Archives of: Kurt Godel Research Center

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.

Moritz Müller: On the relative strength of finitary combinatorial principles

Talk held by Moritz Müller (KGRC) at the KGRC seminar on 2017-12-14 at 4pm .

Abstract: Define a finitary combinatorial principle to be a first-order sentence which is valid in the finite but falsifiable in the infinite. We aim to compare the strength of such principles over a weak arithmetic. We distinguish “weak” and “strong” principles based on their behaviour with respect to finite structures that are only partially defined. The talk sketches a forcing proof of a theorem stating that over relativized $T^1_2$ “weak” principles do not imply “strong” ones.

Maxwell Levine: Forcing Square Sequences

KGRC research seminar on 2017-11-30 at 4pm.

Speaker: Maxwell Levine (KGRC)

Abstract: In the 1970’s, Jensen proved that Gödel’s constructible universe $L$ satisfies a combinatorial principle called $\square_\kappa$ for every uncountable cardinal $\kappa$. Its significance is partially in that it clashes with the reflection properties of large cardinals – for example, if $\mu$ is supercompact and $\kappa \ge \mu$ then $\square_\kappa$ fails – and so it characterizes the minimality of $L$ in an indirect way. Schimmerling devised an intermediate hierarchy of principles $\square_{\kappa,\lambda}$ for $\lambda \le \kappa$ as a means of comparing a given model of set theory to $L$, the idea being that a smaller value of $\lambda$ yields a model that is more similar to $L$ at $\kappa$.

Cummings, Foreman, and Magidor proved that for any $\lambda<\kappa$, $\square_{\kappa,\lambda}$ implies the existence of a PCF-theoretic object called a very good scale for $\kappa$, but that $\square_{\kappa,\kappa}$ (usually denoted $\square_\kappa^\ast$) does not. They asked whether $\square_{\kappa,<\kappa}$ implies the existence of a very good scale for $\kappa$, and we resolve this question in the negative.

We will discuss the technical background of the problem, provide a complete solution, and discuss further avenues of research.

Russell Miller: Isomorphism and Classification for Countable Structures

KGRC research seminar on 2017-11-23 at 4pm.

Speaker:  Russell Miller (Queens College, City University of New York (CUNY), USA)

Abstract: We describe methods of classifying the elements of certain classes of countable structures: algebraic fields, finite-branching trees, and torsion-free abelian groups of rank 1. The classifications are computable homeomorphisms onto known spaces of size continuum, such as Cantor space or Baire space, possibly modulo a standard equivalence relation. The classes involved have arithmetic isomorphism problems, making such classifications possible, and the results help suggest exactly which properties of their elements must be known in order to produce a nice classification.

For algebraic fields, this homeomorphism makes it natural to transfer Lebesgue measure from Cantor space onto the class of these fields, although there is another probability measure on the same class which seems in some ways more natural than Lebesgue measure. We will discuss how certain properties of these fields — notably, relative computable categoricity — interact with these measures: the basic result is that only measure-0-many of these fields fail to be relatively computably categorical. (The work on computable categoricity is joint with Johanna Franklin.)

Paul Ellis: Cycle Reversions and Dichromatic Number in (Infinite) Tournaments

KGRC seminar on 2017-11-16 at 3:30pm

Speaker: Paul Ellis (Manhattanville College, New York, USA)

Abstract: The dichomatic number for a digraph is the least number of acyclic subgraphs needed to cover the graph. In 2005, Pierre Charbit showed that by iterating the operation {select a directed cycle, and reverse the direction of each arc in it} that the dichromatic number in any finite digraph can be lowered to 2. This is optimal, as a single directed cycle will always have dichromatic number 2. Recently, Daniel Soukup and I showed that the same is true for infinite tournaments of any cardinality, and in fact, we proved this by induction. Along the way to proving this, we uncovered some nice structural facts about infinite digraphs that we think are of more general interest. While this talk will be mostly graph theoretic in flavor, we did need to put on our set theory glasses to distinguish between the singular and regular cases in the induction. I should note that the question remains open for arbitrary inifinite digraphs, even those of countable cardinality.

Zoltán Vidnyánszky: Random elements of large groups

KGRC seminar on 2017-11-09 at 3:30pm.

Speaker: Zoltán Vidnyánszky (KGRC)

Abstract: The automorphism groups of countable homogeneous structures are usually interesting objects from group theoretic and set theoretic perspective. The description of typical (with respect to category) elements of such groups is a flourishing topic with a wide range of applications. A natural question is that whether there exist measure theoretic analogues of these results. An obvious obstacle in this direction is that such automorphism groups are often non-locally compact, hence there is no natural translation invariant measure on them. Christensen introduced the notion of Haar null sets in non-locally compact Polish groups which is a well-behaved generalisation of the null ideal to such groups. Using Christensen’s Haar null ideal it makes sense to consider the properties of a random element of the group. We investigate these properties, giving a full description of random elements in the case of the automorphism group of the random graph and the rational numbers (as an ordered set).

 

Monroe Eskew: Global Chang’s Conjecture

KGRC Research seminar on 2017-10-19 at 4pm.

Speaker: Monroe Eskew (KGRC)

Abstract: Instances of Chang’s Conjecture (CC) can be seen as a generalization of the Loweheim-Skolem Theorem to a logic in between those the first and second order. Foreman asked how far the analogy with Lowenheim-Skolem can go, specifically whether a global version of CC is consistent. In joint work with Yair Hayut, the speaker answered Foreman’s question affirmatively, and in the process lowered the known upper bounds on consistency strength for many instances of CC. We will discuss the results, as well as some barriers that singular cardinal combinatorics impose on the possibility of a stronger global CC.

Asaf Karagila: Spectra of Uniformity

The FG1 Seminar of TU Wien

On Friday Oct 6 at 15:00,  we will meet in the “Dissertantenraum” of the Institute for Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstr 8-10, 8th floor, green area.

Speaker:  Asaf Karagila

Title: Spectra of Uniformity

Abstract: The existence of uniform ultrafilters on infinite sets is predicated
on the axiom of choice. We discuss some related results, and prove an
Easton-like theorem showing the class of cardinals which carry uniform
ultrafilters has very few provable properties without the axiom of choice.

This is a joint work with Yair Hayut, and can be found on
https://arxiv.org/abs/1709.04824

Mohammad Golshani: Some properties of Cohen and random reals

Kurt Gödel Research Center – Research Seminar
Thursday, October 5 – 4:00pm in the KGRC lecture room

“Some properties of Cohen and random reals”

Mohammad Golshani
(Institute for Research in Fundamental Sciences (IPM), Tehran, Iran)

We discuss some properties of Cohen and random reals. We prove a general combinatorial fact about them which in particular implies that they are large, they contain arbitrary large arithmetical or geometrical progressions, satisfy the Hindman’s conclusion and so on. We also show that they are wild in the sense of o-minimal theory. We also investigate how badly approximable are they by algebraic numbers.

This is a work in progress with Will Brian.

 

Witold Marciszewski: On factorization properties of function spaces

KGRC Research seminar on 2017-06-29 at 4pm

Speaker: Witold Marciszewski (University of Warsaw, Poland)

Abstract: For a Tychonoff space $X$, by $C_p(X)$ we denote the space of all continuous real-valued functions on $X$, equipped with the topology of pointwise convergence. One of the important questions (due to A.V. Arhangel’skii), stimulating the theory of $C_p$-spaces for almost 30 years and leading to interesting results in this theory, is the problem whether the space $C_p(X)$ is (linearly, uniformly) homeomorphic to its own square $C_p(X)\times C_p(X)$, provided $X$ is an infinite compact or metrizable space.

In my talk I will present some recent developments concerning these type of questions. In particular, I will show a metrizable counterexample to this problem for homeomorphisms. I will also show that, for every infinite zero-dimensional Polish space $X$, spaces $C_p(X)$ and $C_p(X)\times C_p(X)$ are uniformly homeomorphic.

This is a joint research with Rafal Gorak and Mikolaj Krupski.