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.

Garrett Ervin: Decomposing the real line into two everywhere isomorphic

Mathematical logic seminar – Apr 10 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Garrett Ervin
Department of Mathematical Sciences
CMU

Title:     Decomposing the real line into two everywhere isomorphic pieces

Abstract:

A dense linear order is said to be homogeneous if it is isomorphic to its restriction to any of its open intervals. The set of rationals ℚ is an example of a homogeneous order, as is the set of irrationals ℝ ∖ ℚ. In general, if X is a homogeneous suborder of the real line ℝ, then ℝ ∖ X is also homogeneous, and there are many examples where both X and ℝ ∖ X are of size continuum. However, it turns out that a homogeneous X can never be isomorphic to ℝ ∖ X. In fact, if ℝ = A ∪ B is any decomposition of ℝ into two disjoint pieces, there is an open interval I such that A restricted to I is not isomorphic to B restricted to I. We will prove this theorem and discuss some related results.

Menachem Magidor: Compactness and Incompactness for being Corson and Eberline compacta

Place: Fields Institute (Room 210)

Date: April 6, 2018 (13:30-15:00)

Speaker: Menachem Magidor

Title: Compactness and Incompactness for being Corson and Eberline compacta

Abstract:

The problems we shall discuss are examples of compactness (or dually a
reflection) problems. A typical compactness property is a statement that a
given structure has a certain property, provided smaller some structures have
this property. Reflection property is a dual statement, namely if a structure
has the property then there is a smaller substructure having the property. The
notion of “smaller substructure ” may depend on the domain we talk about. Thus
for an algebraic structure “smaller substructure” typically means a subalgebra
having a smaller cardinality. For topological spaces “smaller substructure ”
may mean a continuous image of the space of smaller weight.

Compactness problems tend to form clusters, which share the same pattern. For
instance sharing the same cardinals which are compact for the given property.
In this talk we shall survey some of these patterns. But we shall concentrate
on the problem of compactness for a compact space being Corson.

A compact space is a Corson compact if it can be embedded into $\Sigma\left(
R^{k}\right) $ where $\Sigma\left( R^{k}\right) $ is the the subspace of $R^{k}$ (with the product topology) of those sequences which are non zero only
on countably many coordinates. The compactness problem for Corson compacta is whether a space is Corson compact if compact given that all its continuous images of small weight are Corson. We shall report on some ongoing work about this problem.

Yair Hayut: Filter compactness and squares

BIU seminar in Set Theory

April 8, 2018

Speaker: Yair Hayut (TAU)

Title: Filter compactness and squares

Abstract. Strongly compact cardinals are characterized by the property that any $\kappa$-complete filter can be extended to a $\kappa$-complete ultrafilter. When restricting the cardinality of the underlying set, we obtain a nontrivial hierarchy. For example, when requiring the extension property to hold only for filters on $\kappa$, we obtain Gitik’s $\kappa$-compact cardinals, which are known to be consistently weaker than $\kappa$ being $\kappa^+$-strongly compact.

In this talk I will focus on the level by level connection between the filter extension property and the compactness for $L_{\kappa,\kappa}$. Using the compactness, I will show that if $\kappa$ is $\kappa$-compact then $\square(\kappa^{+})$-fails.

Wojciech Wołoszyn: Laver-type posets

Dear all,

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

Program: Wojciech Wołoszyn will be presenting results about Laver type
posets from the paper of J. Zapletal and myself.

Best,
David

Wednesday seminar: Andrew Brooke-Taylor

Dear all,

The seminar meets on Wednesday April 4th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

We will have a guest speaker Andrew Brooke-Taylor.

Best,
David

Andy Zucker: Maximal equivariant compactifications of categorical metric structures

Mathematical logic seminar – April 3 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Maximal equivariant compactifications of categorical metric structures

Abstract:

Any completely regular space embeds into a compact space. But suppose G is a topological group and X is a completely regular G-space. There is a largest G-map αX: X → Y where Y is compact and αX has dense image, but αX need not be an embedding. Recently, Pestov has constructed an example of a topological group G and non-trivial flow X for which αX is the map to a singleton.

In this talk, we consider automorphism groups of categorical metric structures, which include the Urysohn sphere, the unit sphere of the Banach lattice Lp, and the unit sphere of the Hilbert space L2. We show that if G is the group of automorphisms of a categorical metric structure X, then αX is the embedding of X into the space of 1-types over X.

(Joint work with Itai Ben Yaacov)