Explaining topology to non-mathematicians

[matt grime]

do you know what a ring is? let R be a (commutative) ring with unit and let R[x] be the set of all polynomials in x with coefficients in R. we can differentiate any element of R[x] formally, ie without reference to limits. D(ax^2)=2ax this is well defined since R has a unit, 1, and 2=1+1. the usual rules follow: D(pq)=pD(q)+D(p)q and D(p+q)=D(p)+D(q) etc. Example R= integers mod 2, then D(x^2)=0, and this is a very important algebraic fact as anyone who's done an intro to galois theory will remember, though if yuo have a memory like mine you won't recall exactly what the property is though separability seems to be a good bet.

as for your second question my answer is: eh? don't understand what you want to know. what does "similar" mean in this context? they are both things to do with tangent spaces i suppose, and how the local tangent spaces glue together globally ie although we can on each small area of the surface of the tennis ball comb the hair in the same direction since locally the tennis ball loooks a lot like a flat object, and on a small flat disc we can certainly, at each point, pick a direction vector and let them vary smoothly, eg if we think of it as a disc in the xy plane just have the vector pointing in the x direction with constant length 1 at each point, but we cannot take these local things and patch them together to get a globally smooth one. hopefully that answers the question.

 

[roger]

matt,

but what's the advantage of defining derivatives without reference to limits ?

 

[matt grime]

because it applies to non-normed spaces and it has simply proved useful. if it hadn't then it would have been forgotten. example. suppose f is a poly in k[x] and it splits over k, then if f has a repeated root then f and f' are not coprime. this is a result of use in galois theory, for example. we also care about tangent spaces for abstract algebraic varieties too (lie algebras).

 

[PhilG]

in R^n f is differentaible at x if there is a linaer map Df(x) satisfying

f(x+h)=f(x)+Df(x)h + ho(|h|)

this can be extended to any place where there is a notion of linear map or a map such as |?| to the reals, or some other ordered space.

Is that last term "the vector h, scaled by a number that goes to zero faster than |h|"? What if h and f(h) live in spaces of different dimensions, say n and m?

 

[Maxos]

That's a typo, the general Taylor's polinomium with Peano's remainder is:

$$f(\vec{x_0}+\vec{h})= \sum_{k=0}^n \frac {{d^k}f|_{\vec{x_0}}(\vec{h})} {k!} + o(|\vec{h}|^n)$$
for $$\vec{h} \rightarrow \vec{0}$$

where differentials should be considered as functions $${d^k}f:\mathbb{R}^n \rightarrow \mathbb{R}$$ of the vector $$\vec{h}$$ evaluated at the point $$\vec{x_0}$$.

Saying: $$g(x)=o(f(x))$$ for $$x \rightarrow x_0$$ means that
$$\lim_{x\rightarrow x_0} \frac {g(x)} {f(x)} = 0$$

 

[fourier jr]

back to the beginning, topology is useful for proving that there are infinitely many primes!

consider the following topology on the integers. for $$a, b \in \mathbb{Z}, b>0$$ set $$N_{a,b}$$ = {$$a + nb | n \in \mathbb{Z}$$}.

Each set $$N_{a,b}$$ is a 2-way arithmetic sequence. a set G in this topology is open if either G is empty or if for every $$a \in G$$ there exists some b>0 with $$N_{a,b}$$ a subset of G. (not hard to check that unions & finite intersections are still open, and that Z & the empty set are all open so this makes a topology on the integers)

2 facts:
1) any non-empty open set is infinite
2) any $$N_{a,b}$$ is closed also. since $$N_{a,b} = \mathbb{Z}$$ \ $$\cup_{i=1}^{b-1} N_{a+i,b}$$ $$N_{a,b}$$ is the complement of an open set it's closed

now to use primeness. since any number except -1 or 1 has a prime divisor p & is therefore in $$N_{0,p}$$ we get that $$\mathbb{Z}$$ \ {-1,1} = $$\cup_{p\in\mathbb{P}}N_{0,p}$$

if the set of primes were finite, then $$\cup_{p\in\mathbb{P}}N_{0,p}$$ would be a finite union of closed sets & therefore closed. thus {-1,1} would be open, contradicting 1) above. thus there are infinitely many primes.

 

[Maxos]

Very interesting but not so important for Physicists.

 

[HallsofIvy]

I would think Euclid's proof would be sufficient.

 

[matt grime]

Very interesting but not so important for Physicists.

is it not? surely it is good to know things like this since ther are links between random matrices (spectral stuff) and the riemann zeta function which is very heavily dependent on prime numbers.

 

[Maxos]

Yes you are right.

 

[krab]

Ahem! Back to the question of how to explain to NON-mathematicians. My experience is that it is useless to describe concepts with which we are already familiar; their eyes just glaze over. Simpler is to motivate it with the 7 bridges of Konigsberg. Then state that topology allows one to prove that the goal of crossing each bridge exactly once is impossible, so no one wastes time trying. Everyone understands about the value of not wasting time.

 

[matt grime]

toplogy? i thought that was elementary graph theory: it contains 3 odd nodes or is it only 1?

 

[Maxos]

Again, right.

 

[fourier jr]

toplogy? i thought that was elementary graph theory: it contains 3 odd nodes or is it only 1?

it's a topological thing because the solution doesn't depend on the distances of the bridges from each other, nor the lengths of the bridges. the only thing to be concerned with is the connectivity properties. that's what the wikipedia thing said anyway.

 

[matt grime]

Hmm, most odd. It is only a personal view but I've never considered graphs as anything other than combinatorics. Indeed I have never seen a graph theory theorem refer to any topology (perhaps the author is using it a nontechnical sense?) of the graph (ie not defining open subsets of the graph). Labels are largely unhelpful I admit, and one can certainly use topological ideas in graph theory as one can in many parts of mathematics.

Part of graph theory is concerned with "rigidity" and if a graph may be embedded in the plane.

I really can't agree with it, on reflection. I can't disagree with the assertion that graph theory is not bothered with how things are embedded in some space, merely the data of vertices and edges, but I don't see how that makes it topology. It seems to merely that it is merely a statement that we have realized these problems can be described in mathematical terms.

If you look at the graph theory (mathematics) link at the bottom of the page you'll find that the word topology doesn't appear at all in the description.