# Category Archives: Seminars

## Russell G. Miller – Hilbert’s Tenth Problem for Subrings of the Rational Numbers

Talk held by Russell Miller (Queens College, City University of New York (CUNY), USA) at the KGRC seminar on 2018-11-15.

Abstract: When considering subrings of the field $\mathbb{Q}$ of rational numbers, one can view Hilbert’s Tenth Problem as an operator, mapping each set $W$ of prime numbers to the set $HTP(R_W)$ of polynomials in $\mathbb {Z}[X_1, X_2, \ldots]$ with solutions in the ring $R_W = \mathbb{Z}[W^{-1}]$. The set $HTP(R_{\emptyset})$ is the original Hilbert’s Tenth Problem, known since 1970 to be undecidable. If $W$ contains all primes, then one gets $HTP(\mathbb{Q})$, whose decidability status is open. In between lie the continuum-many other subrings of $\mathbb{Q}$.

We will begin by discussing topological and measure-theoretic results on the space of all subrings of $\mathbb{Q}$, which is homeomorphic to Cantor space. Then we will present a recent result by Ken Kramer and the speaker, showing that the HTP operator does not preserve Turing reducibility. Indeed, in some cases it reverses it: one can have $V <_T W$, yet $HTP(R_W) <_T HTP(R_V)$. Related techniques reveal that every Turing degree contains a set $W$ which is \emph{HTP-complete}, with $W’ \le_1 HTP(R_W)$. On the other hand, the earlier results imply that very few sets $W$ have this property: the collection of all HTP-complete sets is meager and has measure $0$ in Cantor space.

## Konstantin Slutsky: Orbit equivalences of Borel flows

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 14 November 2018, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Konstantin Slutsky

Title: Orbit equivalences of Borel flows

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

Abstract:
The purpose of this talk is to provide an overview of the orbit
equivalence theory of Borel R^n-flows.
An orbit equivalence of two group actions is a bijective map
between phase spaces that maps orbits onto orbits.
Such maps are often further required to posses regularity properties
depending on the category of group actions that is being considered.
For example, Borel dynamics deals with Borel orbit equivalences,
ergodic theory considers measure-preserving maps, topological dynamics
assumes continuity, etc.

Since its origin in 1959 in the work of Henry Abel Dye,
the concept of orbit equivalence has been studied quite extensively.
While traditionally larger emphasis is given to actions of
discrete groups, in this talk we concentrate on free actions
of R^n-flows taking the viewpoint of Borel dynamics.

For a free R^n-action, an orbit can be identified
with an “affine” copy of the Euclidean space, which allows us
to transfer any translation invariant structure from R^n
onto each orbit. The two structures of utmost importance will be
that of Lebesgue measure and standard Euclidean topology.
One may then consider orbit equivalence maps that furthermore
preserve these structures on orbits. Resulting orbit equivalences
are called Lebesgue orbit equivalence (LOE) and time-change
equivalence respectively.

It turns out that properties of LOE maps correspond most closely to
those of orbit equivalence maps between their discrete
counterparts – free Z^n actions.
We illustrate this by outlining a proof of the analog for
R^n-flows of Dougherty-Jackson-Kechris classification
of hyperfinite equivalence relations.
Orbit equivalences of flows often arise as extensions of maps between
cross sections – Borel sets that intersect each orbit in a
non-empty countable set. Furthermore, strong geometric restrictions
on cross-sections are often necessary. As a concrete example,
we explain why one-dimensional R-flows posses
cross sections with only two distinct distances between adjacent
points, and show how this implies classification of R-flows
up to LOE.

We conclude the talk with an overview of time-change equivalence,
emphasizing the difference between Borel dynamics and ergodic theory
and mentioning several open problems.
The interest reader is referred to the technical report on
http://arxiv.org/abs/1504.00958.

## Jing Zhang: A Ramsey theorem for (repeated) sums

Mathematical logic seminar – Nov 13 2018
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Jing Zhang
Department of Mathematical Sciences
CMU

Title: A Ramsey theorem for (repeated) sums

Abstract:

The motivation is the question: for any finite coloring f: R -> r
does there exist an infinite X such that X + X is monochromatic under f?
Hindman, Leader and Strauss showed the answer is negative if CH holds.
Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky showed the
positive answer is consistent relative to the existence of a certain large
cardinal. I will demonstrate how to eliminate the use of large cardinals.
Other variations of the statement will also be discussed, including some
ZFC results.

## Stefan Geschke: There are no universal minimal metric flows for countable discrete groups

Wednesday, November 14, 2018, 11:00
Prague – IM AS CR, Zitna 25, seminar room, front building, third floor

Speaker: Stefan Geschke

Title: There are no universal minimal metric flows for countable discrete groups

Abstract:

Let $G$ be a topological group. A $G$-flow is a compact space $X$ together with a continuous action of $G$. The morphisms between $G$-flows are continuous maps that respect the group action. A flow is minimal if it has no proper (nonempty) subflows. A flow $X$ is universal in a class $\mathcal C$ if it is in $\mathcal C$ and for every flow $Y \in \mathcal C$ there is an epimorphism from $X$ onto $Y$. Using Fürstenberg’s structure theorem for distal flows, Foreman and Beleznay showed that there are no universal minimal metric $\mathbb Z$-flows.

Every group $G$ acts in a natural way on the space $2^G$. Gao and Jackson showed that for every countable discrete group $G$, $2^G$ has a perfect set of minimal subflows. We show that this implies that there are no universal minimal metric flows for any countable discrete group $G$.

## Fulgencio Lopez: A capturing construction scheme from the diamond

Seminar: Working group in applications of set theory, IMPAN

Thursday, 15.11. 2018, 10:15, room 105, IMPAN

Speaker: Fulgencio Lopez (IM PAN)

Title: “A capturing construction scheme from the diamond principle” Continuation from 8.11.2018.

Abstact: “S. Todorcevic introduced the concept of a capturing construction scheme and showed it is consistent with the diamond principle. A construction scheme is a well-founded family of finite subsets of ω1. We give a quick presentation of the history and motivation for this tool and show that it follows from the diamond principle. “.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Noé de Rancourt: Gowers spaces: unifying standard and strategical Ramsey theory

Place: Fields Institute (Room 210)
Date: November 16 , 2018 (13:30-15:00)
Speaker: Noé de Rancourt
Title: Gowers spaces: unifying standard and strategical Ramsey theory
Abstract:

Strategical Ramsey theory was developed in the nineties by Gowers in the
setting of Banach spaces; in this setting where the natural pigeonhole
principle does not always hold, this theory is an alternative to standard
infinite-dimensional Ramsey theory.

In this talk, I will present the formalism of Gowers spaces, an abstract
formalism unifying both strategical and standard infinite-dimensional
Ramsey theory. In this formalism, we can prove an abstract Ramsey theorem
implying both Gowers’ Ramsey-type theorem in Banach spaces, and more
standard Ramsey results like Galvin-Prikry’s theorem. I will also present
a result unifying infinite-dimensional Ramsey theory and determinacy.

I will then introduce a new family of Gowers spaces that arose from a
recent work in progress with Wilson Cuellar-Carrera and Valentin Ferenczi.
These examples from functional analysis are based on local properties of
subspaces of Banach spaces. We hope that examples of the same kind could
be found in other areas of mathematics.

## Shimon Garti: Cardinal invariants for singular cardinals

HUJI Set Theory Seminar

November 14, 2018

Speaker: Shimon Garti

Abstract. We shall try to prove the consistency of d_lambda > r_lambda (and even d_lambda > u_lambda) for a singular cardinal lambda.
This is a joint work with Saharon.

## Jialiang He: Selection covering properties and Cohen reals

BIU seminar in Set Theory

November 12, 2018

Speaker: Jialiang He (BIU)

Title: Selection covering properties and Cohen reals

Abstract: We will introduce four selection covering properties (Menger,
Hurewicz, Rothberger and Gamma property) and consider preserving results
on these properties under Cohen forcing.

## Robert Rałowski: Images of Bernstein sets via continuous functions

Tuesday, November 13, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Robert Rałowski (Wroclaw University of Science and Technology)

Title: Images of Bernstein sets via continuous functions

Abstract:

We examine images of Bernstein sets via continuous mappings. Among other results we prove that there exists a continuous function $f:\mathbb{R}\to\mathbb{R}$ that maps every Bernstein subset of $\mathbb{R}$ onto the whole real line. This gives the positive answer to a question of Osipov. This talk is based upon joint paper with Jacek Cichoń and Michał Morayne.

## Fulgencio Lopez: A capturing construction scheme from the diamond

Seminar: Working group in applications of set theory, IMPAN

Thursday, 08.11. 2018, 10:15, room 105, IMPAN

Speaker: Fulgencio Lopez (IM PAN)

Title: “A capturing construction scheme from the diamond principle”

Abstact: “S. Todorcevic introduced the concept of a capturing construction scheme and showed it is consistent with the diamond principle. A construction scheme is a well-founded family of finite subsets of ω1. We give a quick presentation of the history and motivation for this tool and show that it follows from the diamond principle. “.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/