# 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.

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.