Computability theory (also known as Recursion Theory) formalizes what computation is. In some of my work, I study the "balance scales" used to compare computational information. But I also use these calibration tools to measure the computational power of specific problems about mathematical structures -- such work is termed Computable Structure Theory. I particularly focus on algebraic and discrete structures (such as linear orderings, real and algebraically closed fields, and integer parts).
These talks of mine highlight the big ideas of computability theory:
Climbing (or finding paths through) trees: Computing the difficulty of mathematical problems (47 minutes) SUMS@JMU 2021 Conference (aimed at undergraduates)
Different Problems, Common Threads: Computing the difficulty of mathematical problems (56 minutes) JMM 2020 Conference (aimed at mathematicians)
See also Math Horizons articles in "Work with students" below.
Classifications of definable subsets, S. Boyadzhiyska, K. Lange, A. Raz, R. Scanlon, J. Wallbaum, and X. Zhang, Algebra Logic 58(5) (2019), 383–404. (also listed above)
For more student opportunities, click here to visit Student Resources.