User muad - MathOverflow

Most striking applications of category theory?

2010-03-25T16:15:51Z

What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:

Joyals Combinatorial Species
Grothendieck's Galois Theory
Programming (unification as computing a coequalizer, Tatsuya Haginos categorical construction of functional programming)

I am sure that these only touch on the surface so I would be most grateful to hear of more examples, thank you!

edit: To try and be more precise. "Application" in the context of this question means that it makes use of slightly deeper results form category theory in a natural way. So we are not just trying to make a list of 'maths that uses category theory' but some of the results which exemplify it best, and might not have been possible without it.

What are the connections between pi and prime numbers?

2010-03-03T18:51:00Z

I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more connections between $\pi$ and prime numbers?

Symmetries of the Julia sets for $z^2+c$

2010-09-09T22:20:09Z

The julia set seems to have symmetries roughly corresponding to translation, rotation and scaling.

In the following image

You can see the horizontal translation, which leaves the extremal left and right endpoints fixed is a symmetry.
The 21-fold rotational symmetry about any one of the 'whorls' is visible also - (I think that the number 21 corresponds to the denominator of the mandelbrot bulb the julia set comes from but I don't yet know how to compute this).
There is also a twofold rotation about the center (and any other part similar to it)
The scales are more difficult to describe and I don't think I have found them all so I will just avoid going into detail on this unless someone would like me to do so.

Do these have a mathematical interpretation, for example as automorphisms of the Julia set in some appropriately understood sense? Or is there some other way to describe these kinds of symmetries in mathematical terms?

Thank you.

What's a mathematician to do?

2010-10-26T16:53:25Z

I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.

My question is: what can one (such as myself) contribute to mathematics?

I find that mathematics is made by people like Gauss and Euler - while it may be possible to learn their work and understand it, nothing new is created by doing this. One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work; which would be the creation of original mathematics. It seems entirely plausible that, with all the tremendously clever people working so hard on mathematics, there is nothing left for someone such as myself (who would be the first to admit they do not have any special talent in the field) to do. Perhaps my value would be to act more like cannon fodder? Since just sending in enough men in will surely break through some barrier.

Anyway I don't want to ramble too much but I really would like to find answers to this question - whether they come from experiences or peoples biographies or anywhere.

Thank you.

Answer by muad for Awfully sophisticated proof for simple facts

2010-10-17T15:44:25Z

Another example from Math Underflow:

We can prove Fermats Last Theorem for $n=3$ by a simple application of Nagell-Lutz (to compute the torsion subgroup) then Mordells Theorem (to see that the group must be $\mathbf{Z}^r \times \mathbf{Z}/3\mathbf{Z}$) then to finish Gross-Zagier-Kolyvagin theorem (which gives $r = 0$) - and that shows it has no nontrivial solutions. I beleive a similar approach works for $n=4$.

Gosper's Mathematics

2010-10-04T21:50:44Z

Sometimes I bump into more of the astonishing results of Gosper (some examples follow) and I gather that a lot of them come from hypergeometrics and special functions.

Have there been any attempts to try and collect together these?
What papers/books and such are there which study collections of results like this?

$$\prod_{n=1}^{\infty} \left(\begin{matrix} -\frac{n}{2(2n+1)} & \frac{1}{2n(2n+1)} & \frac{1}{n^4} \\ 0 & -\frac{n}{2(2n+1)} & \frac{5}{4n^2} \\ 0 & 0 & 1 \end{matrix}\right) = \left(\begin{matrix} 0&0&\zeta(5)\\0&0&\zeta(3)\\0&0&1 \end{matrix}\right)$$

from A third-order Apery-like recursion for $\zeta(5)$

$$W(x)=a+\sum_{n=0}^\infty \left\{{\sum_{k=0}^n {S_1(n,k)\over \left[{\ln\left({x\over a}\right)-a}\right]^{k-1}(n-k+1)!}}\right\} \left[{1-{\ln\left({x\over a}\right)\over a}}\right]^n$$

from a page about Lambert's W-Function

$$\prod_{n=1}^{\infty} \frac{1}{e}\left(\frac{1}{3n}+1\right)^{3n+1/2}= \sqrt{\frac{\Gamma(\frac{1}{3})}{2 \pi}} \frac{3^{13/24}\exp\left(1+\frac{2\pi^2-3\psi_1\left(\frac{1}{3}\right)}{12 \pi \sqrt{3}}\right)}{A^4}$$

from Mathworld

$$\sum_{n=1}^{\infty} \frac{(-1)^2}{n^2}\cos(\sqrt{n^2 \pi^2 - 9}) = - \frac{\pi^2}{21 e^3}$$

from On some strange summation formulas

Work on independence of pi and e

2010-07-29T17:56:46Z

It is an open problem to prove that $\pi$ and $e$ are algebraically independent (over $\mathbb{Q}$).

What are some of the important results leading toward proving this?
What are the most promising theories and approaches for this problem?

What is the advantage of inverting elliptic integrals?

2010-09-18T19:17:19Z

In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this integral or derive the addition theorem.

$$\arcsin(y) = \displaystyle \int_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^2}} (1)$$

In the case of the lemniscate $(2)$ we (Fagnano) can derive a doubling theorem by trying out substitutions in analogy with those which rationalize the first integral. (I learned this from Siegel - Topics in Complex Function Theory). One result (of the sort I wish I could find more) which does come nicely from this integral is via the substitution $y \mapsto iy$ we notice it has double periodicity.

$$\text{sl}^{-1}(y) = \displaystyle \int_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^4}} (2)$$

Euler and others were able to produce theorems about the elliptic integral $(3)$ by analogy with the lemniscate (I read this in Stillwell - Mathematics and its History) - Just as ideas from $\arcsin$ helped to produce theorems about the lemniscate integral. Still, these theorems are very hard earned and it appears that you have to be a master like Euler to derive them.

$$F(y,k) = \displaystyle\int_0^y\frac{\mathrm dy}{\sqrt{(1-k^2 y^2)(1-y^2)}} (3)$$

What I would really like to know is, can we derive more results about the functions from these integrals - maybe using integral manipulation techniques I just don't know about?

Another question that has been bothering me deeply is the integrands (which I believe are called invariant differentials of the lie groups for the algebraic curve) - What sort of coincidence is it that allows the integrands to be of this particular form?

Thank you!

Answer by muad for Proofs without words

2010-09-14T07:44:10Z

$$2 \pi > 6$$

Is there a classification of possible linear actions?

2010-06-13T16:06:49Z

In a vector space, linear transforms can act on points of the space by the usual matrix multiplication rule, but in this note I am reading they use a different action (The Möbius transformation). It's easy to multiply everything out and see that $A(Bx) = (AB)x$ holds but that doesn't explain why this works.

So my question is this, Is there a classification of all possible actions?

edit: just to clarify - I'm more interested in the general case than just the Möbius one but I do appreciate the answers so far!

Edit II: I think I have realized what's really happening here is that the matrices are just representations of, for example rationals, So they act on rationals in a way predetermined by what they represent.

Answer by muad for Work on independence of pi and e

2010-07-29T19:50:12Z

There is a proof of the algebraic independence of $\pi$ and $e^\pi$ in Introduction to Algebraic Independence Theory and a detailed exposition of methods created in last the 25 years although I have not read it.

Answer by muad for Slick ways to make annoying verifications

2010-07-22T19:17:44Z

My example is perhaps not exactly what you had in mind, but I hope that will make it more interesting, thus my reason for posting it. I thought of mentioning the (very elementary) but delightfully clever method of proving polynomials equal by comparing them evaluated at finitely many points (which is applied in, for example in A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan) but instead I thought I would point out the notion of reflective proof from dependent type theory:

To write a formal (computer checkable proof) you must justify every single step of reasoning all the way down to the axioms of the foundation. Of course that would make proving simple things like $((ab)cd)e = a(bc)(de)$ a terrible chore requiring repeated applications of the associativity (infact proving something as simple as $1 + 100 = 101$ could require $100$ applications of the definition of plus!). The idea of reflection is to reduce trivial reasoning steps to computation, This is described in Section 4 of Henk Barendregt's Proofs of correctness in Mathematics and Industry, but it was also applied heavily in a recent formal proof of the four color theorem.

Answer by muad for What's your favorite equation, formula, identity or inequality?

2010-07-15T08:01:47Z

$196884 = 196883 + 1$

Answer by muad for Strengthening the Induction Hypothesis

2010-07-13T18:01:56Z

There is a detailed exposition on this proof technique in Knuth's book Surreal Numbers. The characters of the book develop the idea gradually.

In Algebra with Galois theory by Emil Artin (and probably all the other books), any two splitting fields of of f(x) over F are isomorphic is proved neatly in by generalizing the predicate to say that if two fields are isomorphic then they are extended to isomorphic splitting fields. The same sort of generalization repeatedly occurs in correctness arguments for simple recursive programs.

This proof technique also begets a programming technique: Adding a new parameter to a recursive function (which corresponds to generalizing the induction predicate) will often let you give more efficient algorithms. As a simple example consider a converting leaves of a tree to a list from Deforestation: transforming programs to eliminate trees.

Another example from computing is the method of reducibility candidates (which is explained in Proofs and Types) used to prove strong normalization for various forms of lambda calculus.

Answer by muad for Examples of undergraduate mathematics separation from what mathematicians should know

2010-07-06T17:07:18Z

One theme I found repeatedly occurring in education is that teachers delight themselves in not telling anyone directly about the fundamentally important things. For example,

The idea of unification is not taught explicitly - Instead, students are given a table of Laplace transforms and enough exercises that the process is anonymously substituted into their heads.
Taking the quotients of a set by an equivalence relation - Having never seen how to construct $\mathbb{Z}$ from $\mathbb{N}$ or $\mathbb{Q}$ from $\mathbb{Z}$, the ritual of constructing $\mathbb{R}$ (if it is mentioned at all) appears completely alien and is forgotten immediately. The same student will likely forget (if they are able to understand in the first place) the first isomorphism theorem.
Logical language and the deduction rules for proving statements are not mentioned - One is expected to aquire this language as you do with Listen and Repeat cassette tapes. If it is not clear exactly what a proof is, creating one is a great deal more intimidating and difficult! Teaching basic methods of computing science like structural induction on data types should remedy this.
Differential forms are mentioned explicitly but we treat the fickle beasts with great caution - If these unreal quantities are allowed to freely mix with numbers and variables, why must we be constantly told that dividing them is "purely formal"? Despite that various foundations of analysis have been made rigorous beginners to this subject do not benefit. Instead they are troubled by it and develop an allergic reaction to $\epsilon$.

Treating Differential Operators as Numbers

2010-06-26T12:17:31Z

In Penrose's book (The Road to Reality, chapter 21) he gives an example of Oliver Heaviside's observation that you can treat differential operators like numbers:

The differential equation $(1+D^2)y = x^5$ can be solved by dividing by $(1+D^2)$ then taking the power series expansion: $$y = (1-D^2+D^4-D^6+\cdots)x^5$$ which evaluates to $y = x^5 - 20x^3 + 120x$.

Apparently this can be made perfectly rigorous!

How is this done? and do you know where I can read more details about this idea?

Answer by muad for How should one present curl and divergence in an undergraduate multivariable calculus class?

2010-04-21T12:42:18Z

When I did this course myself I was deeply confused and distressed by it all. I know that everyone learns differently but when we are shown all these mnemonics for remembering formula that treat partials as if there were really numbers - it's not so bad, until they start getting used in proofs.. then it drags up all these memories about people saying "well it's not really a fraction but lets treat it that way anyway". You can pass this class (with a very high mark) by memorizing these formula and I suppose that is all that really matters for beginner classes like that (being able to solve lots of problems, without necessarily knowing why or what) but it can be a bit stomach churning and unpleasant to sit through a semester of not having a clue what any of this means while still getting all the right answers. There is a terrible sense of being lied to, when people try to dumb down things in the hope that it makes it simpler and easier to learn. It's only when I found a book on differential forms (which unified all these different concepts in one of the "applications" chapters) that I started to get the impression this was real mathematics and not just a strange act of going through the motions of writing lots of integral signs and so on. Although I do appreciate the remark in the preface of one of the many text books I read cover to cover in a worried haste to try and make sense of all this notational juggling, which said roughly that having illustrative notations like this fuels the intuition (and gave an example of that curl formula coming from the visual that the determinant had two equal rows in it), I didn't personally find this reassuring. Maybe there is value in teaching it this way to scare people into studying hard for it but I don't think I personally got anything out those months of work myself except for the vague geometric intuitions about div, grad and curl (which you could explain in a day by showing a video). If people learned about differential forms earlier on (rather than being told "it's what Leibniz did so we do to") maybe this course could be taught by developing the abstract setting a bit more and then specializing it down for the 3 dimensional case - which you could then get a got grip on by applying it to physics problems. I've not taught any class myself so I just wanted to give an account of what it can feel like sitting (or being dragged through backwards) this type of class.

Proofs that require the existence of large finite numbers

2010-04-11T19:06:01Z

I know some proofs require the existence of large infinite ordinals, they give the fuel that drives induction principles. An example of this is the use of ε₀ to give a consistency proof of peano arithmetic.

What I would like to find is proofs that require the existence of a large finite ordinal. thank you!

Answer by muad for Proofs without words

2010-03-25T19:00:35Z

If we have 3 circles on the plane with tangent lines, we can notice they have colinear intersection!

To prove it, we can visualize the same configuration in 3D, the balls lay on a surface and rather than tangent lines we take cones: The colinearity comes from the fact that if we lay a plane ontop of this configuration it will intersect the table in a line!

This is from 'curious and interesting geometry' and the proof is attributed to John Edson Sweet. I really like this proof because it gives a vivid example of the general idea that sometimes, to solve a problem in the most simple way you need to view it as a part of some bigger whole.

Answer by muad for What are the worst notations, in your opinion ?

2010-03-18T16:36:45Z

I have struggled with 'dx'. I've spent years trying to study every different approach to calculus that I could find to try and make sense of it. I read about the limit definitions in my first book, vector calculus with them as pullbacks of linear transformations or flows/flux, differential forms from the bridge project, k-forms, nonstandard analysis which enlarges $\mathbb{R}$ to give you infinitesimals (and unbounded numbers) but the same first order properties and lets integral be defined as a sum, constructive analysis using a monad to take the closure of the rationals to give reals... but I am still just as confused as ever, I understand that the mathematical notation doesn't have a compositional semantics but still don't really get it - one of the problems is despite not really understanding it, or having any abstract definition of it.. I can still get correct answers and I really hope this doesn't become a theme as I study more topics in mathematics.

Is my (size 200) subset sum proof good?

2010-03-03T16:50:58Z

I'm trying to prove that given 200 integers (not necessarily all distinct), you can pick 100 of them have a sum divisible by 100. I'm wondering if this is a good proof and if you would give me some advice on how to write it better. Thanks!

(A) For any 3 element set, there exists a 2 element subset which sums to an even number. This is clear if we consider cases, the 3 element set must be of the form {even,even,-} or {odd,odd,-} and these two elements sum even.

(B) For any 9 element set, there exists a 5 element subset which sums to a multiple of 5. This is verified by exaustive computation, We only have to consider the sets of numbers mod 5, so that is 5^9 cases to test.

(C) So if we are now give a 200 (= 2*2*2*5*5) element set will we always be able to find a 100 element subset, which sums to a multiple of 100? By (A) we can reduce the problem to a stronger one, take out two elements which sum even and sum then and halve the result - we iterate this until it can't be done any more. This gives us a new set of 99 arbitrary integers (we discard the remaining two integers from the original 200 element set). If we can now find a size 50 (= 100/2) subset which sums to a multiple of 50 we have proved (C). We do this factoring out 2 operation again to get a 49 element set and again to get a 24 element set, let us consider the same operation factoring out 5 instead of 2. Now 24 = 5+5+5+5+4 and 5+4 is 9, so we are given a set of 4 integers which must sum to a multiple of 5, this is not likely -- we really do need 9 arbitrary integers, well there are 4 left over from the factoring out 5 and there was that one left over from factoring out 2 the first time.. So we do have 9 arbitrary integers and thus a subset of size 5 summing to something divisible by 5 and hence (C) is proved.

Comment by muad

2010-12-19T06:34:02Z

I would love to 'see' the Leech lattice somehow.

Comment by muad

2010-12-19T06:32:27Z

I would like to know what can I read to get an appreciation for this absolute galois group?

Comment by muad

2010-11-03T18:37:57Z

I thought we needed axiom of choice to show that there are non-measurable sets? (unless the axiom was used in counting the size of the algebras and I missed it)

Comment by muad

2010-10-24T18:31:03Z

Obviously it's $q^-1 m$ but you should ask this on math.stackexchange.com/questions?sort=newest because it's not really the right sort of question for MO.

Comment by muad

2010-10-23T00:05:48Z

@Alexis Shaw, this sort of question is good on math.stackexchange.com/questions though.

Comment by muad

2010-10-20T14:45:01Z

@Harry Altman, the integrand should be $\frac{x^4 \cdot (1 - x^4)}{1 + x^2}$.

Comment by muad

2010-10-17T17:32:49Z

The six color theorem as a corollary of the four color theorem.

Comment by muad

2010-10-17T16:50:14Z

It's in the comment of Steve Huntsman.

Comment by muad

2010-10-03T13:36:47Z

I don't understand his objection to Gentzen's proof at 29:00. Why would someone be skeptical about well foundedness of $\epsilon_0$?

Comment by muad

2010-10-02T17:36:32Z

This is a lot like the Champernowne constant, if you study that maybe you will be able to adapt the proof toward this number too.

Comment by muad

2010-09-29T20:22:13Z

Can you please provide a link to the proof.

Comment by muad

2010-09-27T14:47:12Z

I recently asked a similar question - mathoverflow.net/questions/33817/…

Comment by muad

2010-09-25T18:30:57Z

This question is not really appropriate on math stack exchange either..

Comment by muad

2010-09-24T07:19:07Z

+1 super question! Indeed, this is something which has been rattling around in my head for a very long time.

Comment by muad

2010-09-21T20:50:57Z

It's because they correspond to the north pole when considered as holomorphic functions onto the riemann sphere.