Miguel Moreno

Postdoctoral researcher, Faculty of Mathematics, University of Vienna .

Miguel Moreno

Postdoctoral researcher, Institute of Mathematics, University of Vienna .

Picture of my Ph.D. thesis

Publications and Projects


In preparation


1. On co-analytic sets. With Lyubomyr Zdomskyy.
2. Fake reflection reload. With Assaf Rinot.
3. 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. Preprint.
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


Articles accepted for publication in peer review journals


8-A. On unsuperstable theories in GDST. Accepted for publication in the Journal of Symbolic Logic.
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 - DOI: 10.48550/arXiv.2203.14292


Published articles in peer review journals


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


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).
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).

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


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 / ongoing

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