## Departmental PhD Thesis Exam – Kenneth Chiu

Wednesday, June 1, 2022 at 11:00 a.m. (sharp)

PhD Candidate: Kenneth Chiu
Supervisor: Jacob Tsimerman
Thesis title: Functional transcendence in mixed Hodge theory

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Ax-Schanuel theorem is a function field analogue of the Schanuel’s conjecture in transcendental number theory. Building on the works of Bakker, Gao, Klingler, Mok, Pila, Tsimerman, Ullmo and Yafaev, we extend the Ax-Schanuel theorem to mixed period mappings. Using this together with the Ax-Schanuel theorem for foliated principal bundles by Blázquez-Sanz, Casale, Freitag, and Nagloo, we further extend the Ax-Schanuel theorem to the derivatives of mixed period mappings. The linear subspaces in the Ax-Schanuel theorem are replaced by weak Mumford-Tate domains, which are certain group orbits of mixed Hodge structures. In particular, we prove that these domains have complex structures, and that their real-split retractions can be decomposed into semisimple and unipotent parts. We prove that the image of a mixed period mapping is contained in the weak Mumford-Tate domain that arises from the monodromy group of the variation. O-minimal geometry, namely the definable Chow theorem and the Pila-Wilkie counting theorem, are used in the proof of our extension of the Ax-Schanuel theorem.

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The draft of the thesis can be found here:Thesis