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

 

[blue_raver22]

Absolutely, there are numerous examples of how algebra is applied in various fields of science and mathematics. In fact, algebra is a fundamental tool in many areas of research and has numerous interesting and important applications.

One example is in the study of dynamical systems, which is a branch of mathematics that studies the behavior of systems that change over time. Algebra is used to model and analyze these systems, allowing us to make predictions and understand their behavior. For instance, in population dynamics, algebraic equations can be used to model the growth and decline of a population over time.

In the field of analysis, algebra is used to study and solve differential equations. These equations describe the relationship between a function and its derivatives and are used to model various physical phenomena such as motion, heat transfer, and fluid dynamics. By using algebraic techniques, we can find solutions to these equations and understand the behavior of these systems.

Another interesting application of algebra is in cryptography, which is the study of secure communication. Algebraic structures such as groups, rings, and fields are used to develop encryption algorithms that protect sensitive information and ensure secure communication.

Algebra is also used in computer science, particularly in the design and analysis of algorithms. By using algebraic structures and techniques, researchers are able to develop efficient algorithms for solving complex problems in various fields such as data analysis, machine learning, and optimization.

In summary, algebra has a wide range of applications in various fields of science and mathematics. Its use in dynamical systems, analysis, cryptography, and computer science highlights its versatility and importance in modern research. As a scientist, understanding and utilizing algebraic techniques is crucial in solving real-world problems and advancing our knowledge in various fields.