Publications and Projects
In preparation

1.
On coanalytic sets.
With Lyubomyr Zdomskyy.

2.
Fake reflection reload.
With Assaf Rinot.

3.
On Generalized EhrenfeuchtMostowski models.
With Ido Feldman.
Articles submitted to peer review journals and preprints

9Pr.
Shelah's Main Gap and the generalized Borelreducibility.
Submitted.

Abstract.
We answer one of the main questions in generalized descriptive set
theory, FriedmanHyttinenKulikov conjecture on the Borelreducibility of the
Main Gap. We show a correlation between Shelah’s Main Gap and generalized
Borelreducibility 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 Borelreducibility. We also show that the following can
be forced: for any countable firstorder theory in a countable vocabulary, T,
the isomorphism of models of T is either Δ11 or analiticallycomplete.

PDF

arXiv

DOI:
10.48550/arXiv.2308.07510
Published articles in peer review journals

8A.
On unsuperstable theories in GDST.
The Journal of Symbolic Logic, 124.

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.

PDF

arXiv

Journal

DOI:
10.1017/jsl.2023.82

7A.
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.

PDF

arXiv

Journal

DOI:
10.1016/j.apal.2021.103044

6A.
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/s1185602122132

5A.
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 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).

PDF

arXiv

Journal

DOI:
10.1007/s00605020014316

4A.
On Σ_1^1completeness of Quasiorders 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 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.

PDF

arXiv

Journal

DOI:
10.4064/fm67912020

3A.
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 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.

PDF

arXiv

Journal

DOI:
10.1215/0029452720190024

2A.
A generalized Borelreducibility 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 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.

PDF

arXiv

Journal

DOI:
10.1007/s0015301705213

1A.
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 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 / 092023

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