Maryanthe Malliaris

Maryanthe Elizabeth Malliaris is an American mathematician specializing in model theory and a professor at the University of Chicago .

Life

Maryanthe Malliaris is the daughter of computer science professor Mary Malliaris and economics professor AG (Tassos) Malliaris, who both work at Loyola University Chicago . She graduated from Harvard University with a bachelor's degree (AB) in 2001 and received her PhD from Thomas Scanlon at the University of California, Berkeley , in 2009 (Persistence and Regularity in Unstable Model Theory). She was a post-doctoral student at the University of Chicago and the Hebrew University of Jerusalem. She is an Associate Professor at the University of Chicago.

She deals in particular with the classification of theories in the context of model theory, taking up a classification by Howard Jerome Keisler on ultra products from 1967. Before Malliaris' work, the theory was long considered difficult to handle from the standpoint of Saharon Shelah's theory of stability . She also made connections from model theory to graph theory.

In 2017 she received the Hausdorff Medal of the European Set Theory Society with Saharon Shelah . You received the award for proving two long-open fundamental problems in set theory and model theory:

• (Set theory): They proved that the cardinal characteristics of the continuum and are equal. Both give the minimum cardinality of an infinite set F of infinite subsets of the natural numbers that meet certain additional conditions. It was known that and and . But if there were an infinite set with a cardinality between that of the natural numbers and that of the real numbers, the continuum hypothesis would be refuted. However, according to Paul Cohen, it is undecidable within the framework of the Zermelo-Fraenkel set theory, so that two possibilities remain: either both cardinalities are the same or the question of how exactly their size relationship looks like is undecidable. Until Malliaris and Shelah were proven to be the same, the second option was generally preferred. ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {t}}}$ ${\displaystyle {\mathfrak {p}}>\aleph _{0}}$ ${\displaystyle {\mathfrak {t}}>\aleph _{0}}$ ${\displaystyle {\mathfrak {p}}\leq {\mathfrak {t}}}$ ${\displaystyle {\mathfrak {p}}<{\mathfrak {t}}}$ ${\displaystyle \aleph _{0}}$
• (Model theory): They showed that a theory with the weak order property results in maximality in the order of Keisler's 1967 classification. Keisler divided mathematical theories into complexity classes and it was known that there were at least two such classes: minimal complexity (simple theories) and maximal complexity. The latter included ordered mathematical theories. Malliaris and Shelah examined the question of whether a weakening of the order would change anything in this division. The problem is the problem of the theory of equality of the cardinal characteristics , in combination, as would be produced if the attenuated order maximum complexity equal . In their treatise they showed both the equality of and and that the ordering property and its weakening both result in maximum complexity. ${\displaystyle SOP_{2}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {t}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {t}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {t}}}$

Malliaris and Shelah also proved that Keisler's classification contains an infinite number of classes (something Keisler had already suspected).

In 2017 she is at the Institute for Advanced Study . For 2018 she is invited speaker at the International Congress of Mathematicians in Rio (Model theory and ultraproducts).

She was a Sloan Fellow and a Gödel Research Prize Fellow.

Fonts (selection)

Except for the works cited in the footnotes

• Hypergraph sequences as a tool for saturation of ultrapowers, J Symb Logic, Volume 77, 2012, pp. 195-223
• Independence, order, and the interaction of ultrafilters and theories, Ann Pure Appl Logic, Volume 163, 2012, pp. 1580-1595.
• with Shelah: A dividing line within simple unstable theories, Advances in Mathematics, Volume 249, 2013, pp. 250–288, Arxiv
• with Shelah: Regularity lemmas for stable graphs, Trans. Amer. Math Soc, Volume 366, 2014, pp. 1551-1585, Arxiv
• with Shelah: Constructing regular ultrafilters from a model-theoretic point of view, Trans. Amer. Math. Soc., Volume 367, 2015, pp. 8139-8173, Arxiv
• with Shelah: Keisler's order has infinitely many classes, Arxiv 2015

Web links

• Homepage
• Article by Malliaris in The Harvard Crimson (the Harvard student newspaper)

References and comments

1. ↑ Homepage of Mary Malliaris , accessed April 16 , 2019
2. ↑ dedication of AG Malliaris in his book Economic Uncertainty, Instabilities and Asset Bubbles , World Scientific 2005
3. ↑ Information in your dissertation
4. ↑ Maryanthe Malliaris in the Mathematics Genealogy Project (English)
5. ↑ Partially published in: Malliaris, The characteristic sequence of a first-order formula, J Symb Logic, Volume 75, 2010, pp. 1415-1440
6. ↑ ESTS, Hausdorff Price 2017
7. ↑ Malliaris, Shelah, General topology meets model theory, on p and t. Proc. Natl. Acad. Sci. USA, Volume 110, 2013, pp. 13300-13305
8. ↑ Malliaris, Shelah, Cofinality spectrum theorems in model theory, set theory, and general topology. J. Amer. Math. Soc., Volume 29, 2016, pp. 237-297, Arxiv
9. ↑ Many different cardinal characteristics (invariants) of the continuum can be defined. They lie in their cardinality between that of the natural and real numbers (both included) and are used to investigate properties that would have to have sets that would violate the continuum hypothesis.
10. ↑ The intersection of two sets from F is infinite and no infinite subset A of the natural numbers is completely contained in all sets from F. When there is the additional constraint that the amounts can be arranged. ${\displaystyle {\mathfrak {t}}}$
11. ↑ Kevin Hartnett, From Infinity to Infinity , Spektrum.de, October 13, 2017
12. ↑ Entry on IAS
13. ↑ Arxiv 2018