Will Johnson

I'm a recent graduate of the UC Berkeley math department. I finished my PhD in 2016, studying model theory under Tom Scanlon.

CV

Teaching

Research interests

• Model theory of valued fields
• Applications of model theory to arithmetic geometry and number theory
• Theories without the second tree property (NTP₂)
• Stable groups
• Pseudofinite fields and ACFA
• Questions related to Hilbert's 10th problem

Thesis work

My PhD dissertation Fun with Fields, was a compilation of several projects I worked on in graduate school, mostly related to the model theory of fields.

• Dp-minimal fields: I constructed a canonical field topology on any unstable dp-minimal field, and classified dp-minimal fields up to elementary equivalence of pure fields. See chapter 9 of the dissertation, or an earlier draft here.
• Fields with several valuations: chapter 11 contains, among other things, a proof that algebraically closed fields with several independent valuations have NTP₂ (they do not have the second tree property), as well as an analysis of forking and dividing in these structures. An older draft is here. This result probably holds without the independence assumption, as well as when "algebraically closed" is replaced with "real closed" or "p-adically closed." Ultimately I would like to extend these results to a continuous logic setting: my hope is that there is a continuous logic variant of Rumely domains, with NTP₂.
• O-minimal imaginaries: I found a counterexample to the hypothesis that any interpretable set in a (densely ordered) o-minimal structure can be put in definable bijection with a definable set, even after naming parameters. This is a strong failure of elimination of imaginaries. On the other hand, I showed that interpretable sets look "locally" like definable sets, in a precise sense. See chapter 8 of the dissertation for full details. Alternatively, earlier drafts of the counterexample are here or in these slides, and the "local" discussion is here.
• Elimination of imaginaries in ACVF: Building off work of Hrushovski, I helped shorten the proof of elimination of imaginaries in ACVF (in the geometric sorts). See chapter 6 of my dissertation, or an earlier version, or Jim Freitag's summary of a talk I gave in the Berkeley Model Theory seminar.
• Counting mod N in pseudofinite fields: the non-standard mod N size of definable sets in pseudofinite fields is definable in families. See chapter 12 of the thesis, or an earlier draft here.

Other projects and drafts

• A shorter proof that geometric irreducibility is definable in families, in ACF. A more polished version is in chapter 10 of my dissertation, as well as the appendix to Freitag, Li, and Scanlon's upcoming paper Differential Chow varieties exist.
• I'm trying to apply techniques from homotopy theory to study groups of finite Morley rank. I've defined a notion of etale cohomology of a group interpretable in an almost strongly minimal theory, and shown that it generalizes usual etale cohomology for ACF. The current definition is unwieldy and very hard to prove anything about, so I've been trying to learn more homotopy theory to better understand the "correct" definition. So far, I've shown that the first cohomology group is bounded and is related to central extensions.
• In C-minimal expansions of ACVF, T^eq eliminates ∃^∞. From the proof, one can extract a general criterion for when T^eq eliminates ∃^∞, which might be useful in other settings. (A more complete discussion is in chapter 5.1 of my dissertation.)
• For global invariant types in C-minimal expansions of ACVF, orthogonality to the value group is equivalent to generic stability is equivalent to stable domination, and the class of stably dominated types is strictly pro-definable. A better version of these notes is in chapter 7 of my dissertation. Hrushovski has proven more general results in some unpublished notes.
• A quick proof of quantifier elimination in ACVF, based on notes of Hrushovski if I recall correctly. Another version is in chapter 2 of my dissertation.
• Notes from a talk I gave in the Berkeley Model Theory seminar on the Stabilizer Theorem from Hrushovski's paper Stable Group Theory and Approximate Subgroups.

For my undergraduate senior thesis (at the University of Washington), I studied the combinatorial game theory of a certain class of "scoring games," specifically the ones in which the total length of the game is fixed at the outset. My very bloated senior thesis is available on the arXiv here. A more condensed version is my article The Combinatorial Game Theory of Well-tempered Scoring Games. A longer preprint arXiv version is here. I worked on this at the University of Washington Mathematics REU, where I also did work related to inverse problems in in nonlinear resistor networks.