# Category Archives: Seminars

## 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.

## 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.

## 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.

## Victoria Gitman – Virtual large cardinal principles

Research Seminar, Kurt Gödel Research Center, Thursday, April 12

Speaker: Victoria Gitman, (Graduate Center, City University of New York (CUNY), USA)

Abstract: Given a set-theoretic property $\mathcal P$ characterized by the existence of elementary embeddings between some first-order structures, we say that $\mathcal P$ holds virtually if the embeddings between structures from $V$ characterizing $\mathcal P$ exist somewhere in the generic multiverse. We showed with Schindler that virtual versions of supercompact, $C^{(n)}$-extendible, $n$-huge and rank-into-rank cardinals form a large cardinal hierarchy consistent with $V=L$. Sitting atop the hierarchy are virtual versions of inconsistent large cardinal principles such as the existence of an elementary embedding $j:V_\lambda\to V_\lambda$ for $\lambda$ much larger than the supremum of the critical sequence. The Silver indiscernibles, under $0^\sharp$, which have a number of large cardinal properties in $L$,are also natural examples of virtual large cardinals. With Bagaria, Hamkins and Schindler, we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle. Time permitting, I will give an overview of Woodin’s new results on virtual large cardinals in cardinal preserving extensions.we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle.

## Osvaldo Guzman Gonzalez: On weakly universal functions

Thursday, April 12, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Osvaldo Guzman Gonzalez (York University)

Title: On weakly universal functions

Abstract:

A function U:[omega_1]^2 —> 2 is called universal if for every function F:[omega_1]^2 —> omega there is an injective function h:omega_1 —> omega_1 such that F(alpha,beta)=U(h(alpha),h(beta)) for each alpha,betain omega_1. It is easy to see that universal functions exist assuming the Continuum Hypothesis, furthermore, by results of Shelah and Mekler, the existence of such functions is consistent with the continuum being arbitrarily large. Universal functions were recently studied by Shelah and Steprans, where they showed that the existence of universal graphs is consistent with several values of the dominating and unbounded numbers. They also considered several variations of universal functions, in particular, the following notion was studied: A function U:[omega_1]^2 —> omega is (1,omega_1)-weakly universal if for every F:[omega_1]^2 —> omega there is an injective function h:omega_1 —> omega_1 and a function e:omega —> omega such that F(alpha,beta)=eU(h(alpha),h(beta)) for every alpha,betain omega_1. Shelah and Steprans asked if (1,omega_1)-weakly universal functions exist in ZFC. We will study the existence of (1,omega_1)-weakly universal functions in Sacks models and provide an answer to their problem.