Category Archives: Seminars

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

Joseph Zielinski: Roelcke precompact sets in Polish groups II

Mathematical logic seminar – May 8 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Joseph Zielinski
Department of Mathematical Sciences
CMU

Title:     Roelcke precompact sets in Polish groups II

Abstract:

In these talks we first recall the uniform structures associated to a topological group. We then present J. Roe’s notion of a coarse space, and consider compatible coarse structures on groups with emphasis on the ‘left-coarse structure’ of a topological group introduced by C. Rosendal. Associated to this notion are the ‘locally bounded Polish groups’: those for which the left-coarse structure is the bounded coarse structure of some compatible, left-invariant metric.

Next, we introduce the Roelcke precompact subsets of a Polish group, which admit equivalent natural definitions both in terms of the lower uniformity on the group and as a subideal of the bounded sets in the left-coarse structure. Through this we define the ‘locally Roelcke precompact Polish groups’ — a subfamily of the locally bounded Polish groups — and present various examples, applications, and several characterizations of these groups.

Vincenzo Dimonte: The sensitive issue of iterability

Talk held by Vincenzo Dimonte (University of Udine, Italy) at the KGRC seminar on 2018-05-03.

Abstract: In the momentous years when the community of set theorists was reaching the definite answer for the problem of the consistency of the Axiom of Determinacy, Martin wrote a small paper in the Proceedings of the International Congress of Mathematicians, 1978, in which he proved that the iterable version of I3, a very large cardinal, implied the determinacy of $\Pi^1_2$ sets of reals. Later it was proved that AD had much lower consistency, and iterable I3 fell into oblivion. In the last decade interest on I3 re-emerged, but iterable I3 is still elusive, and the small paper by Martin is not helpful, as it is terse and full of gaps. Even the definition of iterable I3 is not convincing. In this seminar we will bring back to life this abandoned hypothesis, clean it up to modern standards, and reveal the existence of a new hierarchy of axioms that was previously overlooked.

Joseph Zielinski: Roelcke precompact sets in Polish groups

Mathematical logic seminar – May 1 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Joseph Zielinski
Department of Mathematical Sciences
CMU

Title:     Roelcke precompact sets in Polish groups

Abstract:

In these talks we first recall the uniform structures associated to a topological group. We then present J. Roe’s notion of a coarse space, and consider compatible coarse structures on groups with emphasis on the ‘left-coarse structure’ of a topological group introduced by C. Rosendal. Associated to this notion are the ‘locally bounded Polish groups’: those for which the left-coarse structure is the bounded coarse structure of some compatible, left-invariant metric.

Next, we introduce the Roelcke precompact subsets of a Polish group, which admit equivalent natural definitions both in terms of the lower uniformity on the group and as a subideal of the bounded sets in the left-coarse structure. Through this we define the ‘locally Roelcke precompact Polish groups’ — a subfamily of the locally bounded Polish groups — and present various examples, applications, and several characterizations of these groups.

Andreas Blass: Well-ordered choice implies dependent choice

Thursday, May 3, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Andreas Blass (University of Michigan)

Title: Well-ordered choice implies dependent choice

Abstract:

The axiom of well-ordered choice is a weak form of the axiom of choice. It says that every well-ordered family of nonempty sets has a choice function. The axiom of dependent choice is another weak form of the axiom of choice. It says that, given any directed graph in which every vertex has at least one outgoing arrow, and given any vertex v in that graph, there exists an infinite sequence of vertices that starts at v and then follows the arrows. I’ll prove the old but probably insufficiently well-known theorem of Jensen that well-ordered choice implies dependent choice.

Asaf Karagila: What can you say about critical cardinals?

HUJI Logic Seminar

Tuesday 1/5 at Math 209 13:30-15:30
Speaker: Asaf Karagila
Title: What can you say about critical cardinals?
Abstract. We isolate the property of being a critical point, and prove some basic positive properties of them. We will also prove a lifting property that allows lifting elementary embedding to symmetric extensions, and outline a construction that shows that it is consistent that a successor of a critical cardinal is singular. This is a recent work with Yair Hayut.

David Fernandez Breton: Partition theorems on uncountable abelian groups

Place: Fields Institute (Room 210)

Date: April 27, 2018 (13:30-15:00)

Speaker: David Fernandez Breton

Title: Partition theorems on uncountable abelian groups

Abstract:

In the past two years, a number of Ramsey-theoretic results concerning the additive structure of uncountable abelian groups have been investigated by diverse subsets of the set {Komjáth, Rinot, D. Soukup, W. Weiss, myself} (among others). From Ramsey results generalizing Hindman’s theorem for certain groups and colourings, to anti-Ramsey statements asserting the existence of ”rainbow colourings” for sets of finite sums, I plan to provide a panoramic overview of this exciting line of research, and point towards possible future lines of enquiry.

Filippo Calderoni: The bi-embeddability relation for countable abelian groups

Talk held by Filippo Calderoni (Università di Torino, Italy and Politecnico di Torino, Italy) at the KGRC seminar on 2018-04-26.

Abstract: We analyze the Borel complexity of the bi‑embeddability relation for different classes of countable abelian groups. Most notably, we use the Ulm theory to prove that bi‑embeddability is incomparable with isomorphism in the case of p‑groups, and torsion groups. As I will explain, our result contrasts the arguable thesis that the bi‑embeddability relation on countable abelian p‑groups has strictly simpler complete invariants than isomorphism.

This is joint work with Simon Thomas.

Anthony Bonato: The new world of infinite random geometric graphs

Place: Fields Institute (Room 210)

Date: April 20, 2018 (13:30-15:00)

Speaker: Anthony Bonato

Title: The new world of infinite random geometric graphs

Abstract:

The \emph{infinite random} or \emph{Rado graph} $R$ has been of interest to graph theorists, probabilists, and logicians for the last half-century. The graph $R$ has many peculiar properties, such as its \emph{categoricity}: $R$ is the unique countable graph satisfying certain adjacency properties. Erd\H{o}s and R\'{e}nyi proved in 1963 that a countably infinite binomial random graph is isomorphic to $R$.

Random graph processes giving unique limits are, however, rare. Recent joint work with Jeannette Janssen proved the existence of a family of random geometric graphs with unique limits. These graphs arise in the normed space $\ell _{\infty }^{n}$, which consists of $\mathbb{R}^{n}$ equipped with the $L_{\infty }$-norm. Balister, Bollob\'{a}s, Gunderson, Leader, and Walters used tools from functional analysis to show that these unique limit graphs are deeply tied to the $L_{\infty }$-norm. Precisely, a random geometric graph on any normed, finite-dimensional space not isometric $\ell _{\infty}^{n}$ gives non-isomorphic limits with probability $1$.

With Janssen and Anthony Quas, we have discovered unique limits in infinite dimensional settings including sequences spaces and spaces of continuous functions. We survey these newly discovered infinite random geometric graphs and their properties.

Marek Bienias: About universal structures and Fraisse theorem

Tuesday, April 24, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marek Bienias (Łódź University of Technology)

Title: About universal structures and Fraisse theorem

Abstract:

For a given structure D of language L we can consider age of D, i.e. the family of all finitely generated L-substructures od D. It turns out that age has property (HP) and (JEP). Fraisse theorem let us revers the procedure: if K is nonempty countable family of finitely generated L-structures having properties (HP), (JEP) and (AP), then there exists exactly one (up to isomorphism) L-structure D (so called Fraisse limit) which is countable ultrahomogenous and has age K.
The aim of the talk is to define basic notions from Fraisse theory, proof the main theorem and show some alternative way of looking at the construction of Fraisse limit.