I am a postdoc at UCLA, where I work on computability theory and descriptive set theory.

**Office:** MS 7336

**Email:** pglutz “at” math.ucla.edu

**BA:** Mathematics, UC Berkeley, 2012-2016

**PhD:** Mathematics, UC Berkeley, 2016-2021

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

**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 and measure preserving functions**

with Benny Siskind [arXiv] [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] [slides]**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 [slides]**Coding information into all infinite subsets of a dense set**

with Matthew Harrison-Trainor [arXiv] [slides]

**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)

**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]

Here are some expository documents I’ve written. I hope to add more soon.

**Introduction to Steel forcing**

[notes]

For two of the courses I taught at UCLA, I compiled handwritten notes:

- Notes for Math 182: Introduction to Algorithms
- Notes for Math 114S: Introduction to Set Theory

**Current Teaching:** I am not teaching this quarter.

**Past Teaching:** Here is a list of courses I’ve taught in the past. Also, here are websites I made for some of those classes:

**Spring 2022:**Math 285D, Introduction to Weihrauch Reducibility**Spring 2021:**Math 54, Linear Algebra and Differential Equations**Spring 2018:**Math 10B, Math for Biology Majors**Summer 2017:**Math 54, Linear Algebra and Differential Equations**Spring 2017:**Math 10B, Math for Biology Majors**Fall 2016:**Math 54, Linear Algebra and Differential Equations

**Formalizing math in Lean**

During Summer 2020, Thomas Browning, Rahul Dalal and I organized a seminar on the Lean proof assistant. The website for the seminar is here. As an outgrowth of this seminar, Thomas Browning, Jordan Brown and I formalized parts of Galois theory in Lean. I am still interested in formalization, but no longer actively involved.**The Qual**

Once upon a time I took a qualifying exam and wrote up (most of) a transcript. I’m mostly proud of the picture I made to accompany it (and not so proud of the silly things I said during my qual).