Abstract

This dissertation brings together results in Hodge theory and the theory of cluster varieties. The second chapter, based on a paper written in collaboration with Devin Akman and Matt Kerr, uses admissible normal functions to establish the first finiteness result for zero-dimensional components of the Hodge locus. The third chapter, based on joint work with Matt Kerr, shows that weak polar- ized relations constrain possible adjacencies of mixed Hodge structures across boundary strata in geometric compactifications. The fourth chapter gives a geometric construction of a 6-dimensional cluster variety with a disconnected mutation graph, as discovered by Yan Zhou. These three chap- ters employ different techniques, but they are tied together by the common threads of algebraic cycles and the Hodge theory of degenerations.

Committee Chair

Matthew Kerr

Committee Members

Alessio Corti; Charles Doran; Roya Beheshti Zavareh; Wanlin Li

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

5-2-2025

Language

English (en)

Author's ORCID

https://orcid.org/0000-0002-9748-2020

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Mathematics Commons

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