Currently I am deciding whether I want to do my thesis on graduate algebra next year and I'd like to study something with applications to either physics or computer science (two relatives of mathematics). I have heard that graduate algebra has found applications in automata theory and to system communications in electrical engineering. Therefore, I'd like to be informed of some examples of the recent applications of advanced algebra (non-commutative algebra or commutative algebra) to other fields of science or engineering or just recent research trend motivated by breakthrough problems in disciplines other than math.
I know little about computer science, but if you include category theory among the subjects of algebra (as Hungerford does), then that subject apparently has a lot of applications to that field. I can say more, though, about applications to physics, if that is among your interests:
Lie groups are used to express symmetries in physics. Symmetries occupy a central role in physics especially in gauge theories in quantum field theory. The basics of gauge theories can be found in chapter 10 of Introduction to Elementary Particles by Griffiths. There is also Quantum Theory, Groups and Representations: An Introduction by Woit, which is more in-depth and discusses more topics (still focused on the idea of symmetry in physics). This of course also has a fair amount of crossover with differential geometry, since Lie groups belong both to algebra and geometry.
There are also applications of C*-algebras and von Neumann algebras to quantum physics. Quantum physics itself, is of course built upon linear algebra and functional analysis.