Model theory

Model theory is a field of mathematical logic that studies (usually, infinite) mathematical structures. The word “model” means a mathematical object with functions, constants, relations defined on the elements. A good example is the field of complex numbers \(\mathbb{C}\): it is a structure with constants \(0\) and \(1\), operations \(+\) and \(\times\) that satisfies (or “models”) certain axioms: for example, that addition is commutative, that every non-zero element has a multiplicative inverse, that every non-constant polynomial has a root, etc.

Model theorists are interested both in the behavior of entire classes of models and in understanding the structure of sets that can be described inside each model by a formal language (and the connections between these two topics). Categoricity spectrum problem is an example of a question about the entire class of structures. The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Suppose we are given the class of all models of some complete first-order theory in a countable language (we don’t know what the theory is and the language is absolutely arbitrary). Can we tell what is the categoricity spectrum of this class?
Remarkably, there are only 4 possible categoricity spectra of such classes:

  • all infinite cardinals;
  • all cardinals greater than or equal to \(\aleph_1\);
  • \(\{\aleph_0\}\);
  • the empty set.

This was established by Michael Morley in 1961. For much more general classes of structures, Sebastien Vasey gave a complete list of all possible categoricity spectra in 2019.

diagram of a chain of three dimensional amalgamation problems
One of the themes in my model theory research is the study of the independence notions and generalized amalgamation properties in various model-theoretic contexts. Generalized amalgamation properties were introduced by Saharon Shelah in 1970s. They were used by him to settle a number of difficult questions and played an important role in Vasey’s results.

  • Boney, Grossberg, Kolesnikov, Vasey. (2016) Canonical forking in AECs. Annals of Pure and Applied Logic, 167(7), 590-613.
  • Kolesnikov, Lambie-Hanson. (2016) The Hanf number for amalgamation of coloring classes. The Journal of Symbolic Logic, 81(2), 570-583.
  • Baldwin, Kolesnikov, Shelah. (2009) The amalgamation spectrum. The Journal of Symbolic Logic, 74(3), 914-928.
  • Kim, Kolesnikov, Tsuboi. (2008) Generalized amalgamation and n-simplicity. Annals of Pure and Applied Logic, 155(2), 97-114.
  • Baldwin, Kolesnikov. (2007) Categoricity, amalgamation, and tameness. Israel Journal of Mathematics, 170, 411–443.
  • Kolesnikov, Krishnamurthi. (2006) Morley rank in homogeneous models. Notre Dame Journal of Formal Logic, 47(3), 319-329.
  • Kolesnikov. (2005) \(n\)-simple theories. Annals of Pure and Applied Logic, 131(1-3), 227-261.

Another substantial project, in collaboration with John Goodrick and Byunghan Kim, had been on describing the amalgamation properties in stable first-order theories in homology-theoretic terms and determining what canonical, in a certain sense, objects witness the failure of the properties. Here are the papers where we formalize the results:

  • Dobrowolski, Kim, Kolesnikov, Lee. (2021) The relativized Lascar groups, type-amalgamation, and algebraicity. The Journal of Symbolic Logic, 86(2), 531-557.
  • Goodrick, Kim, Kolesnikov. (2017) Homology groups of types in stable theories and the Hurewicz correspondence. Annals of Pure and Applied Logic, 168(9), 1710-1728.
  • Goodrick, Kim, Kolesnikov. (2015) Type-amalgamation properties and polygroupoids in stable theories. Journal of Mathematical Logic, 15(1) 45pages.
  • Goodrick, Kim, Kolesnikov. (2015) Characterization of the second homology group of a stationary type in a stable theory. Proceedings Of The 13th Asian Logic Conference, 93-104.
  • Goodrick, Kim, Kolesnikov. (2013) Amalgamation functors and boundary properties in simple theories. Israel Journal of Mathematics, 193(1), 169-207.
  • Goodrick, Kim, Kolesnikov. (2013) Homology groups of types in model theory and the computation of \(H_2(p)\). The Journal of Symbolic Logic, 78(4), 1086-1114.
  • Goodrick, Kolesnikov. (2010) Groupoids, covers, and 3-uniqueness in stable theories. The Journal of Symbolic Logic, 75(3), 905-929.

More recently, I became interested in applications of model theory to theoretical questions about machine learning. There are a couple of preprints on arXiv of my work with Vince Guingona and students. I will update the page when these appear in print.