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by teleforce·5y ago·view on hn ↗
>Mathematical Challenge Eight: Beyond Convex Optimization * Can linear algebra be replaced by algebraic geometry in a systematic way?

I think this is probably by far the most useful, practical and relevant challenge to be solved for science and engineering, and perhaps closely follows by the stochastic and duality challenges.

For computer science and engineering it is the popular "Gimbal lock" problem in 3-D environment in which linear algebra cannot comprehensively represent but easily represented by geometric algebra or quaternions.

Similarly in electromagnetics (EM) wave propagation, due to the prominent effect of polarization (other waves like sound does not has polarization), comprehensively modeling polarization with linear algebra is close to impossible. I kind of liken the geometric algebra unpopularity and conundrum similar to 18th mathematicians suspicious views when complex number was originally introduced and looks how far we have got now by embracing it [1]. Basically the discovery and utilization of complex number provide us with WiFi 6 and 5G. But if we want to move forward with robust and reliable wireless similar to wired (or close to wired connection reliability) we need to take control of EM polarization by embracing geometric algebra.

[1]https://en.m.wikipedia.org/wiki/Complex_number#History

2 comments
It seems that what it meant here by algebraic geometry is different compared to the terminology of geometric algebra. In this case geometry algebra is just one of the tools for algebraic geometry albeit a very useful one. But the points on its importance is still valid IMHO.

[1]https://www.reddit.com/r/math/comments/ddqt6f/algebraic_geom...

Any good resources on linear algebra and algebraic geometry? I think I've read about it before but when I look it up now there's few hits coming up for me for some reason.