What's so special about topology?
>>7701995
What does it mean for two objects to be close?
>>7702002
>
Not a topologist, but if you ever study non-foundational math, you'll find that topology is ubiquitous.
A solid introductory graduate course in topology will quickly reveal how much of the, say analysis, you've done was actually topology, with only a few concepts being truly unique to analysis, like, say, integration.
I know very little of algebraic topology, but I do know that whenever you couple algebra with something, you're looking to find a way to perform computations with a new type of object. Hence, algebraic topology applies algebraic methods to topological problems. Why is this useful? I'd imagine because once you can apply algebra, you can apply computers.
Beyond that, Topology is beginning to find a place in applied mathematics, particularly statistics where high dimensional data is involved. Topological methods are being developed to try and extract information about the shape of the data..
You need to remember that this is a very new branch of math, and that it wasn't even in a condition to be formally taught at most universities in the early/mid 1900s.
>>7701995
> Topological Derivative
> Neural Networks
Solving Artificial Intelligence
Basically it solves traditionally intractable problems. The first victory was the classification theorem of closed surfaces.
>>7702020
As simple a problem as proving two euclidian spaces of different dimensions non-homeomorphic can be painful with just topology. Applying some algebra, one constructs a homology theory proving this and many other negative claims with ease.
You can assign a topological invariant to a physically measurable quantity (e.g. the first Chern is related to quantized Hall conductance) as is studied in topological quantum field theories (TQFT's). Further, certain phases of matter exhibit a ground state degeneracy that is dependent on the topology the system is placed upon. For example, in the toric code the degeracy goes like 4^g where g=genus; in the fractional quantum hall laughlin states it goes like m^g where m is an odd integer greater than 1.
It turns out this degenerate ground state manifold can be used for quantum computing in a scheme known as topological quantum computing. I can elucidate on this a bit more if people want.
>>7701995
Very little to do with "omg how many holes does this object have?? Like woah a coffee mug is the same as a hula hoop! Isn't that crazy? I love math, lol xD"
>>7702983
I am physics major
I do not understand a large part of that
Tell me where to study topology pls
>>7702990
The simplest model of a topological phase would be either Kitaev's majorana 1d p-wave superconducting majorana chain or his toric code model. His original 1997 paper is decent or there are lecture notes by him and Laumann on the arxiv. If you know about spins and fermions they should both be accessible.
>>7702986
I disagree. Just because there are other more complicated topological spaces, doesn't mean it's not about counting silly things and thereby classifying them.
>>7702990
He speaks of field theories (the most simple relevant one being Maxwells) and how, as objects upon a space like spacetime, they give rise to coordinate invariant indices (let's say) and how they can be used to classify solutions.
When people (Witten, basically) recognized that much of the physics is there even if you drop a metric distance (that always induces a topology) and just work with a topology directly (see topological (quantum) field theory), the mathematicians jumped on it too.
Yang-Mills and Chern-Simons theory are key words you'll necessarily come across in these contexts. But maybe forgetting about it and making mad dosh is another good life plan, pic related.
>>7702983
topological quantum computing. Interested, can you gib links ?
>>7703020
See Nayak et al. "Topological Quantum Computation", 2007 or 2008 I think. The wiki article on it should also suffice.
Basically, manipulating qubits is hard because of decoherence with the environment. However, it is difficult for the environment to affect the entire system as a whole--it acts only locally to a good approximation. Thus, storing or manipulating the quantum information in a topological way (i.e. non-local) will provide robust protection.
>>7702020
>reveal how much of the, say analysis, you've done was actually topology
This. Exploring the notion of closeness without necessarily defining distance is as fulfilling as the move from calculus on the real line to basic analysis concepts on arbitrary metric spaces.
>>7702990
http://arxiv.org/pdf/0810.0344v5.pdf
>>7703056
While this is not false, topology only makes up about 2 weeks of a Real Analysis 1 class. Considering there's Analysis 1-4 and Functional Analysis 1-4, 2 weeks of topology isn't that much.
>>7703070
Based anon
>>7702990
at your local math department. I'm majoring in physics as well, and am pursuing at least a bachelors in math just because
>>7703105
Well, by "exploring" I mean reaching the highly nontrivial consequences of general assumptions you can get by a little more abstraction, like the jump from calc to analysis.
Not metrics induce topologies and "omg compactness hurr durr"
>>7703105
This isn't a very safe statement to make.
Analysis isn't universally treated the same way.
At my undergraduate institution there was advanced calculus 1 and 2. The first consisted of the first 4 chapters of Pugh's: Real Mathematical Analysis.. Yeah.. It's a pompous title.You can amazon the table of contents. Advanced Calculus 2 was multivariable calculus in full generality, with a bit of Lebesgue stuff at the end.
Analysis I,II,III,IV, etc.. Were advanced topics in analysis. Graduate measure theory, complex analysis, functional analysis, harmonic analysis, geometric blah.. Depending upon your interests.
The first part of advanced calculus is essentially all point-set topology. By this, I mean most of the major results aren't analytical in nature.. They're results of the topological structure on a given space: Extreme value theorem, IMVT, limits, continuity, Arzela-ascoli, Heine-borel, etc. etc.. The only analytical things are derivatives, integration, and probably a few others. I'm being very general here.
