Patrick Lutz

Research

Here is a list of research questions I am interested in at the moment (right now it is very short, but I plan to add to it soon).

Papers

• Incompleteness and jump hierarchies
  with James Walsh [arXiv] [journal]
• Local Martin’s conjecture, revisited
  with Vittorio Bard [in preparation]
• Part 1 of Martin’s conjecture for order preserving functions
  with Benny Siskind [draft available on request] [slides]
• Martin’s conjecture for regressive functions on the hyperarithmetic degrees
  [draft]
• A note on a question of Sacks
  with Kojiro Higuchi [draft] [slides]
• Formalizing Galois theory
  with Thomas Browning [arXiv] [journal]
• Conway can divide by three, but I can’t
  [draft] [slides]
• The Solecki dichotomy and the Posner-Robinson therem are almost equivalent
  [arXiv] [slides]
• Counterexamples concerning the Hausdorff dimension of continuous images
  with Joe Miller

Theses

• Results on Martin’s conjecture
  PhD thesis, UC Berkeley 2021 [pdf]
  Comment: Liang Yu has recently 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]
• Division by two without choice
  [Sock division vs. shoe division]
• Formalizing Galois theory in Lean
  [Progress report from January 2021]