This is a soft question. I play classical guitar and I find stringed instrument industry a very fascinating art. I know that, at least for classical guitar, this industry is still developing and exploring new techniques, so it is a very alive research field.
I know that it must involve at least some basic mathematics, but I was wondering if anyone has tried to apply deep mathematical tools and concepts in this field, for example from geometric analysis.
On the other hand I guess that these techniques could inspire very interesting mathematical questions.
What is the state of art of the interaction between maths and stringed instrument industry?
Not sure of what your question is. There are the 3 major applied.maths/music topics :
Musical signal processing (automatic transcription, recognition).
A.I. and machine learning (automatic improvisation, computer assisted composition)
Physics & acoustic of musical instruments and sound synthesis.
The last one is special because the human voice is a musical instrument, and voice synthesis and analysis is a very active research field since more than 40 years.
All those subjects involve many mathematical tools : harmonic analysis (Fourier transform), differential equations & operator theory, convex optimization, probability and statistics, graph or group theory..
See this for a good start on the physics of stringed instruments.