In preparation

1. On Generalized Ehrenfeucht-Mostowski models. With Ido Feldman.

Articles submitted to peer review journals and preprints

9-Pr. Shelah's Main Gap and the generalized Borel-reducibility. Submitted.

Abstract.

We answer one of the main questions in generalized descriptive set theory, Friedman-Hyttinen-Kulikov conjecture on the Borel-reducibility of the Main Gap. We show a correlation between Shelah’s Main Gap and generalized Borel-reducibility notions of complexity. For any k satisfying k = λ^+ = 2^λ , and 2^c ≤ λ = λ^ω1 , we show that if T is a classifiable theory and T' not, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, T, the isomorphism of models of T is either Δ11 or analitically-complete.

PDF - arXiv - DOI: 10.48550/arXiv.2308.07510

Published articles in peer review journals

8-A. On unsuperstable theories in GDST. The Journal of Symbolic Logic (2023) 1-24.

Abstract.

We study the K-Borel-reducibility of isomorphism relations of complete first order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T', if T is classifiable and T' is unsuperstable, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to K-Borel-reducibility.

PDF - arXiv - Journal - DOI: 10.1017/jsl.2023.82

7-A. The isomorphism relation of theories with S-DOP in generalized Baire spaces. Annals of Pure and Applied Logic (2022) 173: 103044.

Abstract.

We study the Borel-reducibility of isomorphism relations in the generalized Baire space K^K. In the main result we show for inaccessible K, that if T is a classifiable theory and T'is superstable with the strong dimensional order property (S-DOP), then the isomorphism of models of T is Borel reducible to the isomorphism of models of T'. In fact we show the consistency of the following: If K is inaccessible and T is a superstable theory with S-DOP, then the isomorphism of models of T is Σ^1_1-complete.

PDF - arXiv - Journal - DOI: 10.1016/j.apal.2021.103044

6-A. Fake reflection. With Gabriel Fernandes and Assaf Rinot - Israel Journal of Mathematics (2021) 245: 295 -- 345.

Abstract.

We introduce a generalization of stationary set reflection which we call filter reflection, and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations over the higher Cantor and Baire spaces.

PDF - arXiv - Journal - DOI: 10.1007/s11856-021-2213-2

5-A. Inclusion modulo nonstationary. With Gabriel Fernandes and Assaf Rinot - Monatshefte für Mathematik (2020) 192: 827 -- 851.

Abstract.

A classical theorem of Hechler asserts that the structure (ω^ω,≤^*)is universal in the sense that for any 𝜎-directed poset P with no maximal element, there is a ccc forcing extension in which (ω^ω,≤^*) contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue (︀K^K,≤^𝑆)︀.

Theorem. Assume GCH. For every regular uncountable cardinal K, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over K^K and every stationary subset 𝑆 of K, there is a Lipschitz map reducing Q to (K^K,≤^𝑆).

PDF - arXiv - Journal - DOI: 10.1007/s00605-020-01431-6

4-A. On Σ_1^1-completeness of Quasi-orders on K^K. With Tapani Hyttinen and Vadim Kulikov - Fundamenta Mathematicae (2020) 251: 245 -- 268.

Abstract.

We prove under V=L that the inclusion modulo the non-stationary ideal is a Σ^1_1-complete quasi-order in the generalized Borel-reducibility hierarchy (K > ω). This improvement to known results in L has many new consequences concerning the Σ^1_1-completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in literature. Additionally the theorem is applied to prove a dichotomy in L: If the isomorphism of a countable first-order theory (not necessarily complete) is not Δ^1_1, then it is Σ^1_1-complete. We also study the case V is different from L and prove Σ^1_1-completeness results for weakly ineffable and weakly compact K.

PDF - arXiv - Journal - DOI: 10.4064/fm679-1-2020

3-A. Reducibility of equivalence relations arising from nonstationary ideals under large cardinal assumptions. With David Asperó, Tapani Hyttinen, and Vadim Kulikov - Notre Dame Journal of Formal Logic (2019) 60: 665 -- 682.

Abstract.

Working under large cardinal assumptions such as supercompactness, westudy the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal K. We show the consis-tency of E^(λ++,λ++)_(λ-club), the relation of equivalence modulo the non-stationary ideal restricted to S^(λ++)_λ in the space (λ++)^(λ++), being continuously reducible to E^(2,λ;++)_(λ+-club), the relation of equivalence modulo the non-stationary ideal restricted to S^(λ++)_(λ+) in the space 2^(λ++). Then we show that for K ineffable E^(2,K)_(reg), the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space 2^K, is Σ^1_1-complete. We finish by showing, for Π^1_2-indescribable K, that the isomorphism relation between dense linear orders of cardinality K is Σ^1_1-complete.

PDF - arXiv - Journal - DOI: 10.1215/00294527-2019-0024

2-A. A generalized Borel-reducibility counterpart of Shelah's main gap theorem. With Tapani Hyttinen and Vadim Kulikov - Archive for Mathematical Logic (2017) 56: 175 -- 185.

Abstract.

We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations areconsidered on models of some fixed uncountable cardinality obeying certain restrictions.

PDF - arXiv - Journal - DOI: 10.1007/s00153-017-0521-3

1-A. On the Reducibility of Isomorphism Relations. With Tapani Hyttinen - Mathematical Logic Quarterly (2017) 63: 175 -- 192.

Abstract.

We study the Borel reducibility of isomorphism relations in the generalized Baire space K^K. In the main result we show for inaccessible K, that if T is a classifiable theory and T' is stable with OCP, then the isomorphism of models of T is Borel reducible to the isomorphism of models of T'.

PDF - arXiv - Journal - DOI: 10.1002/malq.201500062

Grants and Awards

Austrian Science Fund (FWF)

Lise Meitner Programme, project M3210

Generalized Baire spaces, structures and combinatorics

177,980.00 EUR, 2 years at the University of Vienna

08-2021 / 09-2023

Finnish Academy of Science and Letters

Säätiöiden Postdoc Pool 2018, Vilho, Yrjö and Kalle Väisälä Foundation

The Borel-reducibility hierarchy and generalized

35,000.00 EUR, 1 year at the University of Vienna

12-2019 / 11-2020

Department of Mathematics and Statistics - University of Helsinki

DOMAST, Doctoral Researcher

Finding the main gap in the Borel-reducibility hierarchy

Salaried position, 4 years at the University of of Helsinki

01-2014 / 12-2017

Theses

Ph.D.

Miguel Moreno, FINDING THE MAIN GAP IN THE BOREL-REDUCIBILITY HIERARCHY, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics; Doctoral dissertation (article-based), advisor Tapani Hyttinen. Unigrafia, Helsinki (2017).

Prologue.

This thesis is constituted by two parts. The first part is divided in four chapters, these are: Introduction, Structure of the thesis, Summary, and Conclusions. The second part contains five research articles.

The first part is intended to give a smooth explanation of each of the five articles of the second part. At the same time, the first part is an attempt to motivate the reader over the second part. It gives a motivation for the reader to study the generalized descriptive set theory, the roll of the author on every chapter of the second part, the motivation and a summary of the results behind each article of the second part, using the least amount of technical language.

Each article of the second part has no modifications from the submitted version of the respective article. The reader is advised that the notation between articles might differ. The articles are presented in the chronological order in which them were produced.

PDF - Slides public examination - Opponent's lecture - Picture

M.Sc.

Miguel Moreno, The stationary tower forcing, University of Bonn; Master thesis, advisor Peter Koepke (co-advise by Philipp Schlicht). Bonn (2013).

Abstract.

The purpose of this work was to understand the stationary tower forcing, introduced by W. Hugh Woodin, and the possible suborders that satisfy the basic properties, like projection, lifting and normality. Another aim was to study how large cardinal properties influence the forcing with these orders, and finally to study the applications of these forcings.

PDF

BSc.

Miguel Moreno, Matroides Representables, National University of Colombia, Bogotá; Bachelor thesis, advisor Humberto Sarria. Bogotá (2010).

Introduction.

Con este trabajo se pretende mostrar las nociones básicas de la Teoría de Matroides, haciendo énfasis en los distintos tipos de matroides, los cuales se distinguen por sus representaciones, y dar unas caracterizaciones parciales de las matroides representables, las cuales son, por lo general, el ejemplo ideal para explicar la teoría de Matroides.

PDF - Slides examination