The growth of an infinite-dimensional algebra is a fundamental tool to measure its "size." The growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics, homological stability results, and more. 

We analyze the space of growth functions of algebras, answering a question of Zelmanov on the existence of certain 'holes' in this space, and provide evidence for the ampleness of the possible growth rates of algebras with prescribed properties; we conclude a strong quantitative solution of the Kurosh Problem on algebraic algebras.

Utilizing new layers of the interplay between noncommutative algebra and symbolic dynamics, we exhibit surprising pathologies in the prime spectrum and tensor product structure of algebras with polynomial growth, thereby providing counterexamples to questions of Bergman, Krause, Lenagan, and others; applying our methods to algebras of faster growth types, we resolve a conjecture of Bartholdi on amenable representations in exponential growth.

Finally, the largest objects (groups, algebras, Lie algebras) are, in many contexts, those containing free substructures. We discuss the coexistence of this phenomenon with finiteness properties -- in particular, "almost algebraicity" of algebras and "almost periodicity" of groups -- from algebraic, geometric, and probabilistic perspectives.

This talk is partially based on joint works with Bell, Goffer, and Zelmanov.