# Category Archives: Seminars

## Dilip Raghavan: An application of PCF theory to cardinal invariants above the continuum

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 24 January 2018, 17:00 hrs

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

Speaker: Dilip Raghavan

Title: An application of PCF theory to cardinal invariants above the continuum

Abstract: It will be proved in ZFC that if kappa
is any regular cardinal greater than beth_omega, then
d(kappa) leq r(kappa). Here d(kappa) is the
smallest size of dominating family of functions from kappa
to kappa and r(kappa) is the smallest size of a family
of subsets of kappa which decide every other subset of kappa.
This result partially dualizes an earlier result
of myself and Shelah. The proof uses the revised GCH,
which is an application of PCF theory.
This is joint work with Shelah.

## Fulgencio Lopez: Adding Cohen reals also adds a capturing Construction Scheme

Place: Fields Institute (Room 210)

Date: January 19, 2018 (13:30-15:00)

Speaker: Fulgencio Lopez

Abstract: We show that adding $\kappa\geq \omega_1$ Cohen reals adds a capturing construction scheme.

## Jing Zhang: Rado’s Conjecture and its Baire Version

Mathematical logic seminar – Jan 23 2018
Time:     3:30pm – 4:30 pmRoom:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     Rado’s Conjecture and its Baire Version

Abstract:

Rado’s Conjecture is a reflection/compactness principle formulated by Todorčević, who also showed its consistency relative to the existence of strongly compact cardinals. One of its equivalent forms asserts that any nonspecial tree of height ω1 has a nonspecial subtree of size less or equal to ℵ1. Although it is incompatible with Martin’s Axiom, Rado’s Conjecture turns out to imply a lot of consequences of forcing axioms, for example Strong Chang’s Conjecture, failure of square principles, the semi-stationary reflection principle, the Singular Cardinal Hypothesis etcetera. In fact, almost all known consequences of Rado’s Conjecture are consequences of a weaker statement, the Baire version of it which asserts any Baire tree of height ω1 has a nonspecial subtree of size less or equal to ℵ1.

We will show that in the forcing extension by countable support iteration of Sacks forcing of strongly compact length, the Baire version of Rado’s Conjecture holds. Using a classical Mitchell style model, we show Rado’s conjecture along with not-CH does not imply ω2 has the super tree property, answering a question by Torres-Pérez and Wu. We will also see that in general the Baire version of Rado’s Conjecture does not imply Rado’s Conjecture.

## Clinton Conley: Unfriendly colorings of measure-preserving graphs of finite cost

Mathematical logic seminar – Jan 16 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Unfriendly colorings of measure-preserving graphs of finite cost

Abstract:

We show that any measure-preserving Borel graph on a standard probability space with finite average degree admits a Borel unfriendly coloring on a conull set. This generalizes the results for group actions discussed last semester, and is joint work with Omer Tamuz.

## Osvaldo Guzman Gonzalez: On (1,w_1)-weakly universal functions

Place: Fields Institute (Room 210)

Date: January 12, 2018 (13:30-15:00)

Speaker: Osvaldo Guzman Gonzalez

Title: On (1,w_1)-weakly universal functions

Abstract: We will study a very weak notion of universality of functions in Sacks models. We will answer a question of Shelah and Steprans by showing that there are no (1,w_1)-weakly universal functions after adding uncountably many Sacks reals side by side.

## Egbert Thümmel: Ramsey analytic ideals

The seminar meets on Wednesday January 17th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Egbert Thümmel — Ramsey analytic ideals
It is an open problem whether analytic tall ideals can be Ramsey. We
show that this is connected to another problem of Hrušák: If an analytic
tall Ramsey ideal exists, then it is nowhere Katětov above the ideal
conv and has no F-sigma extension.

## David Schrittesser – On the Complexity of Maximal Cofinitary Groups

Talk held by David Schrittesser (KGRC) at the KGRC seminar on 2018-01-11.

Abstract: A maximal cofinitary group is a subgroup of the group of permutations of the set of natural numbers $\mathbb N$ such that any group element has only finitely many fixed points, and no strictly larger group of permutations of $\mathbb N$ has this property. Improving a result of Horowitz and Shelah, we show that there is a closed maximal cofinitary group.

## Dana Bartošová: Ellis problem for automorphism groups

The seminar meets on Wednesday January 10th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Dana Bartošová — Ellis problem for automorphism groups

Abstract: It is an old problem of Ellis to determine whether two
prominent dynamical system of a given topological group are isomorphic.
For discrete groups, only the integers are known to be a counterexample
by a complex result of Glasner and Weiss. Trivially, groups acting with
a fixed point under any action are counterexamples. We extend the class
of counterexamples to a few automorphism groups and we will have a
closer look at the full permutation group, $S_{\infty}$. This leads us
to questions about the existence of certain ultrafilters on natural
numbers. This is a joint work with Andy Zucker.

## Jan Grebik: Borel selector for Hypergraphons

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

Program: Jan Grebik — Borel selector for Hypergraphons
We recall all relevant definitions and show that there is a Borel way of
choosing a concrete representant of a hypergraphon.

## Omer Mermelstein: ​Searching for template structures in the class of Hrushovski ab initio geometries

BGU seminar in Logic, Set Theory and Topology.
Tuesday, ​December ​​​26th, 12:15-13:3, Seminar room -101, Math building 58.
Speaker:​ ​ Omer Mermelstein (BGU)

Title: ​​Searching for template structures in the class of Hrushovski ab initio geometries

Abstract. Zilber’s trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets — disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic “template” — a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries — non-modular, yet prohibiting any algebraic structure.

In this talk we take a step towards defining “template” structures for the class of (CM-trivial) ab initio Hrushovski constructions. After presenting intuitively the standard ab initio Hrushovski construction, we generalize Hrushovski’s predimension function, showing that the geometries associated to certain Hrushovski constructions are, essentially, ab initio constructions themselves. If time permits, we elaborate on how these geometric structures may generate the class of geometries of ab initio constructions under the Hrushovski fusion operation.