity, in the sense that there is some L such that for every epsilon, there is a compact set off which f-L is less than epsilon, leads to the one point or smallest compactifiacation.

then the stone - cech or largest compacitification corresponds to the maximal ideal space of the full algebra.

it is fun to figure out which subalgebra of functions leads to the closed disc compactification of the open disc, e.g.

for instance the open interval is compactified to the circle by functions that have the same limit at both ends, and to the closed interval by functions that have (possibly different) limits at both ends.

 

[mathwonk]

the riemann singularites theorem is rather nice as well: , i.e. riemann constructed for each complex curve of genus g, a compact complex torus of dimension g, "the picard variety", which parametrizes the divisors on the curve of given degree d.

then he addressed the problem of computing, for each non negative divisor D, the dimension of the space of meromorphic functions with poles supported in D.


The famous Riemann roch theorem says this is deg(D) + 1 -g + i(D) where i(D) is the dimension of the space of holomorphic differentials vanishing on D, but this is tricky to compute.

For d = g-1, the space of effective divisors is a hypersurface of dimension g-1 in the picard variety, and riemann proived that the multiplicity of a point on this divisor actually equals exactly the dimension of the corresponding space of meromorphic functions. I.e. at a smooth point of this divisor, the generic case, the multiplicity is one, and the space of functions has only one independent meromorphic function.

at a double point, there are two,. etc... this famous divisor is called the
theta divisor" since the equation riemann gave for it generalizes jacobi's theta function for an elliptic curve. this function is also the basic solution of the heat equation, and has many applications to moduli of curves.

 

[Hurkyl]

Curious you should mention that...


When I first really introduced myself to algebraic geometry (I'm still very much a beginner, of course), I asked about the prime spectrum of the ring of continuous real functions on some nice space (such as a Euclidean space). Once I learned why that was too ugly (the ideal of functions zero on P has proper prime subideals), I thought about the maximal spectrum, and realized it was nice for a compact space, but less nice for other things.


Then, I just started thinking about it again today. IIRC, the maximal ideals are in 1-1 correspondence with maximal filters on the closed sets.

(For example, we might choose a point P in X, then an example of a maximal filter is saying that a closed set is "big" iff it contains P. Another example is to say a closed set is "big" iff its complement is bounded)

So, I know which points in the maximal spectrum correspond to the boundary points of your disk! But my mind is in the wrong gear and I can't figure out the subalgebra this way. I blame TV.


edit: Ack, I lied -- I'm looking for a quotient space of the compactification, not a subspace! All right, I am once again properly horrified by this compactification.

 

[fourier jr]

there is also a nice extension of stone weierstrass:

i.e. stone weierstrass characterizes uniformly dense subalgebras of the algebra of continuous functions on a compact T2 space, as constant - containing subalgebras that separate points, and moreover the points of the space are in one - one correspondence with the maximal ideas of the algebra.

yeah i think that's the version I have in one of my topology books. it says let X be a compact Hausdorff space. if D is the collection of functions in C*(X) which separates points in X and contains the function identically 1, the uniformly closed subalgebra generated by D is all of C*(X). it's a bit over my head but some of that stuff i get. isn't gillman/jerison's book "rings of continuous functions" one of the best books there is on that stuff?

 

[mathwonk]

yes that is a standard, but you are a mature math student now, so start trying to prove this stuff yourself. it will become easier as you go along.

 

[fourier jr]

yes that is a standard, but you are a mature math student now, so start trying to prove this stuff yourself. it will become easier as you go along.

i've kind of been trying to do that but it's hard to decide how to go about it. I'm not really sure if i should go through a whole book completely & onto the next one or work on a couple simultaneously since (general/point-set) topology and analysis are so interdependent. proving everything by myself also sounds a bit daunting but it would be cool to know the stuff well enough to do that.

 

[gravenewworld]

president garfield's proof of the pythagorean theorem

 

[mathwonk]

i like the proof of cayley hamilton, that says, by lagrange expansion of determinants, adj(XI-A).(XI-A) = ch(X).I. in the ring Mat(k[X]) = Mat(k)[X].

But this says ch(X).I ,the characterstic polynomial of A, is right divisible by XI-A. Hence the right value ch(A).I = 0.