Invited Talks

28-IT. The Borel reducibility Main Gap

February 2024. 7th workshop on generalised Baire spaces. Invited main speaker. Bristol, England.
Slides - Conference

27-IT. Borel-reducibility counterparts of Shelah's classification theory

October 2023. Model Theory Seminar, National University of Colombia. Bogota, Colombia. Online.
Slides - Seminar

26-IT. A Borel-reducibility Main Gap

September 2023. Workshop on Generalized Descriptive Set Theory - Annual Meeting of the Swiss Society for Logic and Philosophy of Science. Invited main speaker. Lausanne, Switzerland.
Slides - Conference

25-IT. Classification Theory in Generalized Descriptive Set Theory

September 2023. ÖMG Tagung 2023 - Meeting of the Austrian Mathematical Society. Invited special session. Graz, Austria.
Slides - Conference

24-IT. Shelah's Main Gap and the generalized Borel-reducibility

July 2023. Sixth workshop on generalised Baire spaces. Vienna, Austria.
Slides - Conference

23-IT. An application of combinatorics into music [ESP]

April 2023. DiscreMath Seminar, National University of Colombia. Bogota, Colombia. Online.
Slides / Sounds - Seminar

22-IT. About Filter Reflection

February 2023. Arctic Set Theory Workshop VI. Kilpisjärvi, Finland.
Slides - Conference

21-IT. Finding the main gap in the generalized descriptive set theory

December 2022. Canadian Mathematical Society Winter Meeting. Invited special session. Toronto, Canada.
Slides - Conference

20-IT. Indestructibility and characterization of filter reflection

September 2022. University of Helsinki Logic Seminar. Helsinki, Finland. Online.
Slides - Seminar

19-IT. Reflection of stationary sets and GDST

August 2022. Young Set Theory Workshop 2022. Invited postdoctoral speaker. Novi Sad, Serbia.
Slides - Conference

18-IT. The isomorphism relation of unsuperstable theories in the generalized Borel-reducibility

April 26th, 2022. KGRC Set Theory Research Seminar. Vienna, Austria.
Seminar

17-IT. How complex are mathematical theories? [ESP]

March 2022. Internnational Day of Mathematics 2022. Bogota, Colombia.
Slides - Event

16-IT. K-colorable linear orders and unsuperstable theories

February 2022. Helsinki Logic Seminar - Minisymposium. Helsinki, Finland.
Slides - Minisymposium

15-IT. On unsuperstable theories in the GDST

February 2022. Arctic Set Theory Workshop V. Kilpisjärvi, Finland.
Slides - Conference

14-IT. Colouring orders and ordering trees

February 2022. Bar-Ilan Set Theory Colloquium. Ramat Gan, Israel. Online.
Slides - Seminar

13-IT. Filter Reflection and Generalised Descriptive Set Theory

October 2020. University of Helsinki Logic Seminar. Helsinki, Finland. Online.
Slides - Seminar

12-IT. Filter Reflection

May 2020. Bar-Ilan University and Hebrew University Logic Seminar. Ramat Gan - Jerusalem, Israel. Online.
Slides - Video - Seminar

11-IT. Consistency of Filter Reflection

May 2020. Seminario Flotante de Bogotá de Lógica Matemática. Bogotá Logic Group, Colombia. Online.
Slides - Video - Seminar

10-IT. Connections between generalised Baire spaces and model theory

February 2020. Fifth Workshop on Generalised Baire Spaces. Invited main speaker. Bristol, England.
Slides - Conference

9-IT. Diamond sharp and a model theory dichotomy in GDST

January 2019. Arctic Set Theory Workshop IV. Kilpisjärvi, Finland.
Slides - Conference

8-IT. A model theory dichotomy in generalized descriptive set theory

October 2018. UH-CAS Workshop on mathematical logic. Helsinki, Finland.
Slides - Conference

7-IT. La relación de equivalencia modulo conjuntos no estacionarios en los espacios de Baire generalizados

September 2017. Bogotá Logic Seminar. Bogotá, Colombia.
Notes (section 3) - Seminar

6-IT. Conjuntos de indiscernibles en teorias estables

September 2017. Seminario de categorias accessibles y teoria de modelos, Universidad Nacional de Colombia sede Bogotá. Bogotá, Colombia.

5-IT. The equivalence modulo non-stationary ideals

January 2017. Arctic Set Theory Workshop III. Kilpisjärvi, Finland.
Slides - Conference

4-IT. Strong DOP and the Borel hierarchy

September 2016. Workshop on Set-theoretical aspects of the model theory of strong logics. Bellaterra, Barcelona.
Slides - Conference

3-IT. Shelah's Main Gap Theorem in the Borel-reducibility hierarchy

March 2016. 27th Nordic Congress of Mathematicians. Invited special session. Stockholm, Sweden.
Slides - Conference

2-IT. Borel reducibility and the isomorphism relation

January 2016. Finnish Mathematical Days 2016. Invited special session. Turku, Finland.
Slides - Conference

1-IT. On the reducibility of isomorphism relation

January 2015. Arctic Set Theory Workshop II. Kilpisjärvi, Finland.
Conference

Public Examination

FINDING THE MAIN GAP IN THE BOREL-REDUCIBILITY HIERARCHY

December 2017. Exactum, Faculty of Science, University of Helsinki.
PDF - Slides - Opponent's lecture

Posters Presentations

2-PP. Filter Reflection and the Borel reducibility

July 2022. Advances in Set Theory 2O22. Jerusalem, Israel.
Poster - Conference - DOI: 10.13140/RG.2.2.16183.60326

1-PP. Filter Reflection and the Borel reducibility

July 2022. ESI Set Theory Workshop 2022. Vienna, Austria.
Poster - Conference - DOI: 10.13140/RG.2.2.16183.60326

Contributed Talks

5-CT. The Borel reducibility Main Gap

September 2024. European Set Theory Conference 2024. Münster, Germany.
Slides - Conference

4-CT. Ordered trees and the kappa-Borel reducibility of unsuperstable theories

August - September 2022. European Set Theory Conference 2022. Turin, Italy.
Slides - Video - Conference

3-CT. Diamond sharp and the Generalized Baire Spaces

July 2019. 7th European Set Theory Conference. Vienna, Austria.
Slides - Conf

2-CT. Reflection principles and the generalized Baire spaces

July 2017. 6th European Set Theory conference. Budapest, Hungary.
Slides - Conf

1-CT. The equivalence Modulo Non-stationary Ideals and Shelah's Main Gap Theorem in the Borel-reducibility hierarchy

September 2016. Bonn Set Theory Workshop 2016, Generalized Baire spaces. Bonn, Germany.
Slides - Conf

Seminar Talks

26-ST. On the Borel reducibility Main Gap (the models)

20.03.2024. Logic Seminar, University of Helsinki.

Abstract.

One of the biggest motivations in Generalized Descriptive Set Theory has been the program of identifying counterparts of classification theory in the setting of Borel reducibility. The most important has been the division line classifiable vs non-classifiable, i.e. identify Shelah's Main Gap in the Borel reducibility hierarchy. This was studied by Friedman, Hyttinen and Weinstein (ne Kulikov) in their book "Generalized descriptive set theory and classification theory". Their work led them to conjecture the following:

If T is a classifiable theory and T' is a non-classifiable theory, then the isomorphism relation of T is Borel reducible to the isomorphism relation of T'.

In this talk we will prove this conjecture, discuss the objects introduced, and provide a detailed overview of the gaps between classifiable shallow theories, classifiable non-shallow theories, and non-classifiable theories. The main result that will be presented is the following:

Suppose \kappa=\lambda^+=2^\lambda, 2^c \leq \lambda = \lambda^{\omega_1} and T_1 and T_2 are countable complete first-order theories in a countable vocabulary. If T_1 is a classifiable theory and T_2 is a non-classifiable theory, then the isomorphism relation of T_1 is continuously reducible to the isomorphism relation of T_2, and the isomorphism relation of T_2 is strictly above the isomorphism relation of T_1 in the Borel reducibility hierarchy.

This talk will be based on the article Shelah's Main Gap and the generalized Borel-reducibility (https://arxiv.org/abs/2308.07510)

Slides

25-ST. On the Borel reducibility Main Gap (the weak isolated linear order)

24.01.2024. Logic Seminar, University of Helsinki.

Abstract.

One of the biggest motivations in Generalized Descriptive Set Theory has been the program of identifying counterparts of classification theory in the setting of Borel reducibility. The most important has been the division line classifiable vs non-classifiable, i.e. identify Shelah's Main Gap in the Borel reducibility hierarchy. This was studied by Friedman, Hyttinen and Weinstein (ne Kulikov) in their book "Generalized descriptive set theory and classification theory". Their work led them to conjecture the following:

If T is a classifiable theory and T' is a non-classifiable theory, then the isomorphism relation of T is Borel reducible to the isomorphism relation of T'.

In this talk we will prove this conjecture, discuss the objects introduced, and provide a detailed overview of the gaps between classifiable shallow theories, classifiable non-shallow theories, and non-classifiable theories. The main result that will be presented is the following:

Suppose \kappa=\lambda^+=2^\lambda, 2^c \leq \lambda = \lambda^{\omega_1} and T_1 and T_2 are countable complete first-order theories in a countable vocabulary. If T_1 is a classifiable theory and T_2 is a non-classifiable theory, then the isomorphism relation of T_1 is continuously reducible to the isomorphism relation of T_2, and the isomorphism relation of T_2 is strictly above the isomorphism relation of T_1 in the Borel reducibility hierarchy.

This talk will be based on the article Shelah's Main Gap and the generalized Borel-reducibility (https://arxiv.org/abs/2308.07510)

Slides

24-ST. On the Borel reducibility Main Gap (the shallow - non-shallow gap)

17.01.2024. Logic Seminar, University of Helsinki.

Abstract.

One of the biggest motivations in Generalized Descriptive Set Theory has been the program of identifying counterparts of classification theory in the setting of Borel reducibility. The most important has been the division line classifiable vs non-classifiable, i.e. identify Shelah's Main Gap in the Borel reducibility hierarchy. This was studied by Friedman, Hyttinen and Weinstein (ne Kulikov) in their book "Generalized descriptive set theory and classification theory". Their work led them to conjecture the following:

If T is a classifiable theory and T' is a non-classifiable theory, then the isomorphism relation of T is Borel reducible to the isomorphism relation of T'.

In this talk we will prove this conjecture, discuss the objects introduced, and provide a detailed overview of the gaps between classifiable shallow theories, classifiable non-shallow theories, and non-classifiable theories. The main result that will be presented is the following:

Suppose \kappa=\lambda^+=2^\lambda, 2^c \leq \lambda = \lambda^{\omega_1} and T_1 and T_2 are countable complete first-order theories in a countable vocabulary. If T_1 is a classifiable theory and T_2 is a non-classifiable theory, then the isomorphism relation of T_1 is continuously reducible to the isomorphism relation of T_2, and the isomorphism relation of T_2 is strictly above the isomorphism relation of T_1 in the Borel reducibility hierarchy.

This talk will be based on the article Shelah's Main Gap and the generalized Borel-reducibility (https://arxiv.org/abs/2308.07510)

Slides

23-ST. Isolation notions and construction of models (S-DOP and the construction of models)

17.05.2023. KGRC Model Theory Research Seminar, University of Vienna.

Abstract.

During this talk we introduce the notions of F-constructible sets and F-primary models. We use the F-primary models to show that the isomorphism relation of any classifiable theory T is continuously reducible to the isomorphism relation of any non-classifiable theory with S-DOP.

Notes

22-ST. Isolation notions and construction of models (example and strong types)

10.05.2023. KGRC Model Theory Research Seminar, University of Vienna.

Abstract.

During this talk we give three examples of isolation notions (Ft, Fs, Fa). We introduce the concept of strong type and discuss some of its properties.

Notes

21-ST. Isolation notions and construction of models (motivation and axioms)

3.05.2023. KGRC Model Theory Research Seminar, University of Vienna.

Abstract.

During this talk we will introduce the axioms of isolation notions. We discuss the motivation comming from GDST into the construction of F-pimary models.

Notes

20-ST. Generalised Descriptive Set Theory, part III

28.03.2023. KGRC Set Theory Research Seminar, University of Vienna.

Abstract.

Following part I and part II in this three part series, during this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided into two kinds: classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored ordered trees. The goal of the talk is to see the classifiable theories case and sketch the ideas of non-classifiable theories.

Notes

19-ST. Generalised Descriptive Set Theory, part II

21.03.2023. KGRC Set Theory Research Seminar, University of Vienna.

Abstract.

We have introduced the notions of κ-Borel class, κ-analytic class, κ-analytic-coanalytic class, κ-Borel* class in the previous talk. In descriptive set theory the Borel class, the analytic-coanalytic class, and the Borel* class are the same class, we showed that this doesn't hold in the generalized descriptive set theory.
In this talk, we will show the consistency of "κ-Borel* class is equal to the κ-analytic class". This was initially proved by Hyttinen and Weinstein (former Kulikov), under the assumption V=L. We will show a different proof that shows that this holds in L but also can be forced by a cofinality-preserving GCH-preserving forcing from a model of GCH, but also by a <κ-closed κ+‑cc forcing.

Notes

18-ST. Generalised Descriptive Set Theory, part I

14.03.2023. KGRC Set Theory Research Seminar, University of Vienna.

Abstract.

This is the first of three talks about Generalised Descriptive Set Theory. The aim of this talk is to introduce the notions of κ‑Borel class, κ‑analytic class, κ‑analytic-coanalytic class, κ‑Borel* class, and show the relation between these classes.

Notes

17-ST. Fake Reflection

23.01.2020. KGRC Research Seminar, Kurt Gödel Research Center, University of Vienna.

Abstract.

Motivated from many results in generalized descriptive set theory, Filter Reflection (aka Fake Reflection) is an abstract version of reflection compatible with large cardinals, forcing axioms, but also V=L.
In this talk we will present the motivation and definition of filter reflection, we will explain how to force filter reflection and how to force its failure. We will also show some applications and properties of filter reflection, e.g. the consistency of “E^K_{w_1} filter reflects to a subset of E^K_{w}”. This is a joint work with Gabriel Fernandes and Assaf Rinot.

16-ST. The Main Gap in the generalized Borel-reducibility hierarchy

11.03.2019. Set Theory Colloquium, Bar-Ilan University.

Abstract.

During this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided in two kind:classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored trees. The goal of the talk is to see the classifiable theories case and start the non-classifiable theories case by proving that it is possible to map every element of the generalized Baire, f, into a colored tree, J(f), such that; for every f and g elements of the generalized Baire space, J(f) and J(g) are isomorphic as colored trees if and only if f and g coincide on a club.

15-ST. An introduction to generalized descriptive set theory, part 4

14.01.2019. Set Theory Colloquium, Bar-Ilan University.

Abstract.

Last week, we gave a detailed proof of Lemma 1.13 from the notes: http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf
This week, we shall continue, proving that, if V=L, then $\kappa$-Borel* class is equal to the Sigma1^_1(K) class.

14-ST. An introduction to generalized descriptive set theory, part 3

07.01.2019. Set Theory Colloquium, Bar-Ilan University.

Abstract.

After introducing the notions of K-Borel class, K-Delta_1^1 class, K-Borel^* class we saw some subset relations between them in the previous talk ( http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf ). We finished the previous talk with a sketch of the proof of:
if V=L, then $\kappa$-Borel* class is equal to the $\Sigma1^ 1(\kappa)$ class.
We will see this proof in complete detail, starting from the key lemma, Lemma 1.13 on the notes.

13-ST. An introduction to generalized descriptive set theory, part 2

31.12.2019. Set Theory Colloquium, Bar-Ilan University.

Abstract.

After introducing the notions of K-Borel class, K-Delta_1^1 class, K-Borel^* class in the previous talk ( http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf ), in this talk, we will show the relation between this classes.
In descriptive set theory the Borel class, the Delta_1^1 class, the Borel* class are the same class, this doesn't hold in the generalized descriptive set theory, in particular under the assumption V=L the Borel* class is equal to the Sigma1^_1 class.

12-ST. An introduction to generalized descriptive set theory, part 1

10.12.2018. Set Theory Colloquium, Bar-Ilan University.

Abstract.

This is the first of many of talks in which an overview of the Borel-reducibility hierarchy in the generalized Baire space will be given. The aim of this talk is to introduce the notions of K-Borel class, K-Delta_1^1 class, K-Borel^* class, and show the relation between these classes.

11-ST. Σ_1^1-complete quasi-orders in L

21.03.2018. Logic Seminar, Helsinki Logic Group.

Abstract.

One of the basic differences between descriptive set theory (DST) and generalized descriptive set theory (GDST) is the existence of some analytic sets in GDST that have no counterpart in DST. The equivalence modulo the non-stationary ideal restricted to a stationary set S (EM-S) is an example of these sets.
These relations have been studied in GDST to understand some of the differences between DST and GDST. In particular, EM-S, with S the set of ordinals with cofinality alpha, has been used to study the isomorphism relations in the Borel-reducibility hierarchy. One of the main results related to this is:
If V=L, then EM-S, with the set of ordinals with cofinality alpha, is a complete analytic equivalence relation in the generalized Baire space.
From this, it was possible to prove that the isomorphism relation of theories with OCP or S-DOP are complete analytic equivalence relations in L(under some cardinal assumptions).
In this talk I will show that the inclusion modulo the non-stationary ideal restricted to the set of ordinals with cofinality alphais is a complete analytic quasi-order in L. This result has many corollary, two of which are:
(V=L) The isomorphism relation of a theory is either a complete analytic equivalence relation or a Delta_1^1 equivalence relation (under some cardinal assumptions).
(V=L) If kappa is not the successor of an omega-cofinal cardinal, then the embedability of dense linear orders is a complete analytic relation.

10-ST. Σ_1^1-complete quasi-orders on weakly compact cardinals

18.10.2017. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we prove Σ_1^1-complete properties of some quasi-orders when K is a weakly compact cardinals.

9-ST. Reflection principles and Borel reducibility

29.03.2017. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we explore implications of reflection principles in the Borel-reducibility hierarchy.

8-ST. The isomorphism relation of Theories with S-DOP

14.09.2016. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we explore the Borel-reducibility of the isomorphism relation of superstable theories with S-DOP.

7-ST. A Borel-reducibility counterpart of Shelah's main gap theorem

02.12.2015. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we prove a Borel-reducibility counterpart of Shelah's main gap theorem.

6-ST. The isomorphism relation of classifable theories

04.11.2015. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we explore the Borel-reducibility of the isomorphism relation of classifiable theories.

5-ST. On the reducibility of the isomorphism relation II

28.01.2015. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we continue exploring the Borel-reducibility of the isomorphism relation of stable unsuperstable theories.

4-ST. On the reducibility of the isomorphism relation I

21.01.2015. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we explore the Borel-reducibility of the isomorphism relation of stable unsuperstable theories.

3-ST. More on Stationary tower forcing

02.03.2014. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we continue exploring the definitions and implications of the stationary tower forcing.

2-ST. Stationary tower forcing

22.01.2014. Logic Seminar, Helsinki Logic Group.

Abstract.

In this talk we explore the definitions and implications of the stationary tower forcing.

1-ST. Matroides representables en GF(2)

21.04.2010. Seminario de Estudiantes, Universidad Nacional de Colombia sede Bogotá.

Abstract.

Esta charla tiene como objetivo el dar a conocer algunos métodos para representar matroides en campos finitos. Para ellos se darán a conocer las nociones básicas de la teoría de matroides y a teoremas básicos de representación de una matroide.