Welcome to Miguel A. Cardona's homepage

Mathematical Path 

I am a researcher in set theory and mathematical logic, with a focus on forcing theory, cardinal invariants, and the combinatorics of the real line. My work explores how forcing constructions illuminate the structure of the continuum, especially the interactions between measure, category, and algebraic properties of the reals. I am currently conducting research at the Institución Universitaria Pascual Bravo in Medellín, Colombia.

From April 2024 to September 2025, I was a Post-doctoral Fellow at the Einstein Institute of Mathematics at the Hebrew University of Jerusalem, under the supervision of Prof. Saharon Shelah. My research there focused on the combinatorial and forcing-theoretic aspects of constant evasion and prediction, studying predictors that constantly approximate or evade reals and analyzing their associated cardinal characteristics. During this period, I also developed new notions of linked forcing that provide fine control over the additivity of the null ideal, contributing new tools to the study linked properties.

Between April 2022 and March 2024, I held a Post-doctoral Fellowship at the Pavol Jozef Šafárik University in Košice, Slovakia, as a member of the Set Theory and Topology Seminar. My research in Košice focused on strong measure zero sets, cardinals generated by ideals, and the introduction and analysis of parameterized ideals together with their associated cardinal invariants. These investigations connected ideal theory with the fine structure of the continuum, extending classical approaches to measure and category.

Earlier, from January to March 2022, I was a Post-doctoral Fellow in the Set Theory Group at the Institute of Discrete Mathematics and Geometry, Technical University of Vienna, under the mentorship of Dr. Jakob Kellner.

I earned my Ph.D. in Mathematics at the Technical University of Vienna, working under Dr. Jakob Kellner and Assoc. Prof. Diego A. Mejía. My dissertation, Forcing theory and combinatorics of the real line, investigates how specific forcing notions shape the continuum and the behavior of classical and modern cardinal invariants. The official version of my thesis can be found here.

I also hold an M.Sc. in Mathematics from the Universidad Nacional de Colombia (Medellín), where I worked under Assoc. Prof. Diego A. Mejía on Cardinal invariants associated with ideals between measure zero and strong measure zero, and a B.Sc. in Mathematics from the same institution, supervised by Prof. Andrés Villaveces.

My current research continues to explore the connections between forcing, topology, and combinatorics, with an emphasis on building new frameworks that reveal structural principles governing the continuum. 

Below you will find my current projects and works in preparation. For a complete list of my publications and preprints, please visit the Publications tab. Each project contributes to a broader vision: refining how we understand the continuum through new combinatorial and forcing frameworks.  

News

• December 15. I gave a talk at the Primer Encuentro de Lógica y Fundamentos de las Matemáticas at the University of Antioquia.

• September 5. I was invited to give a talk at the conference held on the occasion of Jörg Brendle’s 60th birthday, Kobe, Japan.

• September 8–12. I was invited to give a talk at the 18th Asian Logic Conference (ALC), Kyoto, Japan.

Recent acceptances:

• • Miguel A. Cardona, Adam Marton, and Jaroslav Supina. Cardinal characteristics associated with small subsets of reals.
    Accepted for publication in Fundamenta Mathematicae.

  • Miguel A. Cardona, Diego A. Mejía, and Ismael E. Rivera-Madrid.
    Directed schemes of ideals and cardinal characteristics I: the meager-additive ideal.
    Accepted for publication in Comptes Rendus Mathématiques.

  • Miguel A. Cardona.
    Soft-linkedness.
    Accepted for publication in Real Analysis Exchange.

Frontiers of the Continuum:

Current projects exploring new combinatorial and forcing approaches to the real line 

   In preparation: 

1. Miguel A. Cardona and Miroslav Repicky. Many Ways to Differ: New Cardinal Invariants of the Continuum:

Developing a new family of cardinal invariants arising from difference phenomena between classical combinatorial properties.

2. Miguel A. Cardona and Adam Marton. Bounded by k-Slaloms:

Generalizing certain relations introduced by Vojtáš, called k-slaloms, which are closely related to cardinals in Cichoń’s diagram. 

3. Miguel A.  Cardona, Miroslav Repicky, and Saharon Shelah. Solving a problem on halfway cardinal invariant:

Introducing a new Knaster‑type property to control the covering number of the null ideal, resolving open questions on halfway invariants originally studied by Brendle, Shelah, and collaborators. 

4. Miguel A. Cardona and Miroslav Repicky.  On non-constant prediction and evasion number:

Introducing non-constant versions of classical prediction and evasion numbers, highlighting new combinatorial behaviors in these cardinal characteristics. 

5. Miguel A.  Cardona, Diego A. Mejía, Miroslav Repicky, Jaroslav Supina y Andrés F. Uribe-Zapata. On Pseudointersection Numbers.  

6. Miguel A. Cardona. Nice linkedness posets do not add dominating reals. 

Ongoing:

7. Miguel A.  Cardona and Saharon Shelah. Long iterations for the continuum: 

We develop an iteration theorem for proper ℵ₂‑cc forcing notions with a form of countable support, and analyze special cases, including quasi-dense forcings preceded by random forcing. 

8.  Miguel A.  Cardona, Miroslav Repicky, and Saharon Shelah  Constant prediction and evasion number, II: not adding predictors: 

Developing forcing notions that preserve constant prediction while controlling the additivity of the null ideal

9. Miguel A. Cardona. Variants of constant evasion and constant prediction numbers:

Introducing new variants related to the constant prediction and evasion cardinals, revealing previously unexplored combinatorial structures of the real line. 

10. Miguel A. Cardona and Saharon Shelah.  Random iteration:

11.  Miguel A.  Cardona, Diego A. Mejía, and Ismael E. Rivera-Madrid.  Directed schemes of ideals and cardinal characteristics, II: the null additive ideal.

Keep in mind:

This is my new and updated page. I no longer have access to the previous one (https://sites.google.com/mail.huji.ac.il/miguel-cardona-montoya/home-page).

I would recommend that you visit this site for the most recent information about my work.