## 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.

## Diana Carolina Montoya: On some ideals associated with independent families

Talk held by Diana Carolina Montoya (KGRC) at the KGRC research seminar on 2018-04-19.

Title: On some ideals associated with independent families

Abstract. The concept of independence was first introduced by Fichtenholz and Kantorovic to study the space of linear functionals on the unit interval. Since then, independent families have been an important object of study in the combinatorics of the real line. Particular interest has been given, for instance, to the study of their definability properties and to their possible sizes.

In this talk we focus on two ideals which are naturally associated with independent families: The first of them is characterized by a diagonalization property, which allows us to add a maximal independent family along a finite support iteration of some ccc posets. The second ideal originates in Shelah’s proof of the consistency of $\mathfrak i\lt \mathfrak u$ (here $\mathfrak i$ and $\mathfrak u$ are  the independence and ultrafilter numbers respectively). Additionally, we study the relationship  between these two ideals for an arbitrary independent family $A$, and define a class of maximal  independent families — which we call densely independent — for which the ideals mentioned above  coincide. Building upon the techniques of Shelah we (1) characterize Sacks indestructibility for  such families in terms of properties of its associated diagonalization ideal, and (2) devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.

This is joint work with Vera Fischer.

## Petr Simon

Dear all,

I regret to inform you that Petr Simon passed away on Saturday April
14th. The funeral will take place on Friday April 20th at noon in Prague – Strasnice.

Best,
David

## Jonathan Verner: Ultrafilters and models of arithmetic

The seminar meets on Wednesday April 18th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jonathan Verner — Ultrafilters and models of arithmetic

## Samuel Alfaro Tanuwijaya: Introduction to surreal numbers

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 18 April 2018, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Samuel Alfaro Tanuwijaya

Title: Introduction to surreal numbers

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

In this talk, I will introduce the basic definitions of the surreal
numbers and their ordering given in the book by Harry Gonshor, and
their relations to the definitions given by Conway and Knuth. I will
then continue with the definitions operations on the numbers, such as
addition, multiplication, and division, and then prove that the
surreal numbers form a field. I will then establish that the surreal
numbers contain the real numbers and the ordinals.

## Appalachian Set Theory workshop: Dilip Raghavan, June 2, 2018

Appalachian set theory

# Dilip Raghavan : “Boolean ultrapowers and iterated forcing”

## Description

In joint work with Saharon Shelah, we develop a new method for proving consistency results on cardinal invariants, particularly results involving the invariant . This method can be used with a wide range of forcing notions, including arbitrary ccc posets. However the method always requires a supercompact cardinal κ in the ground model and produces forcing extensions in which the desired invariants sit above κ. Another feature of our method is that it generalizes to cardinal invariants above ω, and can be used to give uniform consistency proofs that work at any regular cardinal. It can also be used to treat situations where three cardinal invariants must be separated. In particular, our technique solves various long standing open problems about cardinal invariants at uncountable regular cardinals. All the results use Boolean ultrapowers, studied by Keisler and other model theorists in the 1960s. I will aim to give a fairly self contained introduction to this method and to some to its applications to the theory of cardinal invariants.

## Local information

The nearest airport is Pittsburgh International Airport. The Supershuttle shared van service is cheaper but slower than taking a taxi from the airport.

## Lodging

VERY IMPORTANT NOTE ABOUT LODGING: A block of rooms earmarked for attendees has been set aside at a local hotel (the Shadyside Inn). If we are covering your lodging expenses then we will need to make a reservation for you. Please don’t make your own reservation if we have promised you support, this will cause confusion and may make it impossible for us to reimburse you.

## Participant travel support

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to the email address appalachiansettheory@gmail.com

• Your nationality and visa status
• Faculty: please tell us whether you hold a federal research grant

## Alexei Kolesnikov: Homology groups in model theory.

Thursday, April 19, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Alexei Kolesnikov (Towson University)

Title: Homology groups in model theory.

Abstract:

Higher-dimensional amalgamation properties played a key role in settling several questions in classification theory. It turns out that these properties, suitably formulated, are non-trivial even for totally categorical first order theories. The main goal of this project was to understand and characterize the failure of higher-dimensional amalgamation properties in stable theories. We show that the failure of n-dimensional amalgamation is detected by a suitable homology group; this group must be abelian profinite and is isomorphic to a certain automorphism group. Along the way, we establish that the failure of n dimensional amalgamation is witnessed by certain canonical objects, with the higher category-theoretic flavor, that are definable in the models of the theory.

Joint work with John Goodrick and Byunghan Kim.

## Vahagn Aslanyan: Geometry of strongly minimal sets in differentially closed fields

Mathematical logic seminar – Apr 17 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Vahagn Aslanyan
Department of Mathematical Sciences
CMU

Title:     Geometry of strongly minimal sets in differentially closed fields

Abstract:

I will discuss Zilber’s Trichotomy conjecture and some structures (theories) where it holds (the conjecture in its general form was refuted by Hrushovski). In particular, by a result of Hrushovski and Sokolovic differentially closed fields satisfy Zilber’s trichotomy. However, understanding whether a given definable set is strongly minimal or, given a strongly minimal set, understanding the nature of its geometry is not an easy task. I will show how one can use the Ax-Schanuel theorem for the j-function to deduce strong minimality and geometric triviality of the differential equation of the j-function (I will also explain why it is an important example). This result was first proven by Freitag and Scanlon using the analytic properties of the j-function. My approach is completely abstract, I actually prove that once there is an Ax-Schanuel type statement of a certain form for a differential equation E(x,y) then some fibres of E are strongly minimal and geometrically trivial.

## Piotr Borodulin-Nadzieja: Tunnels through topological spaces

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

Speaker: Piotr Borodulin-Nadzieja (University of Wroclaw)

Title: Tunnels through topological spaces

Abstract:

I will show a ZFC example of a compact space (without
isolated points) through which one cannot drill a tunnel. I will discuss
when and when not $\omega^*$ has a tunnel.