Abstract Algebra: Applications & Overview

[heman]

I just started with the course of discrete mathematics,,where we have abstract algebra..I am actually interested in the application of this algebra..i know that this is used in Cryptography,error correcting codes,and theoretical computer science...i just want to have a basic outline of how they do it..so that i can be more inclined towards the course..

 

[matt grime]

RSA being so famous will be explained in greater or lesser detail on any page that you google for.

Computer languages and discrete maths go hand in hand often because they use the same ideas: you are manipulating discrete data with structure such as posets. Not to mention truth tables and logic gates. In some sense the essence of computer science is doing discrete mathematics.

Error correcting codes are kind of striaghtforward and it's not really algebra just common sense to explain, though a good knowledge of linear algebra helps. At one extreme, suppose I want to send either yes or no as an answer, I can encode these as strings of n (eg n=8) bits, I might want to do so so as to ensure that any errors are noticed. It would therefore be silly to encode yes as 10000000 and no as 00000000 cos one error could leave them indistinguishable. Instead send yes as 11111111 and no as 00000000 then if there are 3 or fewer errors we can spot them and know what the original message was. Obviously I can have 5 errors in 11111111 and get something 'closer' to 00000000 so there will be false positives. The key is to choose the bit length and the difference between code words (distance is the hamming metric: the number of bits where two words disagree is the distance between them) and the number of codewords as well as possible. But algebra is far more interesting than that. It is the essence of modern physics: string theory, elementary particles are matrices, even statistical mechanics can use the theory of algebra(s).

 

[heman]

Matt ,,How can you say Abstract algebra is so interesting?

I know its very strong and powerful but don't know why its so interesting.

 

[matt grime]

Matt ,,How can you say Abstract algebra is so interesting?

I know its very strong and powerful but don't know why its so interesting.

Do you actually know what abstract algebra is, though? Or at least my opinion of what abstract algebra is?

Do any of the following mean anything to you? Universal enveloping algebras, cohomology, quantized lie bialgebras, homotopy categories, Hecke algebras? You might, if you search for these terms come up with some interesting and unexpected links to things you probably find more amenable such as physics, though you shouldn't try and learn about them right now. Do not judge the subject upon the small part of it you've been exposed to in a lecture or two. A lecture you probably thought was very boring and as such has coloured your judgement.

Complaining you think it is boring because of your first course in it is a little like saying books are boring because of Spot the Dog, or Janet and John or *insert books given to kindergarten/reception age children in your country here*

 

[heman]

What's your opinion of abstract algebra?why do you have so much fascination for it.

Its very tough to develop interest without seeing any nearby applications ,,which are very close ,which i can observe easily..

 

[Hurkyl]

Error correcting codes are kind of striaghtforward and it's not really algebra just common sense to explain

ECC are more interesting than that!

Not only do you need to be able to come up with a good code in the first place, but you need to have reasonable algorithms for encoding and decoding. Algebra can be useful in such ventures.

 

[devious_]

Something doesn't have to be applicable to be interesting.

 

[heman]

Really...
are there any things which we study and doesn't have any applications but very interesting..

what actually invokes interest i don't know,,why person develops interest for something and why not for other,,

i believe if you become in contact with someone who has fascination for something soon the other guy also develops,i have seen that working..

 

[matt grime]

I didn't say ECC aren't interesting, I just said it was straightforward to understand. Finding them is bloody hard, since it is a packing problem.

And, for heman, why does something have to have an application before you'll find it interesting? Presumably you do not then read fiction or listen to music? It is perfectly possible to find elegance in something abstract, as is almost certainly a prerequisite for doing (non-applied) mathematics.

If you want another analogy, and why not, two analogies are exactly as good as one (2x=x solve for x), then I presume you dismiss all of music in this way because you think learning the violin is boring.

 

[heman]

music and fiction have got a purpose as i believe,their elegance as you said is their purpose as i think...

music is tried to be composed in such a way that people like it..

 

[matt grime]

so why can't that apply to pure mathematics (like music, fiction, or philosophy as well)?Music is composed so as to follow certain set rules and patterns that we have declared that are pleasurable to us, and that we often frequently learn to like, and as such is not a million miles from mathematics where we deduce certain results based upon our preconceived notions of what ought to be correct. If you don't believe me try listening to other cultures' music or reading about the treatment meted out to the atonalists under the Russian Communist Regime, then also look at the evolution of non-Euclidean geometries.

 

[heman]

so is it like saying ,,how far can we go with certain defined characteristics...how much can we dig..how much beauty can we bring with something raw...

 

[TenaliRaman]

I would like to point out one small thing, when abstract algebra was first conceived, there were no applications for it, neither was it conceived with any application in mind.

Now much later in the era (infact much much later), when people started working with Cryptography, ECC and Theoretical Aspects of Computer Science (the very same three fields mentioned above), it was realized that much of the mathematics necessary for it was already there in the form of abstract algebra. Interesting isn't it, that something which had been completely useless and was pursued only out of sheer interest, has suddenly found itself an application!

This is not the only case. As another example, kernel methods in statistics is one very rich field of study as far as statistical learning theory is concerned. All the mathematics of this theory were developed some 50-60 years back. Now, as recent as a decade back, its application was seen in the form of Support Vector Machines in the field of Artificial Intelligence. It has roots in pure mathematical fields like measure theory for instance.

There are several such instances and in all of these, mathematics wasnt developed with any application in mind. It was pursued because people were interested in it.

Each of us is born with a natural urge to explore, merely pushed by interest and having absolutely no motive behind it. A child when given a block of wood, tries to feel it, touch it, taste it and does a lot of other things with it, till it finds it uninteresting. As we grow, some of us tend to lose this urge or rather are influenced by the more practical world at looking at things from the perspective of applications. However, there are those who don't lose that urge to explore and show this inclination in certain specific fields, say art, science, mathematics etc. There is no rational way of explaining this interest as much as a child can't explain why it was fascinated by the block of wood or that cardboard box he/she has been holding onto for several days.

However, as we notice from the above examples, mathematics which may seem useless at the time of its creation, may find its use at some point in the future. I believe that's reason enough to pursue mathematics out of sheer interest, without regards for any application.

-- AI

 

[SqrachMasda]

abstract algebra takes everything that was once fun and interesting about math and changes it to a million vocabulary words and symbols that mean a bunch of other stuff and confuses the hell out of you
haha, i am very frustrated with abstract algebra if you can't tell

 

[heman]

Guys what's this fuss about Reimann hypothesis...is it something which is unsolvable..

 

[shmoe]

Try searching for Riemann Hypothesis or Riemann zeta function in the number theory forum. It comes up often.

 

[matt grime]

What do you mean by fuss? It is a celebrated open problem. One that is probably true (in my opinion) but which we frustratingly can't prove yet. I say 'frustratingly' because the evidence seems to say it should be true, and the corresponding statements over fields of positive characterstic have been known to be true for many years. When it was first conjectured, it was thought that it would be easy to solve and was given to Hardy (I think, though I don't feel confident in saying that) as his PhD subject.

There are also surprising links with many areas of mathematics and physics.

Oh, and it is relatively easy to explain to people, unlike, say, the Hodge Conjecture or the Birch-Swinnerton-Dyer Conjecture. Of course, the easy explanations don't really give the real picture. Quite what catches the general public's imagination about mathematics seems an unpredictable business. Why on Earth do so many posters here seem to think that Goedel's theorem is so interesting, for instance?

 

[shmoe]

When it was first conjectured, it was thought that it would be easy to solve and was given to Hardy (I think, though I don't feel confident in saying that) as his PhD subject.

Close! It was given to Littlewood as a thesis problem by Barnes. It didn't have quite the same reputation in England at that point in time as it did in the rest of Europe. I can't recall exactly what Littlewood accomplished though I believe he independantly produced some results that were already known in continental Europe. Communcation was not so good in those days.

 

[heman]

i have come across this thing Reimann Zeta function once in my Complex number course...okay i try to see what was it..

 

[heman]

What do you mean by fuss? It is a celebrated open problem. One that is probably true (in my opinion) but which we frustratingly can't prove yet. I say 'frustratingly' because the evidence seems to say it should be true, and the corresponding statements over fields of positive characterstic have been known to be true for many years. When it was first conjectured, it was thought that it would be easy to solve and was given to Hardy (I think, though I don't feel confident in saying that) as his PhD subject.

There are also surprising links with many areas of mathematics and physics.

Oh, and it is relatively easy to explain to people, unlike, say, the Hodge Conjecture or the Birch-Swinnerton-Dyer Conjecture. Of course, the easy explanations don't really give the real picture. Quite what catches the general public's imagination about mathematics seems an unpredictable business. Why on Earth do so many posters here seem to think that Goedel's theorem is so interesting, for instance?

i guess from your words ,it will take me a quiet lot of time to realize the beauty of these theorems ...Actually i heard about that in some conference that's why i asked..

 

[kryptyk]

Structure

To me it seems that the essence of abstract algebra is the study of structure for its own sake. While most of mathematics makes use of algebra in some way or another, the algebra is thought to represent specific, concrete objects and operations on those objects (i.e. integers, vectors, matrices, tensors, etc...).

However, abstract algebra studies the inherent structures of these operations on sets without regard for the particulars of representations or physical manifestations of such structures. As such, it allows us to focus on structure abstractly which sometimes leads to surprising results when we can establish correspondences between the structures of seemingly very different systems or representations. I would say perhaps the most fundamental concept of all in abstract algebra is that of ISOMORPHISM.

It is often the case that certain systems or representations of systems reveal certain structures more clearly to our minds than others. For example, the complex numbers under multiplication exhibit the exact same structure as the group of scalings and rotations of vectors on the plane under composition. Complex numbers are easier to work with symbolically but scalings and rotations on the plane are easier to visualize. Establishing an isomorphism between these two systems allows us to freely move back and forth between them giving us the best of both worlds.

For more intricate systems there could potentially be many useful representations or ways to visualize them. Abstract algebra allows us to not only move between representations - it also often allows us to construct entirely new representations to highlight certain aspects of a system's structure or to focus on specific substructures of interest.

Also, by establishing general results for abstract structures we can immediately apply these results to all problems exhibiting similar structure rather than having to prove them for each system separately.

 

[mathwonk]

i agree with what was just said, but not with the remarks that abstract algebra was conceived without applications in mind. this disagrees with what little i know about the origins of abstract methods in algebra by hilbert and his student emmy noether.

hilbert was trying to deal with concrete questions in invariant theory when he developed abstract methods of reasoning about polynomial rings. Noether extended this study to what are now called "noetherian rings", and applied abstract methods to study also homology groups of manifolds, possibly in an attempt to make more precise and invariant, the study of manifolds in topology.

homology theory in topology had originally been a combinatorial theory that had trouble being proved a homeomorphism invariant. Apparently until Noether, homology groups were originally just numerical invariants, torsion numbers and ranks.

abstract algebra also got a boost from emmy noether's father, max noether, who translated riemann's ideas on riemann surfaces in complex analysis into pure algebra, along with brill, to render the theory independent of some foundations of real analysis (dirichlet's principle) which had not yet been provided. hilbert also provided these by the way.

van der waerden and emil artin and zariski and serre and grothendieck continued the development of abstract algebra to provide foundations for algebraic geometry, and render into algebra more tools from analysis, such as sheaf theory and its cohomology. serre's great paper GAGA is a monumental work translating analytic geometry into algebraic geometry and vice versa.

grothendieck and others showed how to use algebra to translate over into algebraic geometry basic concepts of differential topology like chern classes, and provided an abstract generalized riemann roch theorem, first proved by hirzebruch on complex manifolds using cobordism theory from topology.

abstract algebra methods also allow the study of singular spaces on the same footing with smooth spaces and manifolds, originally preferred in topology and differential geometry. this leads to a theory of degenerate varieties, a marriage of algebraic and topological singularity theory, and geometric compactifications of moduli spaces.

so in my opinion abstract algebra was developed as a means to avoid the highly tedious computations in classical algebra that had become unmanageable, as well as questionable foundations in analysis. then it continued as a way to bring the power of calculus and analysis and topology to bear on algebra and number theory, and algebraic geometry, via sheaf cohomology.

with the swing back in computing power, computational algebra is again in vogue. the foundations of analysis have also been rendered more sound, and algebra and analysis now cooperate ever more fully.