Publications and Projects
In preparation

1.
On coanalytic sets.
With Lyubomyr Zdomskyy.

2.
Fake reflection reload.
With Gabriel Fernandes and Assaf Rinot.

3.
On Generalized EhrenfeuchtMostowski models.
With Ido Feldman.

4.
Shelah's Main Gap and the generalized Borelreducibility.
Articles submitted to peer review journals and preprints

1.
Miguel Moreno

On unsuperstable theories in GDST.
Submitted.

Abstract.
We study the KBorelreducibility 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 KBorelreducibility.

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arXiv

DOI:
10.48550/arXiv.2203.14292
Published articles in peer review journals

7.
Miguel Moreno

The isomorphism relation of theories with SDOP in generalized Baire spaces.
Annals of Pure and Applied Logic (2022) 173: 103044.

Abstract.
We study the Borelreducibility 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 (SDOP), 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 SDOP, then the isomorphism of models of T is Σ^1_1complete.

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arXiv

Journal

DOI:
10.1016/j.apal.2021.103044

6.
Gabriel Fernandes

Miguel Moreno

Assaf Rinot

Fake reflection.
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.

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arXiv

Journal

DOI:
10.1007/s1185602122132

5.
Gabriel Fernandes

Miguel Moreno

Assaf Rinot

Inclusion modulo nonstationary.
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 𝜎directed poset P with no maximal element, there is a ccc forcing extension in which (ω^ω,≤^*) contains a cofinal orderisomorphic 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 cofinalitypreserving GCHpreserving forcing extension in which for every analytic quasiorder Q over K^K and every stationary subset 𝑆 of K, there is a Lipschitz map reducing Q to (K^K,≤^𝑆).

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arXiv

Journal

DOI:
10.1007/s00605020014316

4.
Tapani Hyttinen

Vadim Kulikov

Miguel Moreno

On Σ_1^1completeness of Quasiorders on K^K.
Fundamenta Mathematicae (2020) 251: 245  268.

Abstract.
We prove under V=L that the inclusion modulo the nonstationary ideal is a Σ^1_1complete quasiorder in the generalized Borelreducibility hierarchy (K > ω). This improvement to known results in L has many new consequences concerning the Σ^1_1completeness of quasiorders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the nonstationary 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 firstorder theory (not necessarily complete) is not Δ^1_1, then it is Σ^1_1complete. We also study the case V is different from L and prove Σ^1_1completeness results for weakly ineffable and weakly compact K.

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arXiv

Journal

DOI:
10.4064/fm67912020

3.
David Asperó

Tapani Hyttinen

Vadim Kulikov

Miguel Moreno

Reducibility of equivalence relations arising from nonstationary ideals under large cardinal assumptions.
Notre Dame Journal of Formal Logic (2019) 60: 665  682.

Abstract.
Working under large cardinal assumptions such as supercompactness, westudy the Borelreducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal K. We show the consistency of E^(λ++,λ++)_(λclub), the relation of equivalence modulo the nonstationary 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 nonstationary 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 nonstationary ideal restricted to regular cardinals in the space 2^K, is Σ^1_1complete. We finish by showing, for Π^1_2indescribable K, that the isomorphism relation between dense linear orders of cardinality K is Σ^1_1complete.

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arXiv

Journal

DOI:
10.1215/0029452720190024

2.
Tapani Hyttinen

Vadim Kulikov

Miguel Moreno

A generalized Borelreducibility counterpart of Shelah's main gap theorem.
Archive for Mathematical Logic (2017) 56: 175  185.

Abstract.
We study the Borelreducibility 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 Borelreducibility. In fact, we can also ensure that a range of equivalence relations modulo various nonstationary ideals are strictly between those isomorphism relations. The isomorphism relations areconsidered on models of some fixed uncountable cardinality obeying certain restrictions.

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arXiv

Journal

DOI:
10.1007/s0015301705213

1.
Tapani Hyttinen

Miguel Moreno

On the Reducibility of Isomorphism Relations.
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 BORELREDUCIBILITY HIERARCHY
, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics; Doctoral dissertation (articlebased), 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 (coadvise 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

082021 / ongoing

Finnish Academy of Science and Letters

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

The Borelreducibility hierarchy and generalized

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

122019 / 112020

Department of Mathematics and Statistics  University of Helsinki

DOMAST,
Doctoral Researcher

Finding the main gap in the Borelreducibility hierarchy

Salaried position, 4 years at the University of of Helsinki

012014 / 122017