Homepage of M. Malliaris

Department of Mathematics
University of Chicago

Research: My research interests are in model theory, in particular classification of theories.

Teaching: most recently, Math 277, Math 300, and Math 310.

Papers: All papers, abstracts, and other information

Some (not all) links:

  • "Realization of \phi-types and Keisler's order," Ann. Pure Appl. Logic 157 (2009) 220-224. link
  • ''The characteristic sequence of a first-order formula,'' J Symb Logic 75, 4 (2010) 1415-1440. PDF, link.
  • "Edge distribution and density in the characteristic sequence," Ann Pure Appl Logic 162, 1 (2010) 1-19. PDF, link
  • "Hypergraph sequences as a tool for saturation of ultrapowers,'' J Symb Logic 77, 1 (2012) 195-223. PDF, link
  • "Independence, order, and the interaction of ultrafilters and theories," Ann Pure Appl Logic, 163, 11 (2012) 1580-1595. PDF, link
  • (with S. Shelah) "Regularity lemmas for stable graphs.'' Trans. Amer. Math Soc, 366 (2014), 1551-1585. link, PDF
  • (with S. Shelah) "Constructing regular ultrafilters from a model-theoretic point of view." Trans. Amer. Math. Soc. 367 (2015), 8139--8173. link, arxiv
  • (with S. Shelah) "Model-theoretic properties of ultrafilters built by independent families of functions." JSL 79, 1 (2014) 103-134. PDF, arxiv
  • (with S. Shelah) "A dividing line within simple unstable theories." Advances in Math 249 (2013) 250--288. PDF, link
  • (with S. Shelah) "Cofinality spectrum theorems in model theory, set theory, and general topology." J. Amer. Math. Soc. 29 (2016), 237--297. 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) 13300-13305. PDF, link
  • (with S. Shelah) "Existence of optimal ultrafilters and the fundamental complexity of simple theories." Advances in Math. 290 (2016) 614-681. 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) 1761--1765. arxiv
  • (with C. Terry) "On unavoidable induced subgraphs in large prime graphs." J. Graph Theory 88, 2 (2018) 255-270. 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) 642-671. 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. 121-152. 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), 1171-1205. 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) 403-426. journal, arxiv
  • (with S. Shelah) "Some simple theories from a Boolean algebra point of view." to appear, APAL. journal, arxiv
  • (with S. Shelah) "New simple theories from hypergraph sequences." to appear, Model Theory. arxiv
  • (with S. Moran) "Agnostic online learning and excellent sets." arxiv
  • (with S. Shelah) "Shearing in some simple rank one theories." to appear, Israel J. Math. arxiv
  • (with N. Alon, M. Bun, R. Livni, S. Moran) "Private and online learnability are equivalent." J. ACM, 2022. journal
  • (with L. N. Coregliano) "Countable Ramsey." arxiv
  • (with L. N. Coregliano) "Weak randomness in graphons and theons." arxiv
  • (with S. Moran) "The unstable formula theorem revisited via algorithms." arxiv
  • (with L. N. Coregliano) "High-arity PAC learning via exchangeability." 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 DMS-1300634 and DMS-1001666, a Gödel research prize fellowship, a Sloan research fellowship, NSF CAREER award DMS-1553653, and NSF-BSF 2051825.

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