Donovan

Topology. Please discuss.

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On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:
> Topology. Please discuss.

What would you like to know about it ? Are you considering
studying it ?

Topologists are interested in studying topological
invariants of various objects. That is, they look at
properties that are independent of things like bending and
stretching. If you can map one object to another with a 1-1
mapping that is continuous and has a continuous inverse,
then to a topologist, they are the same thing (if two
objects can be mapped such a way, they are homoemorphic, or
one could simply say, topologically equivalent).

So the topologist looks for algebraic and geometric
properties that apply across a homoemorphism class. For
example, Euler characteristic of a surface is topology
invariant (so it's 2 for any polygon but 0 for a torus)

Some of these may involve a certain amount of geometry. For
example, knot theory looks at the space around the knot. The
geometry of the embedding is important, even though the knot
is always topologically a surface (though the way it is
embedded varies)

Another toplogical invariant is the number of connected
components ("connected" means what you think it means. A
connected component is a maximal connected subset)

There are actually algebraic invariants of topological
objects. For example, one could define a formal sum with
integer coefficients assigned to each connected component,
(x1 C1,x2 C2,x3 C3) where C1, ... , C3 are components and x1
are coefficients. Then one can define an operation + by:
(x1,x2,x3) + (y1,y2,y3) = (x1+y1,x2+y2,x3+y3) This algebraic
structure is a topological invariant (a "homology group"
actually). Most of the work I did involved exploring
algebraic invariants of topological objects.

HTH,
--
Donovan Rebbechi http://pegasus.rutgers.edu/~elflord/

WTF did he say???

Donovan Rebbechi <abuse@aol.com> wrote in message
news:<slrncerjoo.esn.abuse@panix2.panix.com>...
> On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:
> > Topology. Please discuss.
>
> What would you like to know about it ? Are you considering
> studying it ?
>
> Topologists are interested in studying topological
> invariants of various objects. That is, <snip gibberish

On Thu, 8 Jul 2004 22:47:52 +0000 (UTC), Donovan Rebbechi <abuse@aol.com> wrote:

>On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:
>> Topology. Please discuss.
>
>What would you like to know about it ? Are you considering
>studying it ?
>

[snip]

Group Theory and Ring Theory was tough enough for me, so I
decided to skip Topology all together and took Diff EQ
instead. But that was more than two decades ago.

Easy enough to understand, but more importantly, how is this
going to help to make the perfect cup of expresso?

Donovan Rebbechi <abuse@aol.com> wrote:

> On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:
> > Topology. Please discuss.
>
> What would you like to know about it ? Are you considering
> studying it ?
>
> Topologists are interested in studying topological
> invariants of various objects. That is, they look at
> properties that are independent of things like bending and
> stretching. If you can map one object to another with a
> 1-1 mapping that is continuous and has a continuous
> inverse, then to a topologist, they are the same thing (if
> two objects can be mapped such a way, they are
> homoemorphic, or one could simply say, topologically
> equivalent).
>
> So the topologist looks for algebraic and geometric
> properties that apply across a homoemorphism class. For
> example, Euler characteristic of a surface is topology
> invariant (so it's 2 for any polygon but 0 for a torus)
>
> Some of these may involve a certain amount of geometry.
> For example, knot theory looks at the space around the
> knot. The geometry of the embedding is important, even
> though the knot is always topologically a surface (though
> the way it is embedded varies)
>
> Another toplogical invariant is the number of connected
> components ("connected" means what you think it means. A
> connected component is a maximal connected subset)
>
> There are actually algebraic invariants of topological
> objects. For example, one could define a formal sum with
> integer coefficients assigned to each connected component,
> (x1 C1,x2 C2,x3 C3) where C1, ... , C3 are components and
> x1 are coefficients. Then one can define an operation +
> by: (x1,x2,x3) + (y1,y2,y3) = (x1+y1,x2+y2,x3+y3) This
> algebraic structure is a topological invariant (a
> "homology group" actually). Most of the work I did
> involved exploring algebraic invariants of topological
> objects.
>
> HTH,

"Donovan Rebbechi" <abuse@aol.com> wrote in message
news:slrncerjoo.esn.abuse@panix2.panix.com...
> On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:

> > Topology. Please discuss.

> Topologists are interested in studying topological
> invariants of various objects.

Fascinating.

I never fail to be surprised by the directions into which
'science' can delve in the most esoteric, baffling and
completely pointless manner. Sure, I know you have to go
climb distant hills to see if the view is worth seeing, but
common sense suggests that we should climb the hills that
*look* like they might provide interesting perspectives
before we wander up each and every lump that presents itself
before us. Our time on this earth is finite, and we are
obliged to use this time in the most beneficial manner for
ourselves and our species.

Clearly, you have a brain well suited to dealing with the
minutia of a subject and you revel in the intellectual
satisfaction thus provided. But I'm forced to ask why a
young, capable, and articulate individual in the prime of
his working life is wasting his time on Usenet partaking
in the most mind-numbingly boring discussions on almost
every contentions aspect of running whilst at the same
time studying a subject that can only - with the best
will in the world - be described as 'of pheripheral
benefit' to humanity?

On Thu, 8 Jul 2004 22:47:52 +0000 (UTC), Donovan Rebbechi <abuse@aol.com>
wrote:

>On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:
>> Topology. Please discuss.
>
>What would you like to know about it ? Are you considering
>studying it ?
>
>Topologists are interested in studying topological
>invariants of various objects. That is, they look at
>properties that are independent of things like bending and
>stretching. If you can map one object to another with a 1-1
>mapping that is continuous and has a continuous inverse,
>then to a topologist, they are the same thing (if two
>objects can be mapped such a way, they are homoemorphic, or
>one could simply say, topologically equivalent).
> . . .

Brother in law is a math kinda guy, so I have learned: the
doughnut is topologically equivalent to the coffee mug! :-)

--
Daniel
deltaechomike@usa.net

On 2004-07-09, Dangling <danglingdingleberrys@hotmail.com> wrote:
> WTF did he say???

Never mind, you don't need to know this stuff to run (or for
that matter, to troll)

Cheers,
--
Donovan Rebbechi http://pegasus.rutgers.edu/~elflord/

But how's learning how to make a perfect cup of expresso(?)
going to help with refueling a Sukhoi Su-30?

Bumper <bobemery@bellsouth.net> wrote:
> Easy enough to understand, but more importantly, how is
> this going to help to make the perfect cup of expresso?
>
> Donovan Rebbechi <abuse@aol.com> wrote:
....

"Bumper" <bobemery@bellsouth.net> wrote in message
> Easy enough to understand, but more importantly, how is
> this going to help to make the perfect cup of expresso?
>

It's spelled espresso - I'm not trying to be a smartass or
anything, its almost never spelled right, even on some signs
in coffee stores! :-)

cheers,
--
David (in Hamilton, ON) www.allfalldown.org
www.absolutelyaccurate.com

On 2004-07-09, Bumper <bobemery@bellsouth.net> wrote:
> Easy enough to understand, but more importantly, how is
> this going to help to make the perfect cup of expresso?

Take up grad school and get an espresso machine. By the time
you graduate, you'll be pretty good at making the stuff (I
speak from experience).

Cheers,
--
Donovan Rebbechi http://pegasus.rutgers.edu/~elflord/

On 2004-07-09, np426z <np426z@btinternet.com> wrote:

> Clearly, you have a brain well suited to dealing with the
> minutia of a subject and you revel in the intellectual
> satisfaction thus provided. But I'm forced to ask why a
> young, capable, and articulate individual in the prime of
> his working life is wasting his time on Usenet partaking
> in the most mind-numbingly boring discussions on almost
> every contentions aspect of running

I don't know. Why do we do it ? (-;

> whilst at the same time studying a subject that can only -
> with the best will in the world - be described as 'of
> pheripheral benefit' to humanity?

I work in a psychology department now. But interestingly
enough, I see more and more advanced math making its way
into that field.

Cheers,
--
Donovan Rebbechi http://pegasus.rutgers.edu/~elflord/

On 2004-07-09, Daniel <deltaechomike@usa.net> wrote:
> On Thu, 8 Jul 2004 22:47:52 +0000 (UTC), Donovan Rebbechi
> <abuse@aol.com> wrote:
>
>>On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:
>>> Topology. Please discuss.
>>
>>What would you like to know about it ? Are you considering
>>studying it ?
>>
>>Topologists are interested in studying topological
>>invariants of various objects. That is, they look at
>>properties that are independent of things like bending and
>>stretching. If you can map one object to another with a
>>1-1 mapping that is continuous and has a continuous
>>inverse, then to a topologist, they are the same thing (if
>>two objects can be mapped such a way, they are
>>homoemorphic, or one could simply say, topologically
>>equivalent).
>> . . .
>
> Brother in law is a math kinda guy, so I have learned:
> the doughnut is topologically equivalent to the coffee
> mug! :-)

Yep. So it won't make a barista of you, but at least you'll
always have a doughnut with your coffee.

Cheers,
--
Donovan Rebbechi http://pegasus.rutgers.edu/~elflord/

>It's spelled espresso - I'm not trying to be a smartass or anything,
>its almost never spelled right, even on some signs in coffee stores! :-)

I was in a coffee shop in Albuquerque (the Double Rainbow,
IIRC) where the employees' shirts said on the back, "There
is no X in espresso."

--
Brian P. Baresch Fort Worth, Texas, USA Professional editing
and proofreading

If you're going through hell, keep going. --Winston
Churchill

Donovan Rebbechi <abuse@aol.com> wrote in message news:<slrnces3mb.i60.abuse@panix2.panix.com>...
> On 2004-07-09, Dangling
> <danglingdingleberrys@hotmail.com> wrote:
> > WTF did he say???
>
> Never mind, you don't need to know this stuff to run (or
> for that matter, to troll)
>
> Cheers,

Oh good, as long as I'm still good at one of the above, I'm
satisfied.