Patrick Lutz

Research

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

Papers (published/submitted)

• Incompleteness and jump hierarchies
  with James Walsh [arXiv] [journal]
• Formalizing Galois theory
  with Thomas Browning [arXiv] [journal] [slides]
• Conway and Doyle can divide by three, but I can’t
  [journal] [slides]
• Martin’s conjecture for regressive functions on the hyperarithmetic degrees
  [arXiv]
• Part 1 of Martin’s conjecture for order preserving and measure preserving functions
  with Benny Siskind [arXiv] [slides]
• A note on a conjecture of Sacks
  with Kojiro Higuchi [arXiv] [slides]
• The Solecki dichotomy and the Posner-Robinson therem are almost equivalent
  [arXiv] [slides]
• Coding information into all infinite subsets of a dense set
  with Matthew Harrison-Trainor and Lu Liu [arXiv] [slides]

Preprints

• A theory satisfying a strong version of Tennenbaum’s theorem
  with James Walsh [draft]

In preparation

• Local Martin’s conjecture, revisited
  with Vittorio Bard
• Counterexamples concerning the Hausdorff dimension of continuous images
  with Joe Miller [slides]

Theses

• 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)

Slides

• Survey of some recent work on Martin’s conjecture
  [Survey]
• Part 1 of Martin’s conjecture for measure preserving functions and order preserving functions
  [Overview] [Measure preserving functions] [Ultrafilters on the Turing degrees] [Order preserving implies measure preserving]
• Embedding partial orders into the Turing degrees: Height 2 vs. height 3 partial orders
  [Overview]
• The Solecki dichotomy and the Posner-Robinson theorem
  [Overview]
• Hausdorff dimension and continuous images
  [Overview]
• Coding information into all infinite subsets of a set
  [Main theorem] [Kolmogorov complexity version]
• Division by two without choice
  [Sock division vs. shoe division]
• Formalizing Galois theory in Lean
  [Progress report from January 2021]