# Category Archives: Seminars

## Miguel Moreno: The Main Gap in the generalized Borel-reducibility hierarchy

BIU Infinite Combinatorics Seminar

Mon, 11/03/2019 – 13:00

Speaker: Miguel Moreno (BIU)

Title: The Main Gap in the generalized Borel-reducibility hierarchy

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.

## Jonathan Verner: Towers in filters, cardinal invariants, and Luzin type families

Dear all,

The seminar meets on Wednesday March 13th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jonathan Verner — Towers in filters, cardinal invariants, and
Luzin type families

Jonathan will present results from his recent paper (with J. Brendle, B.
Farkas);
We investigate which filters on ω can contain towers, that is, a modulo
finite descending sequence without any pseudointersection. We prove the
following results:
(1) Many classical examples of nice tall filters contain no towers.
(2) It is consistent that tall analytic P-filters contain towers of
arbitrary regular height.
(3) It is consistent that all towers generate non-meager filters.
(4) The statement “Every ultrafilter contains towers.” is independent of
ZFC.

Best,
David

## Wong Tin Lok: End extensions and subsystems of second-order arithmetic

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 13 March 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Wong Tin Lok

Title: End extensions and subsystems of second-order arithmetic

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:
Investigations in reverse mathematics reveal that most naturally
occurring theorems in mathematics are equivalent to one of five
arithmetic axioms nowadays known as the BIG FIVE. These provide
strong empirical evidence for the importance of the Big Five.
In the talk, I will attempt to explain their importance mathematically
in terms of the characteristics of their models.

The work to be presented is joint with Stephen G. Simpson (Vanderbilt).

## Serhii Bardyla – Complete topological semigroups

Talk held by Serhii Bardyla (KGRC) at the KGRC seminar on 2019-03-07.

Abstract: The first part of the talk will be devoted to the investigation of completeness in the class of topological semigroups.

Then we shall discuss a topologization of semigroups of partial isomorphisms between principal ideals of a tree.

## Dilip Raghavan: A small ultrafilter number

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 6 March 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Dilip Raghavan

Title: A small ultrafilter number

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract: It is proved to be consistent relative to a measurable
cardinal that there is a uniform ultrafilter on the real numbers which
is generated by fewer than the maximum possible number of sets. It is
also shown to be consistent relative to a supercompact cardinal that
there is a uniform ultrafilter on aleph_{omega+1} which is generated
by fewer than 2^{aleph_{omega+1}} sets.

This is joint work with Saharon Shelah.

## Yann Pequignot: Finite versus infinite: an insufficient shift

Place: Fields Institute (Room 210)
Date: , 2018 (13:30-15:00)
Speaker: Yann Pequignot
Title:    Finite versus infinite: an insufficient shift
Abstract: The Borel chromatic number – introduced by Kechris, Solecki, and Todorcevic (1999) – generalizes the chromatic number to Borel graphs. While the G_0 dichotomy states that there exists a minimal graph with uncountable Borel chromatic number, it turns out that characterizing when a graph has infinite Borel chromatic number is far more intricate. Even in the case of graphs generated by a single function, the situation is quite complicated. The Shift Graph on the space of infinite subsets of natural numbers is generated by the function that removes the minimum element. It is acyclic but has infinite Borel chromatic number. In 1999, Kechris, Solecki, and Todorcevic asked whether the Shift Graph is minimal among the graphs generated by a single Borel function that have infinite Borel chromatic number. I will sketch a proof that the answer is negative using descriptive complexity considerations and a representation theorem for Sigma^1_2 sets due to Marcone (1994). This result has recently been considerably strengthened by Todorcevic and Vidnyanszky who proved that the set of closed subsets of the Shift Graph that have infinite Borel Chromatic number is Pi^1_2 complete, therefore ruling out most interesting basis results for this class of Borel graphs.

## Hector Alonzo Barriga-Acosta: Some combinatorics on the normality of the countable box product of the convergent sequence

Mathematical logic seminar – Mar 5 2019
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Hector Alonzo Barriga Acosta

Title: Some combinatorics on the normality of the countable box product
of the convergent sequence

Abstract:

The normality of □ (ω + 1)^ω is a question raised in the 40’s (it is
known
that consistently this space is normal). Through the years many different
tecniques have been developed, but non of them have solved the question in
ZFC. We’ll take a look to a combinatorial point of view, given by Judy
Roitman, of this problem.

## Michal Doucha: Definable pseudometrics and Borel reductions between them

Dear all,

The seminar meets on Wednesday March 6th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Michal Doucha — Definable pseudometrics and Borel reductions
between them

I will introduce a “continuous generalizatio” of the theory of
definable equivalence relations and Borel reductions between them.
Equivalence relations will be replaced by pseudometrics and reductions
between them will be replaced by certain uniformly continuous maps. I
will explain our motivation and prove some basic results. I will present
some open problems whose solutions may require completely new ideas from
invariant descriptive set theory. It will be based on a joint paper with
Marek Cúth and Ondřej Kurka.

## Alejandro Poveda: Prikry-type forcing and the failure of the Singular Cardinal Hypothesis

BCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR

Prikry-type forcing and the failure of the
Singular Cardinal Hypothesis

Alejandro Poveda
Universitat de Barcelona

Abstract: Prikry-type forcing plays a central role in
combinatorics due to its close links with central set-theoretic
principles such as the Singular Cardinal Hypothesis (SCH) or the
Tree Property. The original Prikry forcing was devised to change
the cofinality of a measurable cardinal to ω, but there are now
many other more sophisticated constructions that yield more
powerful applications. Among them we can find Magidor and
Radin forcing for changing cofinalities to uncountable cardinals.
In this session we will describe the Prikry forcing with collapses
and present a proof of Magidor’s theorem on the consistency,
relative to appropriate large cardinal hypotheses, of the failure
of the SCH at the first singular cardinal.

Date: Thursday 7 March 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.

## Sandra Müller: Projective determinacy for games of length $\omega^2$ and longer

BIU Infinite Combinatorics Seminar

Sandra Müller (KGRC)
25/02/2019 – 13:0015:00

We will study infinite two player games and the large cardinal strength corresponding to their determinacy. For games of length $\omega$ this is well understood and there is a tight connection between the determinacy of projective games and the existence of canonical inner models with Woodin cardinals. For games of arbitrary countable length, Itay Neeman proved the determinacy of analytic games of length $\omega \cdot \theta$ for countable $\theta\> \omega$ from a sharp for $\theta$ Woodin cardinals.

We aim for a converse at successor ordinals. In joint work with Juan P. Aguilera we showed that determinacy of $\boldsymbol\Pi^1\_{n+1}$ games of length $\omega^2$ implies the existence of a premouse with $\omega+n$ Woodin cardinals. This generalizes to a premouse with $\omega+\omega$ Woodin cardinals from the determinacy of games of length $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1\_1$ payoff.

If time allows, we will also sketch how these methods can be adapted to, in combination with results of Nam Trang, obtain $\omega^\alpha+n$ Woodin cardinals for countable ordinals $\alpha$ and natural numbers $n$ from the determinacy of sufficiently long projective games.