## Departmental PhD Thesis Exam – David Urbanik

Thursday, August 11, 2022 at 10:00 a.m. (sharp)

PhD Candidate: David Urbanik
Supervisor: Jacob Tsimerman
Thesis title: Algebraic Cycle Loci at the Integral Level

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Suppose f is a smooth projective family from X to S defined over the ring of integers I of a number field K. For each prime of I with residue field k, we consider the algebraic loci in S_k above which cohomological cycle conjectures predict the existence of non-trivial families of algebraic cycles, generalizing the Hodge loci of the generic fibre S_K. We develop a technique for studying all such loci, together, at the integral level. As a consequence we give a non-Zariski density criterion for the union of non-trivial ordinary algebraic cycle loci in S. The criterion is quite general, depending only on the level of the Hodge flag in a fixed cohomological degree w and the Zariski density of the associated geometric monodromy representation.

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A draft of the thesis can be found here: David Urbanik – Math PhD Thesis