I will give an overview of recent developments in the study of Hodge-theoretic aspects of Alexander-type invariants associated with smooth complex algebraic varieties. Our results are motivated by (and can be regarded as global analogues of) similar statements for the Milnor fiber cohomology of complex hypersurface singularity germs. (Joint work with E. Elduque, C. Geske, M. Herradon-Cueto and B. Wang).

The classical Gabriel-Ulmer duality asserts a dual adjoint equivalence between finitely complete categories, and a full sub-2-category of the complete, filtered-cocomplete categories which are known as locally finitely presentable (LFP). The definition of LFP category involves ``exactness conditions'' asserting compatibility between limits and filtered colimits, together with a different sort of condition which amounts to admitting enough structure-preserving functors to Set; removing this last condition yields the continuous locally presentable (CLP) categories in the sense of Johnstone-Joyal. We prove an analog of Gabriel-Ulmer duality for all CLP categories, by replacing the dualizing category Set with the category CPUMet of complete partial ultrametric spaces. As with Gabriel-Ulmer duality, this result has a logical interpretation, as a strong conceptual completeness theorem for the ``lex fragment'' of a continuous first-order logic for partial ultrametric structures.

Central objects in the study of elliptic curves are the fields of definition of the points of order n. Relatively little is known about the way in which a fixed elliptic curve's division fields can intersect. In this talk, we lay the foundations for a systematic study of the entanglements of division fields of elliptic curves from a group theoretic perspective. In particular, we classify the way in which two prime-level division fields can intersect for infinitely many elliptic curves. This is joint work with Jackson S. Morrow.