Homepage of M. Malliaris
Professor
Department of Mathematics
University of Chicago
Research:
My research interests are in model theory, in particular classification of theories.
Teaching: most recently, Math 277 and Math 309.
Papers: All papers, abstracts, and other information
Some (not all) links:
"Realization of \phitypes and Keisler's order,"
Ann. Pure Appl. Logic 157 (2009) 220224.
link
''The characteristic sequence of a firstorder formula,'' J Symb Logic 75, 4 (2010) 14151440.
PDF,
link.
"Edge distribution and density in the characteristic sequence,"
Ann Pure Appl Logic 162, 1 (2010) 119.
PDF,
link
"Hypergraph sequences as a tool for saturation of ultrapowers,'' J Symb Logic 77, 1 (2012) 195223.
PDF,
link
"Independence, order, and the interaction of ultrafilters and theories," Ann Pure Appl Logic,
163, 11 (2012) 15801595. PDF,
link
(with S. Shelah) "Regularity lemmas for stable graphs.'' Trans. Amer. Math Soc,
366 (2014), 15511585.
link, PDF
(with S. Shelah) "Constructing regular ultrafilters from a
modeltheoretic point of view."
Trans. Amer. Math. Soc. 367 (2015), 81398173.
link, arxiv
(with S. Shelah) "Modeltheoretic properties of ultrafilters built by independent families of functions." JSL 79, 1 (2014) 103134. PDF,
arxiv
(with S. Shelah) "A dividing line within simple unstable theories." Advances in Math
249 (2013) 250288.
PDF, link
(with S. Shelah)
"Cofinality spectrum theorems in model theory, set theory, and general
topology." J. Amer. Math. Soc. 29 (2016), 237297.
arxiv,
journal
(with S. Shelah)
"General topology meets model theory, on p and t."
This announces the results of the paper above.
Proc Natl Acad Sci USA 110, 33 (2013) 1330013305.
PDF,
link
(with S. Shelah)
"Existence of optimal ultrafilters and the fundamental complexity of simple theories."
Advances in Math. 290 (2016) 614681. journal,
arxiv
(with S. Shelah)
"Keisler's order has infinitely many classes." Israel J. Math. 224 (2018), no. 1, 189–230.
pdf, journal
(with A. Pillay)
"The stable regularity lemma revisited."
Proc. Amer. Math. Soc. 144 (2016) 17611765. arxiv
(with C. Terry) "On unavoidable induced subgraphs in
large prime graphs." J. Graph Theory 88, 2 (2018) 255270. arxiv, journal
(with D. Casey) "Notes on cofinality spectrum problems."
arxiv
(with S. Shelah) "A new look at interpretability and saturation."
Ann Pure Appl Logic 170, 5 (2019) 642671.
arxiv,
journal
(with A. Peretz) "What simplicity is not."
Simplicity: ideals of practice in mathematics and the arts. 51–58, Math. Cult. Arts, Springer, Cham, 2017.
link
"Model theory and ultraproducts." Proceedings of the 2018 ICM, Rio de Janeiro.
arxiv
(with S. Shelah) "An example of a new simple theory."
Contemp. Math. vol 752. pps. 121152.
arxiv
(with N. Alon, R. Livni, S. Moran) "Private PAC learning implies finite Littlestone dimension."
51st Symposium on the Theory of Computing (STOC), 2019. arxiv
(with S. Shelah) "A separation theorem for simple theories."
Trans. Amer. Math. Soc. 375 (2022), 11711205.
arxiv,
journal
(with S. Shelah) "Keisler's order is not simple (and simple theories may not be either)."
Adv. Math. 392 (2021) paper 108036, 94pp.
arxiv,
journal
(with P. Diaconis) "Complexity and randomness in the Heisenberg groups (and beyond)."
New Zealand J. Math. 52 (2021) 403426. journal,
arxiv
(with S. Shelah) "Some simple theories from a Boolean algebra point of view."
APAL, 175, 1, part B (2024) paper 103345.
journal,
arxiv
(with S. Shelah) "New simple theories from hypergraph sequences."
Model Theory 3 (2024), no. 2, 449–464. journal,
arxiv
(with S. Moran) "Model theory and agnostic online learning via excellent sets."
Trans. AMS (2024), no. 11, 7753–7776.
journal,
arxiv
(with S. Shelah) "Shearing in some simple rank one theories."
Israel J. Math. vol 257, 481–518 (2023). journal,
arxiv
(with N. Alon, M. Bun, R. Livni, S. Moran) "Private and online learnability
are equivalent." J. ACM (2022), 69, 4, article 28, pages 134. journal
(with L. N. Coregliano) "Countable Ramsey." 76 pages, 2022.
arxiv.
(with S. Moran) "The unstable formula theorem revisited via algorithms." 21 pages, 2022.
arxiv.
(with L. N. Coregliano) "Weak randomness in graphons and theons."
To appear, Random Structures and Algorithms. 68 pages, revised 2024.
arxiv.
(with L. N. Coregliano) "Higharity PAC learning via exchangeability." 150 pages, 2024.
arxiv
Thesis: Persistence and Regularity in Unstable Model Theory.
PDF; here is a brief overview
Contact information:
Department of Mathematics
University of Chicago
5734 S. University Avenue
Chicago, IL 60637
Email: (my three initials) at math; see URL of this page.
Office: 406
Research partially supported by:
NSF grants DMS1300634 and DMS1001666,
a Gödel research prize fellowship,
a Sloan research fellowship,
NSF CAREER award DMS1553653, and NSFBSF 2051825.
 space 
