Thread: Algebra in Applied Math

I like to divide math into 3 disjoint sets: Algebra, Analysis (Calculus), Geometry (Topology). *

Now the most useful one is Analysis. The use of partial differencial equation and Fourier series is one of the big uses in applied math. As well as integration.

It is also true that Geometry is used in applied math. For instance, Differencial Geometry and I am sure certain concepts from Topology are used as well.

But how about Algebra? That is my favorite out of the 3 (or 4) and is sad to see that it is possibly the smallest one by far. I cannot possibly image how Field and Galois theory or Cyclotomic Extensions, Abelian Varieties are used. Can someone give examples?

This is true for all except Linear Algebra. I seen it used heavily by JakeD. And I have seen online lectures by Professor Strang from MIT University. I am convinced it is extremely important. There are many strange concepts about matrices: transposes, orthogonality, decompositions .... I never used them nor have any idea what they are since Linear Algebra never interested me**.
So my question is what is so special about this "strange" linear algebra that is used by applied mathematicians?

*)Sometimes 4. We can think of Combinatorics as the other fourth one. In fact, it was a great honor when I saw MathWonk on PhysicsForums make the same classification.

**)Except for Vector Spaces and Inner Product Spaces. Which is used a lot in pure math.

Originally Posted by ThePerfectHacker

So my question is what is so special about this "strange" linear algebra that is used by applied mathematicians?

Linear Algebra is about linear operators on (usualy finite dimensional) vector
spaces, well almost all of applied maths is set in vector spaces of one sort or
another, and often finite dimensional.

Also as linear problems are so much easier in general that non-linear often
the first thing one does with a problem is linearise it.

RonL

Hacker, you are so biased But in general terms, that's a good classification.

For me also, there three kinds of Maths. The little I understand, those I don't really understand but can deal with, and those I can't understand for the life of me

The latter includes algebraic geometry, algebraic logic and aspects of algebraic topology
(funny, it all includes algebraic in it!!! guess I found my weakness )

Originally Posted by Rebesques

The latter includes algebraic geometry, algebraic logic and aspects of algebraic topology
(funny, it all includes algebraic in it!!! guess I found my weakness )

How good are you in Field theory? Perhaps your weakness is that your knowledge of this is not so strong? Is it?

Note: I am planning to study algebraic geometry soon. But I am afraid, lest their be topology in it. From books I gazed at it seems so (the Zarski Topology). I am at a point where I need a topic to self-study after this following semester is over. All of them are algebraic . Either algebraic number theory, projective geometry or algebraic geometry. But I just cannot decide. I hate when that happens.

I believe Munkres Topology is the classic intro textbook. Linear algebra has many uses in graph theory and economics/ linear programming. Also numerical linear algebra has its uses (i.e. the problem of decomposing a given matrix to the following . That problem had to do with image compressing and embedding. Here

Munkres Topology

Loved it. Any other classics we share?

How good are you in Field theory

Funny u should say that. I remember three major classes I had to take in the first semester of my 3rd year as an undergraduate: Real Analysis, Differential Geometry, Field Theory. Did bad at the first two and really bad at the third But I did well at Ring Theory, I still have bad dreams about exact sequences and Noetherian chains.

Here are the books I like:

Spivak Calculus
Courants Differential and Integral Calculus vols. 1 & 2
Courant and John Intro to Analysis vols. 1 &2
Apostol Calculus with linear algebra vol 1 & 2
Linear Algebra by Hoffman/Kunze
Linear Algebra by Sheldon Axler
An Introduction to Mathematical Reasoning by Peter J. Eccles
Problem Solving Strategies by Engel

Topology by Munkres
Abstract Algebra by Dummitt/Foote
Principles of Mathematical Analysis by Rudin
Analysis by Pugh
Intro Analysis by Ross
Mathematical Analysis by Tom Apostol
An Introduction to Probability Theory by Feller
Ordinary Differential Equations by V'Arnold
Combinatorics by R. Merris

Originally Posted by Rebesques

Funny u should say that. I remember three major classes I had to take in the first semester of my 3rd year as an undergraduate: Real Analysis, Differential Geometry, Field Theory.

Maybe you do not like Field Theory, but it is one of the bestest things I ever learned. And I have learned a lot, so it is fair for me to make such a judgement.

Why? Where you taking Graduate courses as an Junior? You must be really good.

Have you ever done any of the three algebraic ones I mentioned in my other post? Which ones should I do?

Originally Posted by tukeywilliams

Abstract Algebra by Dummitt Foote

Did you ever buy that book? I like it, especially the additional sections like Algebraic Geometry and Representation Theory. I would buy it, it is just it I know everything except those two additional sections. And it is expensive so it would be a waste of money to me.


Here is what I want.
And more.
Even more.
And still more.
And the last one.

Just sharing my fantasies.

I checked it out of the library (currently self studying abstract algebra from that text).

You may want to take a look at the abstract algebra books by Artin or Hungerford.

Originally Posted by tukeywilliams

I checked it out of the library (currently self studying abstract algebra from that text).

You may want to take a look at the abstract algebra books by Artin or Hungerford.

Oh and I forgot.

Beachy and Blair.

tukey:

Spivak, Courant, Apostol I know Dang good books! The other ones I regret not having opened (besides V' Arnold, I just can't read it to the end ) Thanks for that list!

Artin

Oh! Had to study some pages out of that one for an exam, a green and slender nerve-wrecker, isn't it?

hacker:

Why? Where you taking Graduate courses as an Junior? You must be really good.

I am tempted to say yes but not really. The courses were there for the undergraduates, and stupidly hard on purpose. I know this, because I later
took the Graduate courses.

Either algebraic number theory, projective geometry or algebraic geometry

I am honoured you trust my judgement. Well, projective geometry is not a big subject in itself, and there's much more to its structures if you get there by the Differential Geometry path. Algebraic Number Theory is nice, but still rather limited.

...And Algebraic Geometry is just huge. There's just so many topics, and some very hard.

So if you are just looking for somethinh hard, why don't you skip all three and try Lie Groups and Algebras?

I wanted to study that... *Sob*

Originally Posted by Rebesques

Well, projective geometry is not a big subject in itself, and there's much more to its structures if you get there by the Differential Geometry path.

It also appears in the study of elliptic curves. Thus, I assumed it would be easier for me to start with Projective Goemetry.

Algebraic Number Theory is nice, but still rather limited.

I think you are wrong about that. Modern Number Theoresits do work with this area a lot.

So if you are just looking for somethinh hard, why don't you skip all three and try Lie Groups and Algebras?

I am not looking for something that is hard. I am looking for something I would like. I know that Algebraic Number Theory is something I would really like.

Dang, there goes my devious attempt to make a proper Geometer out of you!

I am looking for something I would like. I know that Algebraic Number Theory is something I would really like.

I guess you are right. Just don't want to see you make a living by breaking Russian/Chinese/North Korean codes.