Patrick Lutz

Research

Here is a list of research questions which I think are interesting.

Current projects (with slides)

• Hyperfinite partial orders
  with Matthew Harrison-Trainor
  [slides]
• Borel diagonalization
  [slides]
• Measure hyperfiniteness and lossless expansion
  with Jan Grebík
  [slides]
• Complexity of oracles for packing dimension
  [slides]
• Hausdorff dimension and continuous images
  with Joe Miller
  [slides]

Preprints

1. Nonembeddings of combinatory algebras
   with Paul Shafer and Sebastiaan Terwijn
   [arXiv]
2. A Borel graphable equivalence relation with no Borel graphing of diameter two
   [arXiv]
3. Algorithmic information bounds for distances and orthogonal projections
   with Peter Cholak, Marianna Csörnyei, Neil Lutz, Elivra Mayordomo and D. M. Stull
   [arXiv]
4. Ranking theories via encoded β-models
   with Hanul Jeon, Fedor Pakhomov and James Walsh
   [arXiv]
5. A theory satisfying a strong version of Tennenbaum’s theorem
   with James Walsh
   [arXiv] [slides]
6. The Solecki dichotomy and the Posner-Robinson theorem are almost equivalent
   [arXiv] [slides]
7. A note on a conjecture of Sacks
   with Kojiro Higuchi
   [arXiv] [slides]

Papers

8. Bounding the dimension of exceptional sets for orthogonal projections
   with Peter Cholak, Marianna Csörnyei, Neil Lutz, Elivra Mayordomo and D. M. Stull
   [arXiv]
9. Borel graphable equivalence relations
   with Tyler Arant and Alexander Kechris
   [arXiv] [journal] [slides]
10. Coding information into all infinite subsets of a dense set
    with Matthew Harrison-Trainor and Lu Liu
    [arXiv] [slides 1] [slides 2]
11. Conway and Doyle can divide by three, but I can’t
    [arXiv] [journal] [slides]
12. Formalizing Galois theory
    with Thomas Browning
    [arXiv] [journal] [slides]
13. Part 1 of Martin’s conjecture for order preserving and measure preserving functions
    with Benjamin Siskind
    [arXiv] [journal] [slides]
14. Martin’s conjecture for regressive functions on the hyperarithmetic degrees
    [arXiv] [journal]
15. Incompleteness and jump hierarchies
    with James Walsh
    [arXiv] [journal]

Thesis

• Results on Martin’s conjecture
  PhD thesis, UC Berkeley 2021 [pdf]
  Comment: Liang Yu has found a counterexample to Conjecture 5.36 (a.k.a. question 9.10)

Other slides

• Martin’s conjecture
  [Survey of some recent work on Martin’s conjecture]
• Part 1 of Martin’s conjecture
  [Overview] [Measure preserving functions] [Ultrafilters on the Turing degrees] [Order preserving implies measure preserving]
• Part 2 of Martin’s conjecture
  [Martin’s conjecture above the hyperjump]