Applications of Topology to real life problems.

Forum for the GRE subject test in mathematics.
fawazshitu
Posts: 4
Joined: Sat Apr 06, 2013 7:49 pm

Applications of Topology to real life problems.

Postby fawazshitu » Thu Apr 11, 2013 5:05 am

I studied topology for 2 semesters.It's fun but the biggest problem i had is to prove theorems and lemmas on my own.
I do understand partially When my professor teaches me but very difficult for me to prove his assignments and homework.
Infact on a scale of 10, I am able to prove 2 ; Which is very bad.How can i ameliorate this situation ?
I also want to have ideas of the applications of Topology to real life problems.

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Gasquet
Posts: 237
Joined: Thu Nov 15, 2012 5:00 am

Re: Applications of Topology to real life problems.

Postby Gasquet » Thu Apr 11, 2013 5:30 am

This may help. In addition to the things mentioned in the link, applications of Topology to Physics.

A comment from there:
Topology is useful to mathematics as a whole. Mathematics is useful to humanity. By transitivity, Topology is useful to humanity.


Of course, "real life" problems is a very ambiguous term. Some people may consider proving Tychonoff's theorem to be a real life problem. :lol:

Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

Re: Applications of Topology to real life problems.

Postby Legendre » Thu Apr 11, 2013 10:23 am

More specific examples:

1. This paper uses Topology to analyze complex networks. E.g. social networks, biological networks, internet etc.

http://arxiv.org/pdf/1204.6376.pdf

2. This paper applies Differential Topology to probability to identify multivariate interactions. This was used in neuroscience recently to deduce how neurons are interacting.

http://www.qwone.com/~jason/trg/papers/ ... chy-01.pdf

CoxZucker_Machine
Posts: 18
Joined: Sun Sep 23, 2012 11:38 am

Re: Applications of Topology to real life problems.

Postby CoxZucker_Machine » Thu Apr 11, 2013 12:06 pm

Brouwer's fixed point theorem can be used to conclude that if you swirl around your coffee in your coffee mug, then after you're done swirling, there is one drop of coffee that ended up back where it was before you moved anything around. This is the kinda shit that trips me out

Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

Re: Applications of Topology to real life problems.

Postby Legendre » Thu Apr 11, 2013 12:56 pm

CoxZucker_Machine wrote:Brouwer's fixed point theorem can be used to conclude that if you swirl around your coffee in your coffee mug, then after you're done swirling, there is one drop of coffee that ended up back where it was before you moved anything around. This is the kinda shit that trips me out


I have heard the one that says if you take a map of any size and drop it on the area it represents, there is a point on the map that is touching exactly the point it represents on the area. Fuzzy memory but might need the Intermediate Value Theorem for this.

But the coffee fact is fuuuuuuun!!! :D

Manyfolds
Posts: 53
Joined: Thu Jan 24, 2013 1:22 pm

Re: Applications of Topology to real life problems.

Postby Manyfolds » Thu Apr 11, 2013 12:59 pm

It's the same thing ;) Indeed, Brouwer Fixed Point theorem is a simple corollary of Intermediate Value Theorem in the R -> R case.

RicciFlow
Posts: 18
Joined: Thu Feb 21, 2013 2:24 pm

Re: Applications of Topology to real life problems.

Postby RicciFlow » Thu Apr 11, 2013 1:22 pm

There is also this:

http://paginas.fe.up.pt/~niadr/PUBLICAT ... 237_CR.pdf

That paper discusses using cell phones to actually map out the topology of indoor spaces. I also know one can use topology maps for automated robot navigation. Lots of machine learning applications as well. Another cool application is in the world of chemistry where one can discuss the shape of molecules by an analysis of the topology of a related graph. There is also an application for medical imaging software and technology. I'm pretty sure one can find an example of the application of topology to basically every field of the sciences.

Applied topology has become a pretty hot area of research recently. I know Math Fest in Madison 2012 had some great talks on applied topology, so I imagine there will be other conferences centered around this topic in the future. Maybe try to go to one. Here's a list of conferences that will have such topics.
http://acat.lix.polytechnique.fr/

Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

Re: Applications of Topology to real life problems.

Postby Legendre » Thu Apr 11, 2013 3:12 pm

Oh wow this topic is getting interesting. Never thought "Applied Topology" would be a branch!

UPenn course notes on an "Applied Topology" course: http://www.math.upenn.edu/~ghrist/notes.html

"Scientific and Engineering Applications of Algebraic Topology" workshop with descriptions of areas: http://www.ima.umn.edu/2013-2014/ (including data analysis, communication and biology!)

I am suddenly looking forward to my Topology coursework. :D

korean
Posts: 14
Joined: Sun Dec 09, 2012 7:51 pm

Re: Applications of Topology to real life problems.

Postby korean » Fri Apr 12, 2013 9:24 am

@Legendre Thanks for the nice link!




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