# When does injectivity imply surjectivity?

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Lewis Bowen, University of Texas, Austin & Princeton University
Fine Hall 314

Any injective map from a finite set to itself is surjective. Ax's Theorem extends this to algebraic varieties and regular maps. Gromov invented sofic groups as a way to extend to this result to cellular automata and other settings. We'll re-prove his results via sofic entropy theory. Similarly we use sofic mean dimension to prove non-embeddability results in topological dynamics and $l^p$ dimension to prove non-embeddability results in representation theory.

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