Justin D. Miller

I am a John Wesley Young Research Instructor at Dartmouth College.

My research interests are primarily in mathematical logic and computability theory. They include asymptotic computability, algorithmic randomness and stochasticity, and reverse mathematics. I am also interested in computability as it relates to infinite combinatorial structures, such as families of graphs and combinatorial games.

My office is 315 Kemeny Hall. I can be contacted via email at justin DOT d DOT miller AT dartmouth DOT edu.

My CV is available here.

Teaching

I am happy to meet with students throughout the term. My office hours are Wednesday from 3:30-5pm, and Friday from 1-2pm and 3:30-4:30pm. I am also available by appointment. We will meet in my office, 315 Kemeny Hall, as space permits, or in our classroom 008 Kemeny if the space is available. Depending on attendance, we may need to move elsewhere, which I will do my best to communicate.

I am happy to schedule additional time to meet with students as necessary.

  • During Fall Term 2023, I am teaching Math 3
  • During Spring Term 2023, I taught Math 17 and Math 19
  • During Fall Term 2022, I taught Math 3
  • During Spring Term 2022, I taught Math 29
  • During Winter Term 2022, I taught Math 13
  • During Fall Term 2021, I taught Math 13

Research

Here is a list of my publications, with preprints where available. The status of some resolved open questions can be found here.

  • Computability of chip-firing games on graphs. In preparation. Joint with Liling Ko and David Belanger.
  • Ordinal arithmetic over \( RCA_0^* \). In preparation. Joint with Marcia Groszek and Ben Logsdon.
  • Comparing disorder and adaptability in stochasticity. Submitted. Joint with Liling Ko. ArXiv
  • Intermediate Intrinsic Density and Randomness. Computability. 10(4):327-341, 2021. doi:10.3233/COM-210309. ArXiv
  • Intrinsic Smallness. The Journal of Symbolic Logic, 86(2), 558-576, 2020. doi:10.1017/jsl.2020.39. ArXiv

My PhD dissertation is available here.

Talks

Below is a list of talks I have given at meetings and conferences, with slides where available.

Open Questions

This is a list of solutions to open questions appearing in previous work, along with links and attributions where necessary.

Last Updated: 8/1/2023

Intrinsic Smallness

  • Previously Question 4.2: There is an arithmetically intrinsically small set which is not \(\emptyset^{(\omega)}\)-intrinsically small. In fact, there is an \(\emptyset^{(\omega)}\)-computable set with this property, as \(\emptyset^{(\omega)}\) can compute \(\emptyset^{(n+2)}\) uniformly in \(n\) and therefore can compute all arithmetic permutations uniformly.
  • Question 4.1: For all \(X\), the degrees of \(X\)-intrinsically small sets are exactly the \(X\)-high or \(X\)-DNC degrees. (Corollary 2.27 in my dissertation)

Dissertation

  • Previously Question 4.2.6: The degrees of intrinsic density \(r\) sets cannot be the high or DNC degrees for all \(r\) because there are uncountably many \(r\) but only countably many elements of each degree. This was observed by Denis Hirschfeldt.