David Fernández-Bretón: Variations and analogs of Hindman’s theorem

Mathematical logic seminar – May 22 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         David Fernández-Bretón
Department of Mathematics
University of Michigan

Title:     Variations and analogs of Hindman’s theorem

Abstract:

Hindman’s theorem is a Ramsey-theoretic result asserting that, whenever one colours the set of natural numbers with finitely many colours, there will be an infinite set such that all numbers that can be obtained by adding finitely many elements from the set (no repetitions allowed) have the same colour. I will explore generalizations and extensions of this theorem: replacing “natural numbers” with “abelian group” and varying the number of colours, as well as the size of the desired monochromatic set, yields a plethora of very interesting results.

Harry Altman: Lower sets in products of well-ordered sets and related WPOs

Thursday, May 24, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Harry Altman (University of Michigan)

Title: Lower sets in products of well-ordered sets and related WPOs

Abstract:

Following last week’s talk on maximum order types of well partial orders, we’ll compute the maximum order type of the set of bounded lower sets in N^m, as well as generalizations to finite products of other well-ordered sets, and discuss the maximum order types of some other related well partial orders also.

Damian Sobota: Rosenthal families and ultrafilters

Dear all,

The seminar meets on Wednesday May 23rd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Damian Sobota — Rosenthal families and ultrafilters

Rosenthal’s lemma is a simple technical result with numerous
applications to measure theory and Banach space theory. The lemma in its
simplest form reads as follows: “For every infinite real-entried matrix
(m(n,k): n,k in N) such that every entry is non-negative and the sum of
every row is <=1, and every epsilon>0, there exists an infinite subset A
of N such that for every k in A we have sum_{n in A, n\neq
k}m_n^k<epsilon.” A natural question arises whether we can choose the
set A from a previously fixed family F of infinite subsets of N. If F
has such a property, then we call it Rosenthal. Thus, Rosenthal’s lemma
states that [N]^omega is Rosenthal. During my talk I’ll present some
necessary or sufficient conditions for a family to be Rosenthal and
prove that under MA(sigma-centered) there exists a P-point which is a
Rosenthal family but not a Q-point. (No Banach space will appear during
the talk.)

Best,
David

Alessandro Vignati: Homeomorphisms of Cech-Stone remainders

Place: Bahen Centre Information T (Room BA 2165)
Date: May 18, 2018 (13:30-15:00)
Speaker: Alessandro Vignati
Title: Homeomorphisms of Cech-Stone remainders
Abstract: From a locally compact space X one construct its Cech-Stone remainder X*=beta X minus X. We analyze the problem on whether X* and Y* can be homeomorphic for different spaces X and Y. In the 0-dimensional case, a solution to this problem has been proved to be independent of ZFC, by the work of Parovicenko, Farah, Dow-Hart and Farah-McKenney among others.
We prove, under PFA, the strongest possible rigidity result: for metrizable X and Y, we prove that X* is homeomorphic to Y* only if X and Y are homeomorphic modulo compact subsets. We also show that every homeomorphism X* –> Y* lifts to an homeomorphism between cocompact subsets of X and Y.

François Le Maître: Ample generics and full groups

Mathematical logic seminar – May 15 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         François Le Maître
Institut de Mathématiques de Jussieu-PRG
Université Paris Diderot

Title:     Ample generics and full groups

Abstract:

In this talk, I will explain why the full group of the action of the rationals on the real line is an example of a connected topologically simple Polish group with ample generics, providing a natural answer to a question of A. Kechris and C. Rosendal. If time permits, I will discuss the following open question: is there a Polish group with ample generics which is not quasi non-archimedean ? The talk is based on a joint work with Adriane Kaïchouh.

Yair Hayut: Stationary reflection at $\aleph_{\omega+1}$

Talk held by Yair Hayut (Tel-Aviv University, Israel)
at the KGRC seminar on 2018-05-17.

Abstract: Stationary reflection is one of the basic prototypes of reflection phenomena,
and its failure is related to many counterexamples for compactness
properties (such as almost free non-free abelian groups,
and more). In 1982, Magidor showed that it is consistent, relative to infinitely many
supercomapct cardinals, that stationary reflections holds at $\aleph_{\omega + 1}$.
In this talk I’m going to present a new method for forcing stationary reflection
at $\aleph_{\omega+1}$, which allows to significantly reduce the upper bound for the consistency strength of the full stationary reflection at $\aleph_{\omega+1}$ (below a single partially supercompact cardinal).

This is a joint work with Spencer Unger.

Harry Altman: Well partial orderings and their maximum extending ordinals

Thursday, May 17, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Harry Altman (University of Michigan)

Title: Well partial orderings and their maximum extending ordinals

Abstract:

A well partial order is a partial order all of whose extensions to a total order are well-orders. (These are often studied as well-quasi-orders, where the requirement of antisymmetry is dropped.) In 1976 De Jongh and Parikh showed that for a given WPO X, among the ordinals obtained this way there is always a maximum o(X). We will discuss the theory of WPOs and o(X), several equivalent formulations, and how o(X) can actually be computed for some concrete WPOs.

Osvaldo Guzman: hm and the ultrafilter number

Place: Bahen Centre Information T .   BA 2165
Date: May 11, 2018 (13:30-15:00)
Speaker: Osvaldo Guzman
Title: hm and the ultrafilter number
Abstract: The cardinal invariant $\mathfrak{hm}$ is defined as the minimum size of a family of $\mathsf{c}_{\mathsf{min}}$-monochromatic sets that cover $2^{\omega}$ (where $\mathsf{c}_{\mathsf{min}}\left( x,y\right) $ is the parity of the biggest initial segment both $x$ and $y$ have in common). We prove that $\mathfrak{hm}=\omega_{1}$ holds in the Shelah’s model of $\mathfrak{i<u}$ so the inequality $\mathfrak{hm<u}$ is consistent with the axioms of $\mathsf{ZFC.}$ This answers a question of Thilo Weinert.

Andrzej Starosolski: The Rudin-Keisler ordering of P-points under b=c

Tuesday, May 15, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Andrzej Starosolski (Silesian University of Technology)

Title: The Rudin-Keisler ordering of P-points under $\mathfrak b=\mathfrak c$

Abstract:

M. E. Rudin proved under CH that for each P-point there exists another P-point strictly RK-greater . Assuming $\mathfrak p=\mathfrak c$, A. Blass showed the same; moreover, he proved that each RK-increasing $\omega$-sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-preordering. He also asked what ordinals can be embedded in the set of P-points.
In my talk the results cited above are proved and the mentioned question is answered under a (weaker) assumption $\mathfrak b =\mathfrak c$.

Jonathan Verner: Ultrafilters and nonstandard models of arithmetic

Dear all,

The seminar meets on Wednesday May 9th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jonathan Verner — Ultrafilters and nonstandard models of
arithmetic

(This is a talk postponed from April 18th.)

Best,
David