Real-world applications of fields, rings and groups in linear algebra. 
Real-world applications of fields, rings and groups in linear algebra.

A friend of mine asked me where one could use the definitions of rings, groups, fields etc. I was very embarrassed of the fact that I could only mention cyber security - nothing more (I'm studying IT). That's why I would like to get some more detailed suggestions. I would really appreciate that.
Thank you in advance!
 A: Well, I'd consider the brand new book from Gilbert Strang: ''Linear Algebra and Learning from Data'', Cambridge Univ. Press, 2018. His learning from data culminates into the construction of deep neural networks.
Another application besides cryptography is coding theory, where finite fields are used to define linear codes. There is also a decent generalization to linear codes over the ring of integers modulo 4, which give rises by the Gray map to nonlinear binary codes which are better than any linear code with the same block length.
A: There's the GraphBLAS project [0] which is for graph algorithms expressed as linear algebra (see also [1]). It turns out that many such algorithms can be done this way if you change the semiring over which the matrices live.  See for example this TOMS pre-print [2].
[0] http://graphblas.org
[1] Graph Algorithms in the Language of Linear Algebra, Kepner&Gilbert(ed), https://epubs.siam.org/doi/book/10.1137/1.9780898719918
[2] http://faculty.cse.tamu.edu/davis/GraphBLAS_files/toms_graphblas.pdf
A: Error-Correcting Codes are a nice application. In transmitting data (e.g., email, cellular, to space, etc.), data can get damaged. Recovering the transmission is very much a nice to have. For quantum computing, error-correction is actually a necessity since quantum computing is inherently probabilistic.
See my answer here on the interactions between Group Theory and Error-Correction Codes (What does group theory add to the understanding of error correcting codes?).
Algebraic Structures like Tensors (Multi-linear Algebra) also come up quite a bit in Machine Learning (https://arxiv.org/pdf/2202.06930.pdf).
Another application that comes to mind is Fourier Analysis. The Discrete Fourier Transform is an isomorphism from an Abelian group to its Dual Group. Another way to think about the Fourier transform is a change of basis into the character basis. For Abelian groups, the characters form an orthonormal basis of the vector space of functions from $G \to \mathbb{C}$. These find a whole host of applications in areas like applied math, electrical engineering, and quantum computing. The big reason we care about quantum computing is b/c there is an efficient quantum algorithm to compute a Fourier transform on a finite Abelian group. This is the key piece to obtain Shor's quantum factoring algorithm.
