# Abstract Algebra

1. Dec 29, 2005

### 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 theoritical 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..

2. Dec 30, 2005

### 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 thoery, elementary particles are matrices, even statistical mechanics can use the theory of algebra(s).

3. Feb 20, 2006

### heman

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

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

4. Feb 20, 2006

### matt grime

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*

5. Feb 20, 2006

### 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..

6. Feb 20, 2006

### Hurkyl

Staff Emeritus
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.

7. Feb 20, 2006

### devious_

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

8. Feb 20, 2006

### heman

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

what actually invokes interest i dont 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..

9. Feb 20, 2006

### 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.

10. Feb 20, 2006

### 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..

11. Feb 20, 2006

### 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.

Last edited: Feb 20, 2006
12. Feb 20, 2006

### 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...

13. Feb 21, 2006

### 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 realised that much of the mathematics necessary for it was already there in the form of abstract algebra. Interesting isnt 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 dont 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 cant 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

14. Feb 21, 2006

### 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 cant tell

15. Mar 23, 2006

### heman

Guys whats this fuss about Reimann hypothesis...is it something which is unsolvable..

16. Mar 23, 2006

### shmoe

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

17. Mar 23, 2006

### 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?

18. Mar 23, 2006

### shmoe

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.

19. Mar 23, 2006

### heman

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

20. Mar 23, 2006

### heman

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