Jacob Davis successfully defended his dissertation, “Universal Graphs at ${\mathrm{\aleph }}_{{\omega }_1+1}$ and Set-theoretic Geology,” at Carnegie Mellon University on April 29, 2016, under the supervision of James Cummings. I was on the dissertation committee (participating via Google Hangouts), along with Ernest Schimmerling and Clinton Conley.

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The thesis consisted of two main parts. In the first half, starting from a model of ZFC with a supercompact cardinal, Jacob constructed a model in which $2^{{\mathrm{\aleph }}_{{\omega }_1}}=2^{{\mathrm{\aleph }}_{{\omega }_1+1}}={\mathrm{\aleph }}_{{\omega }_1+3}$ and in which there is a jointly universal family of size ${\mathrm{\aleph }}_{{\omega }_1+2}$ of graphs on ${\mathrm{\aleph }}_{{\omega }_1+1}$.  The same technique works with any uncountable cardinal in place of ${\omega }_1$.  In the second half, Jacob proved a variety of results in the area of set-theoretic geology, including several instances of the downward directed grounds hypothesis, including an analysis of the chain condition of the resulting ground models.