Applied Algebra: Examples & Interesting Uses

[Skrew]

I hope this is appropriate as it is related to algebra but if it is not I pre-apologize.

I was wondering if someone would give examples of algebra being used in interesting ways, either in direct applications or applied to subjects which interest me(analysis/differential equations/dynamical systems)?

 

[henry_m]

Here's a few off the top of my head from your favourite subjects, and one from mine.

Analysis: there's a whole branch of topology that applies abstract algebra, the imaginatively titled subject algebraic topology. As the most basic example, (path connected) topological spaces have associated with them a group, called the fundamental group. This is basically the group of every type of closed loop you can make in the space, and to find the product of two loops you simply follow one and then the other. For example, a circle's fundamental group is the integers under addition: you just count the number of times you go anticlockwise round the circle, and clockwise counts negative. This sort of thing is very useful for classifying topological spaces, so it can be used, for example, to tell us what every closed connected surface looks like.

Dynamical systems/ODEs: elementary linear algebra is used to classify fixed points of systems of ODEs. You linearise the system at the fixed point, so near the point the system looks like x'=Ax for x in Rⁿ and A an nxn matrix. Then if the eigenvalues of A are all negative, the point is a sink, or an attractive or stable fixed point, if they're negative it's a source, and if there's some of each it's a saddle of some sort (and if some of them are zero there's a bit more work to do). If you think of your dynamical system as depending on a parameter, bifurcations usually appear at the parameter value for which an eigenvalue is zero. This happens, for example, in the famous Lorenz system of equations.

Finally, here's my favourite, from particle physics. Physical theories often come with symmetries (or approximate symmetries) depending on some continuous parameters, and these can be studied using algebra, specifically Lie algebras. By studying these structures and their representations (i.e. ways they can be written as matrices) you can find out loads about particles and their interactions (the omega minus baryon is a particle that was predicted to exist basically just using algebra!).

That's a little taster. I find abstract algebra quite hard and it can be a little dry in my opinion, but there's no denying that it's useful!

 

[Skrew]

Here's a few off the top of my head from your favourite subjects, and one from mine.

Analysis: there's a whole branch of topology that applies abstract algebra, the imaginatively titled subject algebraic topology. As the most basic example, (path connected) topological spaces have associated with them a group, called the fundamental group. This is basically the group of every type of closed loop you can make in the space, and to find the product of two loops you simply follow one and then the other. For example, a circle's fundamental group is the integers under addition: you just count the number of times you go anticlockwise round the circle, and clockwise counts negative. This sort of thing is very useful for classifying topological spaces, so it can be used, for example, to tell us what every closed connected surface looks like.

Dynamical systems/ODEs: elementary linear algebra is used to classify fixed points of systems of ODEs. You linearise the system at the fixed point, so near the point the system looks like x'=Ax for x in Rⁿ and A an nxn matrix. Then if the eigenvalues of A are all negative, the point is a sink, or an attractive or stable fixed point, if they're negative it's a source, and if there's some of each it's a saddle of some sort (and if some of them are zero there's a bit more work to do). If you think of your dynamical system as depending on a parameter, bifurcations usually appear at the parameter value for which an eigenvalue is zero. This happens, for example, in the famous Lorenz system of equations.

Finally, here's my favourite, from particle physics. Physical theories often come with symmetries (or approximate symmetries) depending on some continuous parameters, and these can be studied using algebra, specifically Lie algebras. By studying these structures and their representations (i.e. ways they can be written as matrices) you can find out loads about particles and their interactions (the omega minus baryon is a particle that was predicted to exist basically just using algebra!).

That's a little taster. I find abstract algebra quite hard and it can be a little dry in my opinion, but there's no denying that it's useful!

Thanks.

I am taking my last term of undergraduate abstract algebra and I loathe it, both because I felt it was sort of useless and because I am pretty terrible at it. Still I am happy to know it is used for something, it provides a little motivation at least.

 

[jasonRF]

google: coding ind information theory. Very important for communication systems. An example rigorous book:

https://www.amazon.com/dp/3540641335/?tag=pfamazon01-20

I don't know if the book is any good, but look inside and you will see that all your work isn't for nothing. I wish I know algebra!

jason