AP09259295 On the algebraic and definable closures in new strongly minimal theories

Scientific adviser – Verbovsky Victor Valerievich

Relevance

The solution of the issues under consideration will make it possible to obtain new qualitative results in model theory. Also, the implementation of the tasks will make it possible to introduce new scientific mathematical areas in Kazakhstan, through the introduction of special master's and doctoral courses into the curricula.

The research results will be published in international ranking scientific journals. Solving the problems posed in the project will contribute to the development of both national and international mathematical science and its applications.

Objective of the project

The goal of the project is to study the properties of algebraic and definable closures in new strongly minimal theories, including the development of Galois theory in these theories.

Expected results:

- We will construct functions of two independent variables, definable in some new strongly minimal theories;

- We will find conditions, including conditions for the function mu, guaranteeing that there is no function of two independent variables that can be defined in new strongly minimal theories.

- We will find conditions, including conditions for the function mu, guaranteeing that there is no symmetric function of two independent variables that can be defined in new strongly minimal theories

- We will get a description of new strongly minimal theories that do not allow the elimination of imaginary elements.

- We will find conditions, including on the function mu, guaranteeing that there is no definable function of n independent variables.

- We will describe the structure of algebraic closure in new strongly minimal theories.

Results achieved:

The results of the work and their novelty: for the first time in the history of the Kazakhstan architectural school, the model of a specialist architect of the XXI century was formulated and justified, recommendations for adjusting educational programs for the training of architects in Kazakhstan, modern methods of teaching architects at the undergraduate level.

Names and surnames of the study group members with their identifiers:

1. Verbovsky Viktor Valerievich - scientific director
2. Bayzhanov Bektur Sembiuly - Chief Researcher
3. Bekenov Makhsut Iskanderovich - Leading Researcher
4. Zambarnaya Tatyana Sergeevna - Researcher
5. Ershigeshova Aisha Daurenovna - junior researcher
6. Umbetbaev Olzhas Asylbekovich - junior researcher
7. Adil Zhanar Torebekkyzy - junior researcher

List of publications:

• V.V.Verbovskiy, On definability of types and relative stability, Math.Logic.Quart., 65(2019):332-346, DOI:10.1002/malq.201600084 (цитирований 1 Scopus, индексируется в Science Citation Index Expanded) CiteScore-2019:0.8, Процентиль-2019:21,Q4.
• V.V.Verbovskiy, On commutativity of circularly ordered c-o-stable groups, Eurasian Mathematical Journal, ISSN 2077-9879, 4(2018):91-98. DOI:10.32523/2077-9879-2018-9-4-91-98 (цитирований 1 Scopus) CiteScore-2018:0.7, Процентиль-2018:32.
• V.V.Verbovskiy, On ordered groups of Morley o-rank 1, Siberian Electronic Mathematical Reports, 15(2018):314-320. DOI:10.17377/semi.2018.15.028. (цитирований 1 Scopus). CiteScore-2018:0.8. Процентиль-2018:39.
•  B.Sh.Kulpeshov, V.V.Verbovskiy, On weakly circularly minimal groups, Math.Logic.Quart., 61(2015):82-90.  DOI:10.1002/malq.201300076 (цитирований 2 Scopus, индексируется в Science Citation Index Expanded) CiteScore-2018:0.5, Процентиль-2018:10,Q4.
•  V.V.Verbovskiy, On a classification of theories without the independence property, Math.Logic.Quart., 59(2013):119-124. (цитирований 3 Scopus) CiteScore-2013:0.7, Процентиль-2013:19.
•  B.S.Baizhanov, S.V.Sudoplatov, V.V.Verbovskiy, Conditions for non-symmetric relations of semi-isolation, Siberian Electronic Mathematical Reports, 9(2012):161-184. (цитирований 10 Scopus) CiteScore-2012:0.1,         Процентиль-2012:5.
• B.Baizhanov, V.Verbovskiy.  O-stable theories. Algebra and Logic. 50(2011):303-325. (цитирований 6 Scopus) CiteScore-2011:0.6, Процентиль-2012:17.
• V.V.Verbovskii.  O-Stable Ordered Groups. Siberian Advances in Mathematics. 22(2012):50-74. (цитирований 7 Scopus) CiteScore-2012:0.3, Процентиль-2012:10.
• O.Belegradek, V.Verbovskiy, F.Wagner. Coset-Minimal Groups, Ann.Pure.Appl.Logic,  121(2003):113:143. (цитирований 5 Scopus) CiteScore-2019:1.9, Процентиль-2019:71,Q1.
• V.Verbovskiy, I.Yoneda, CM-triviality and relational structures, Ann.Pure.Appl.Logic,  122(2003):175-194. (цитирований 7 Scopus) CiteScore-2019:1.9, Процентиль-2019:71,Q1.
• A.Alibek, B.Baizhanov, B.Kulpeshov, T.Zambarnaya. Vaught's conjecture for weakly o-minimal theories of convexity rank 1 // Ann.Pure.App.Logic. 169(2018):1190-1209. DOI: 10.1016/j.apal.2018.06.003. CiteScore-2018:1.6. Процентиль-2018:62. Q1.
• B.Baizhanov, J.Baldwin, T.Zambarnaya. Finding countable models for ordered theories // Siberian Electronic Mathematical Reports. 15(2018):719-727. DOI: 10.17377/semi.2018.15.057. CiteScore-2018:0.8. Процентиль-2018:39.
• B.Baizhanov, B.Kulpeshov, T.Zambarnaya. A.D.Taimanov and model theory in Kazakhstan // Siberian Electronic Mathematical Reports. 17(2020):1–58. DOI: 10.33048/semi.2020.17.011. CiteScore-2019:0.7. Процентиль-2019:32.
• B.Baizhanov, F.Kobdikbayeva, T.Zambarnaya. On the number of countable models of complete theories with a partial order // Математический журнал. 17(2017):5-12. Рекомендован КОКСОН.
• B.S.Baizhanov, N.S.Tazabekova, A.D.Yershigeshova, T.S.Zambarnaya. Types in small theories. Математический журнал. 15(2015):38-56. Рекомендован КОКСОН.
• B.S.Baizhanov, O.Umbetbayev, T.Zambarnaya. Non-existence of uniformly definable family of convex equivalence relations in an 1-type of small ordered theories and maximal number of models // Kazakh Mathematical Journal. 19(2019): 98-106. Рекомендован КОКСОН.
• B.S.Baizhanov, O.A.Umbetbayev, T.S.Zambarnaya. On a criterion for omissibility of a countable set of types in an incomplete theory // Kazakh Mathematical Journal. 19(2019):22-30. Рекомендован КОКСОН.
• B.Baizhanov, S.Baizhanov. Some questions on external definability // News of the NAS of the Republic of Kazakhstan, physico-mathematical series. 6(2019):146-150. Рекомендован КОКСОН.
• B.S.Baizhanov, S.S.Baizhanov, N.E.Sailaubay, O.A.Umbetbayev, T.S.Zambarnaya Essential and inessential expansions: model completeness and number of countable models // Математический журнал. 17(2017):43-52. Рекомендован КОКСОН.
• B.Baizhanov, S.Baizhanov., T.Saulebayeva, T.Zambarnaya. One-formulas and one-types in ordered theories // Математический журнал. 16(2016):121-141. Рекомендован КОКСОН.
• M.I.Bekenov Properties of Elementary Embeddability in Model Theory, Journal of Mathematical Sciences, 1(2018):10-13. DOI: 10.1007/s10958-018-3721-4.