Patrick Lutz

Research

Currently I’m mostly working on questions related to Martin’s conjecture, but I’m interested much more generally in logic and foundations of math. My goal is to work on problems that are both mathematically appealing and philosophically significant.

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]
• Martin’s conjecture for regressive functions on the hyperarithmetic degrees
  [draft]
• A note on a question of Sacks
  with Kojiro Higuchi [draft]
• Conway can divide by three, but I can’t
  [draft]
• Formalizing Galois theory
  with Thomas Browning [arXiv]

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]
• Division by two without choice
  [Sock division vs. shoe division]
• Formalizing Galois theory in Lean
  [Progress report from January 2021]