**Andrew Marks**

I'm a professor at UC Berkeley. My research interests lie in descriptive set theory and its connections to related areas such as computability theory, combinatorics, ergodic theory, probability, operator algebras, and quantum information.

**Office:** 733 Evans.

**Email:**
marks@math.berkeley.edu

**Publications and preprints:**

- One-ended spanning subforests and treeability of groups (with Clinton Conley, Damien Gaboriau, and Robin Tucker-Drob). Submitted [ pdf | arXiv ]
- On a question of Slaman and Steel (with Adam Day). Submitted. [ arXiv | pdf ]
- Descriptive graph combinatorics (with Alekos Kechris). Preprint [ pdf ].
- Measurable graph combinatorics. Proc. Int. Cong. Math. 2022, Vol. 3, pp. 1488–1502. EMS Press, Berlin, (2023) [ pdf | doi | errata ]
- Borel asymptotic dimension and hyperfinite equivalence relations (with Clinton Conley, Steve Jackson, Brandon Seward, and Robin Tucker-Drob). To appear in Duke Mathematical Journal [ pdf | arXiv ]
- Distance from marker sequences in locally finite Borel graphs (with Clinton Conley) in Samuel Coskey and Grigor Sargysan eds.
*Trends in Set Theory*, Contemp. Math. 752, (2020), 89-92 [ arXiv | pdf | doi ]. - Measurable realizations of abstract systems of congruences (with Clinton Conley and Spencer Unger).
*Forum of Math, Sigma*8, (2020) e10 [ arXiv | pdf | doi ]. - Hyperfiniteness and Borel combinatorics (with Clinton Conley, Steve Jackson, Brandon Seward, and Robin Tucker-Drob).
*J. European Math. Soc.*22, No. 3 (2020), 877-892 [ arXiv | pdf | doi ] - Folner tilings for actions of amenable groups (with Clinton Conley, Steve Jackson, David Kerr, Brandon Seward, and Robin Tucker-Drob).
*Mathematische Annalen*371 (2018), 663-683. [ arXiv | pdf | doi ] - Jump operations for Borel graphs (with Adam Day).
*J. Symb. Log*82 (2018), 13-28. [ arXiv | pdf | doi | errata ]. - Borel circle squaring (with Spencer Unger).
*Ann. of Math.*186 (2017), 581-605. [ arXiv | pdf | doi | pictures ]. - Uniformity, universality, and computability theory.
*J. Math. Logic*17 (2017) no 1. [ arXiv | pdf | doi | errata ]. - The universality of poly-time Turing equivalence.
*Mathematical Structures in Computer Science*(2016) [ arXiv | pdf | doi ]. - Brooks's theorem for measurable colorings (with Clinton Conley and Robin Tucker-Drob).
*Forum of Math. Sigma*4 (2016) [ arXiv | pdf | doi | errata ]. - Baire measurable paradoxical decompositions via matchings (with Spencer Unger).
*Adv. Math.*289 (2016), 397-410. [ arXiv | pdf | doi ]. - A determinacy approach to Borel combinatorics.
*J. Amer. Math. Soc.*29 (2016), 579-600. [ arXiv | pdf | doi | errata ] - Martin's conjecture, arithmetic
equivalence, and countable Borel equivalence
relations (with Theodore Slaman and John Steel). Ordinal definability and recursion theory: The
cabal seminar volume III,
*Lecture Notes in Logic*43, Cambridge University Press, 2016, 200-219. [ arXiv | pdf | doi | errata ] - Minimal Betti Numbers (with Christopher Dodd, Victor Meyerson, and Ben Richert).
*Communications in Algebra*Vol 35 (3), 2007, pp 759-772. [ arXiv | doi ]

**Seminar:**

**Teaching Fall 2024:**

- 225A. TuTh 12:30-2. Graduate model theory. Our main reference will be Marker's book "Model Theory: An Introduction". Prerequisites for the class are 125A (mathematical logic) and (135 (set theory) or 136 (computability and incompleteness))
- 235A. This class will be an introduction to descriptive set theory. Prerequisites for the class are 104 (analysis) and 135 (set theory), or permission of the instructor. Our main reference will be Kechris's book "Classical Descriptive Set Theory". See also Tserunyan's notes on Descriptive Set Theory. Near the end of the class we will also discuss some applications to the study of Polish groups, Borel equivalence relations, and Borel graphs.

**Teaching notes:**

- Notes on effective desciptive set theory from a course I taught in spring 2019.
- Notes on set theory from a course I taught in spring 2020 and spring 2021.
- Notes on computability theory from a course I taught in winter 2021, 2022, 2023.

**Research notes (not intended for publication):**

- Larson-Zapletal's proof of Hjorth's turbulence theorm. December 2022. [ pdf ]
- A short proof of the Connes-Feldman-Weiss theorem. November 2017. [ pdf ]
- A Baire category proof of the Ackerman-Freer-Patel Theorem. May 2016. [ pdf ]
- Structure in complete sections of the shift action of a residually finite group. November 2013. [ pdf ]
- A short proof that an acyclic n-regular Borel graph may have Borel chromatic number n+1. May 2013. [ pdf ]
- Is the Turing jump unique? : Martin's conjecture and countable Borel equivalence relations. December 2011. [ pdf ]