Scott Cramer: Algebraic properties of elementary embeddings

Time: Mon, 12/04/2017 – 4:00pm – 5:30pm
Location: RH 440R

Speaker: Scott Cramer (California State University San Bernardino)

Title: Algebraic properties of elementary embeddings

Abstract. We will investigate algebraic structures created by rank-into-rank elementary embeddings. Our starting point will be R. Laver’s theorem that any rank-into-rank embedding generates a free left-distributive algebra on one generator. We will consider extensions of this and related results. Our results will lead to some surprisingly coherent conjectures on the algebraic structure of rank-into-rank embeddings in general.

Thematic Semester on Descriptive Set Theory and Polish groups, More info

Thematic Semester on Descriptive Set Theory and Polish groups
Bernoulli Center, Lausanne, Switzerland.
January – June, 2018.

During the period January 1st – June 30th, 2018, there will be a thematic semester on Descriptive Set Theory and Polish Groups held at the Bernoulli Center in Lausanne, Switzerland.

The semester is organised around five week long activities, including three conferences and two workshops, along with three Bernoulli Lectures held in connection with these events.
Conference: Borel combinatorics and ergodic theory (organised by C. Conley and D. Gaboriau), February 5-9.
Bernoulli Lecture: Stephen Jackson (Univ. North Texas), February 8.
Conference: Structure and dynamics of Polish groups (organised by A. Thom and T. Tsankov), March 19-23.
Workshop: Large scale geometry of Polish groups (organised by J. Moore and C. Rosendal), March 26-29.
Bernoulli Lecture: Mikhail Gromov (IHES) – What is Probability? March 27.
Workshop: Ideals and exceptional sets in Polish spaces (organised by M. Elekes and S. Solecki), June 4-8.
Conference: Descriptive set theory (organised by B. Miller, A. Kechris and S. Todorcevic), June 18-22.
Bernoulli Lecture: Slawomir Solecki (Cornell) – Projective Fraisse limits, approaching topology through logic. June 21.
In addition to these semester activities, the 11th Young Set Theory Workshop will be held at the Bernoulli Center during the week of June 25-29.

Detailed information about the semester and these event is available at the following link

https://bernoulli.epfl.ch/semesters/32/show

The Bernoulli Center has large capacity and everyone is invited to attend the events of the semester. Registration for the individual events can be done through the above link by clicking at the conference/workshop in the right column.

Funding for US based visitors is secured through an NSF grant, while limited funding for other participants is also available. Requests for funding should be made during online registration.

Please note that registration and funding requests for the events in March should be made before mid-December.

David Chodounsky: A generalization of the Solovay–Tennenbaum theorem

Dear all,

The seminar meets on Wednesday November 29th at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program:
David Chodounsky will present a generalization of the Solovay–Tennenbaum theorem; Assuming a diamond principle and given a suitable class PHI of ccc posets, there is a poset in the class PHI which forces MA(PHI) and c=kappa.

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 Szeptycki: Ladder systems after forcing with a Suslin tree

Place: Fields Institute (Room 210)

Date: November 24, 2017 (13:30-15:00)

Speaker: Paul Szeptycki

Title: Ladder systems after forcing with a Suslin tree

Abstract: Uniformization properties of ladder systems in models obtained by forcing with a Suslin tree S over a model of MA(S) are considered.

Andres Caicedo: Real-valued measurability and the extent of Lebesgue measure (II)

Thursday, November 30, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Andres Caicedo (Math Reviews)

Title: Real-valued measurability and the extent of Lebesgue measure (II)

Abstract:

On this second talk I begin with Solovay’s characterization of real-valued measurability in terms of generic elementary embeddings, and build on results of Judah to prove that if there is an atomlessly measurable cardinal, then all (boldface) Delta-1-3 sets of reals are Lebesgue measurable. This is optimal in two respects: Just from the existence of measurable cardinals we cannot prove that lightface Delta-1-3 sets are measurable, and there are models with atomlessly measurable cardinals where there is a non-measurable Sigma-1-3 set. I will also discuss some related results.

Merlin Carl: Complexity theory for ordinal Turing machines

Monday, November 27, 2017, 16.30
Seminar room 0.008, Mathematical Institute, University of Bonn

Speaker: Merlin Carl (Universität Konstanz)

Title: Complexity theory for ordinal Turing machines

Abstract:

Ordinal Turing Machines (OTMs) generalize Turing machines to transfinite working time and space. We consider analogues of theorems from complexity theory for OTMs, among them the Cook-Levin theorem, the P vs. NP problem and Ladner’s theorem. This is joint work with Benedikt Löwe and Benjamin Rin.

Philipp Schlicht: The Hurewicz dichotomy for definable subsets of generalized Baire spaces

Monday, November 20, 2017, 16.30
Seminar room 0.008, Mathematical Institute, University of Bonn

Speaker: Philipp Schlicht (Universitat Bonn)

Title: The Hurewicz dichotomy for definable subsets of generalized Baire spaces

Marcin Michalski: Bernstein, Luzin and Sierpiński meet trees

Tuesday, November 28, 2017, 17:15
Wrocław University of Science and Technology, 215 D-1

Speaker: Marcin Michalski (Wrocław University of Science and Technology)

Title: Bernstein, Luzin and Sierpiński meet trees

Abstract:

In [2] we have proven that if $\mathfrak{c}$ is a regular cardinal number, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to Marczewski ideal $s_0$. We will generalize this result for other tree ideals – $m_0$ and $l_0$ – using some lemmas on special kind of fusion sequences for trees of respective type.
Let us introduce a following notion. Let $\mathbb{X}$ be a set of trees.
Definition. We call a set $B$ a $\mathbb{X}$-Bernstein set, if for each $X\in\mathbb{X}$ we have $[X]\cap B\neq\emptyset$.
We shall explore this notion for various set of trees, including Sacks, Miller and Laver trees, with the support of technics developed in [1].

[1] Brendle J., Strolling through paradise, Fundamenta Mathematicae, 148 (1995), pp. 1-25.
[2] Michalski M., Żeberski Sz., Some properties of I-Luzin, Topology and its Applications, 189 (2015), pp. 122-135.