/sci/ - Science & Math

No advanced algebra is heavily used in cryptography but intro-grad-level algebra is used in things like the elliptic curve Diffie-Hellman, which is currently one of the most widely used public-key internet protocols. You can check this for yourself by monitoring your PC's traffic with Wireshark. Before some retard says that you don't need graduate-level algebra to understand basic elliptic curves, you definitely need it to formulate and understand attacks against the ECDH, which is a subject matter every cryptographer should be capable of handling.

More complex tools in algebraic geometry have started to appear in crypto literature invoking many pre-Gronthendieck tools, as a response to potential weaknesses in the discrete-log hardness assumption.

In physics, there's heavy use of multilinear algebra (frequently taught in general form in intro-grad-level abstract algebra courses) in physics in the form of exterior algebra. In quantum mechanics, which is predominantly functional analysis, algebra plays a secondary role in the sense that it forms the algebraic structure of actions on the vectors and covectors of study. Gauge theory and gauge invariances are also closely linked to algebraic topology concepts like homology and cohomology, which in theory could be learned without any algebra but you'd be better served learning it with.

Lie groups and Lie algebras heavily feature in robotics, control theory, and physics as well. This too can be "learned" without formally studying algebra past basic definitions of groups and rings, but it'd serve your understanding of the subject not to do this.

That's just off the top of my head, and I'm pretty far removed from algebra. There's likely quite a bit more.