My research is in the branch of mathematical logic known as model theory. More specifically, my research interests are in classification theory, first-order stability theory, dp-rank and strongly dependent structures, and applications to combinatorics.
Publications and preprints
18. A characterization of strongly dependent ordered Abelian groups (with Alfred Dolich), preprint, 2017. We give a new proof of a simple characterization of which ordered Abelian groups, in the pure language of ordered groups, are strongly dependent. This result was recently obtained independently by Halevi and Hasson as well as by Farré, but our proof uses more direct and elementary methods.
17. Parametrized Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior (with Tristram Bogart and Kevin Woods), Discrete Analysis, 2017 #4. We eneralize Erhart's theorem on integer points in dilates of polyhedra to show that all counting functions definable in a certain extension of Presburger arithmetic allowing multiplication by an integer parameter t are "eventually quasi-polynomial": that is, they are eventually polynomial when t is restricted to each class modulo some fixed m. This solves a conjecture of Woods and includes, for example, parametrized versions of the Frobenius coin problem.
16. Tame topology over definable uniform structures: viserality and dp-minimality (with Alfred Dolich), submitted in 2017. We establish a cell decomposition result for definable uniform structures which we call visceral (essentially, that any infinite definable unary set has interior), and use this to study properties of the topological dimension of definable sets. This simultaneously generalizes previously known results for o-minimal structures and P-minimal fields.
15. Left-ordered inp-minimal groups (with Jan Dobrowolski), submitted in 2017. We prove that any left-ordered inp-minimal group is abelian, and we give an exameple of a non-abelian left-ordered group of dp-rank 2. We also give a necessary algebraic condition for a group to have finite burden (inp-rank).
13. Strong theories of ordered abelian groups (with Alfred Dolich), Fundamenta Mathematicae vol. 236 (2017), no. 3, 269-296. This article focuses on the special properties of 1-dimensional finite sets in ordered abelian groups whose theories are strong. We establish various results about discrete definable sets and give some illustrative examples. Here “strong” means that there are no inp-patterns of infinite length, which is more general than being strongly dependent or of finite dp-rank.
11. Type-amalgamation properties and polygroupoids in stable theories (with Alexei Kolesnkiov and Byunghan Kim), Journal of Mathematical Logic, vol. 15 no. 1 (June 2015). We show that in a stable first-order theory, the failure of higher-dimensional type amalgamation properties can always be witnessed by algebraic structures which we call n-ary groupoids. This generalizes a result of Hrushovski that failures of 4-amalgamation in stable theories are witnessed by definable groupoids (which are 2-ary polygroupoids in our terminology). The n-ary polygroupoids are definable in a mild expansion of the language (adding a unary predicate for an infinite Morley sequence).
9. The Schröder-Bernstein property for a-saturated models (with Michael C. Laskowski; Proceedings of the American Mathematics Society, vol. 142 (2014), no. 3, 1013-1023; arxiv:1202.6535). We prove that a first-order theory T has the property thas some expansion of T by constants has the Schröder-Bernstein property if and only if T is both superstable and non-multidimensional. Among the superstable theories, we characterize those theories whose sufficiently saturated (a-saturated) models have the Schröder-Bernstein property: they are those with no “nomadic types” (regular types that are orthogonal to all of their images via iterations of a single automorphism).
4. Dp-minimal theories: basic facts and examples (with Alfred Dolich and David Lippel; Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, 267-288; arXiv:0910.3189v2 [math.LO]). Update: In 2014 Uri Andrews alerted us to a gap in the proof of Corollary 5.6 (the dp-minimality of a certain example), and later in 2015 we were able to repair this gap to show that the example is, indeed, dp-minimal (see preprint number 14 above). None of the other results of this paper (such as the dp-minimality of the p-adic field) are affected by this gap.
2. A monotonicity theorem for dp-minimal densely ordered groups (The Journal of Symbolic Logic, vol. 75, no. 1 (March 2010), pp. 221-238). Important update: In 2015, Pierre Simon alerted me to a serious gap in the proof of Lemma 3.30 of the published version which may not be fixable. In particular, I am no longer sure whether continuous unary functions definable in dp-minimal densely ordered groups are always locally monotonic, as I claimed in the paper. However, I still believe that everything up to and including Theorem 3.27 is correct, which constitutes the majority of the paper (in particular, the result that definable unary functions are finite unions of continuous definable functions).
1. When are elementarily bi-embeddable models isomorphic? (Ph.D. thesis, 2007, under Thomas Scanlon, University of California, Berkeley). Some of the results of this thesis were later published in "The Schroder-Bernstein property for weakly minimal theories" (which significantly strengthens the results of the last chapter) and in "The Schroder-Bernstein property for a-saturated models."
I am open to advising student theses in all areas of mathematical logic.
Undergrate theses supervised
Nicolás Nájar, "Un análisis de los teoremas de incompletitud de Kurt Godel" (as co-advisor, Universidad Pedagógica Y Tecnológica de Colombia, 2013).
Angela Jaramillo, "Some Diophantine forms of Godel's Theorem " (Los Andes, 2013).
I am on sabbatical during the second semester of 2017 and not teaching any courses.
Selected Past Courses
Integral Calculus [MATE 1214], spring 2014 through spring 2017
Foundations of Mathematics ["Matemática Estructural", MATE 1102], spring 2017 and fall 2016
Logic 2 [MATE 3121/4120], spring 2016
Tame topology and o-minimal structures [MATE 3129/4127], spring 2015
Honors Differential Calculus [MATE 1204], fall 2013
Differential Calculus [MATE 1203], fall 2013
Vector Calculus [MATE 1207], spring 2013
Linear Algebra 1 [MATE 1105], spring 2010 through spring 2012
Last update: August 20, 2017