Algebra and logic have a long history together. Early work of Tarski on elimination theory and the decidability of theories of algebraic objects has led to a large and active field today.
Computable model theory now has an extensive body of results on Abelian groups, on rings, and on fields. There is still plenty to do on the difficulty of recognizing a particular structure (up to isomorphism, for instance) in a class of similar structures, and on the existence of computable embeddings of one structure in another.
Homotopy Type Theory is an exciting new area along the borders of algebra, logic, and topology. A recent book, available free in electronic form describes the program in considerable detail. It promises new insights into many areas of mathematics by reformulating them as questions about topology and computation --- which, in the sense of Homotopy Type Theory, are closely related. It would be interesting, for instance, to see what could be learned by stating parts of the Langlands Program in the Homotopy Type Theory context.
Algorithmic randomness also has close connections to certain parts of number theory. In work presently underway with Valentina Harizanov and Alexandra Shlapentokh, I'm trying to understand what a "random" algebraic field looks like. There are various other algebraic structures that may have random examples, and they would be good to know about. In many respects, an algorithmic example of something has enough regularity to be described and explored well, while somehow capturing the essence of what is typical of objects of that kind.
On the other hand, algorithmically random numbers have interesting relationships with transcendental number theory. Normal numbers, UD-random numbers, and other concepts give us a way to explore the very difficult territory around transcendence and independence.
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