2013
Fridays 
Seminars to June 30, 2013 (Seminars
in the 201314 academic year)
at 1:30 p.m. in the Fields Institute, Room 210

June 28 
Asger Törnquist
Talking about a theorem he is proving at the moment and
is not yet ready to be revealed
A wellknown result of Mathias says that an infinite maximal
almost disjoint family of subsets of $\omega$ is never analytic.
The question has been raised if a maximal family of eventually
different functions on $\omega$ can be analytic. In this
talk, I will present a proof that it can't be. We also obtain
a new (and quite different) proof of Mathias' theorem. Towards
the end of the talk, I will discuss how the proofs can be
adapted to show that maximal cofinitary families and groups
of permutations of $\omega$ can't be analytic either.

June 21 
David Fernández
Strongly summable ultrafilters and union ultrafilters are
not the same thing
This is, in some sense, a continuation of my previous talk
(though of course selfcontained). So I will introduce strongly
summable ultrafilters, union ultrafilters, and additive
isomorphisms, and then I will proceed with the construction
(assuming cov(M)=c) of a strongly summable ultrafilter (on
the Boolean group) that is not additively isomorphic to
any union ultrafilter.

June 14 
Frank Tall
A provisional solution to Nyikos’ manifold problem
Peter Nyikos observed that, although the Long Line is a
nonmetrizable, hereditarily normal manifold, it is difficult
to find such a manifold of dimension > 1. Indeed the
only such examples are constructed with extra settheorteic
hypotheses, e.g. CH. He therefore conjectured in 1981 that
it was consistent there were no such higher dimensional
manifolds. Assuming results claimed by Todorcevic and by
Dow, we can prove this, modulo a supercompact cardinal..
Our proof is mainly topological, deriving the nonexistence
of such manifolds from the conjunction of several assertions
known to follow from or asserted to follow from PFA(S){S].

June 7
Fields Institute, Room 230

Assaf Rinot
The fragility of chromatic number of graphs
The chromatic number of a graph $(G,E)$ is the least cardinal
$\kappa$ for which there exists a coloring $c:G\rightarrow\kappa$
with the property that $c(x)\neq c(y)$ whenever $xEy$. How
robust is this notion? Could a graph change its chromatic
number via forcing? via a cofinalitypreserving forcing?
Could the same graph have different chromatic numbers in
different cofinalitypreserving forcing extensions? and
if so, is there a bound for the amount of different chromatic
numbers the same graph can get? and what is the effect of
forcing axioms on this problem?
In this talk, we shall address all of these questions.

May 31 
Dilip Raghavan
Combinatorial dichotomies and cardinal invariants
Assuming the Pideal dichotomy, we attempt to isolate those
cardinal characteristics of the continuum that are correlated
with two wellknown consequences of the proper forcing axiom.
We find a cardinal invariant $\chi$ such that the statement
that $\chi$ > ${\omega}_{1}$ is equivalent to the statement
that $1$, $\omega$, ${\omega}_{1}$, $\omega \times {\omega}_{1}$,
and ${[{\omega}_{1}]}^{lt; \omega}$ are the only cofinal
types of directed sets of size at most ${\aleph}_{1}$.
We investigate the corresponding problem for the partition
relation ${\omega}_{1} \rightarrow ({\omega}_{1}, \alpha)^2$
for all $\alpha$ < ${\omega}_{1}$.
To this effect, we investigate partition relations for pairs
of comparable elements of a coherent Suslin tree $S$.
We show that a positive partition relation for such pairs
follows from the maximal amount of the proper forcing axiom
compatible with the existence of $S$.
As a consequence we conclude that after forcing with the
coherent Suslin tree $S$ over a ground model satisfying
this relativization of the proper forcing axiom, ${\omega}_{1}
~\rightarrow~{({\omega}_{1}, \alpha)}^{2}$ for all $\alpha$
< ${\omega}_{1}$.
We prove that this positive partition relation for $S$ cannot
be improved by showing in ZFC that $S\not\rightarrow ({\aleph}_{1},
\omega+2)^2$.

May 24 
David Fernández
Every strongly summable ultrafilter is sparse!
The concept of a Strongly Summable Ultrafilter was born
from Hindman’s efforts for proving the theorem that
now bears his name (which at the time was known as GrahamRothschild’s
conjecture), although later on it got a life of its own
and started to be studied for its own sake, mostly because
of its nice algebraic properties. At the time the focus
was on ultrafilters over the semigroup (N,+) , but eventually
Hindman, Protasov and Strauss generalized much of this theory
to abelian groups in general in a 1998 paper. In that same
paper, they introduced the notion of a sparse ultrafilter,
one which subsumes that of strongly summable as a particular
case but that has even nicer algebraic properties. In a
2012 paper, Hindman, Steprans and Strauss found a large
class of abelian groups (which included (N,+) ) over which
every strongly summable ultrafilter must be sparse.
In this talk I extend this result to all abelian groups.
Moreover we show that in most cases the strong summability
of these ultrafilters is due to their being additively isomorphic
to a union ultrafilter (I will explain what this means).
However, this does not happen in all cases: I will also
construct (assuming p=c ), on the Boolean group, a strongly
summable ultrafilter that is not additively isomorphic to
any union ultrafilter.

May 17 
Arnie Miller
Countable subgroups of Euclidean Space
In his PhD Thesis Konstantinos Beros proved a number of
results about compactly generated subgroups of Polish groups.
Such a group is Ksigma – the countable union of compact
sets. He notes that the group of rationals under addition
with the discrete topology is an example of a Polish group
which is Ksigma (since it is countable) but not compactly
generated.
Beros showed that for any Polish group G, every Ksigma
subgroup of G is compactly generated iff every countable
subgroup of G is compactly generated. Beros showed that
any Ksigma subgroup of Z^omega (infinite product of the
integers) is compactly generated and more generally, for
any Polish group G, if every countable subgroup of G is
finitely generated, then every countable subgroup of G^omega
is compactly generated.
In unpublished work Beros asked whether finitely generated
may be replaced by compactly generated in his theorem. He
conjectured that the reals R under addition might be an
example such that every countable subgroup of R is compactly
generated but not every countable subgroup of R^omega is
compactly generated. We prove that this is not true. The
general question remains open.
In the course of our proof we came up with some interesting
countable subgroups. We show that there is a dense subgroup
of the plane which meets every line in a discrete set. Furthermore,
for each n there is a dense subgroup of Euclidean space
R^n which meets every (n1)dimensional subspace in a discrete
set. Similarly there is a dense subgroup of R^omega which
meets every finite dimensional subspace of R^omega in a
discrete set.

May 10 
*no seminar  ASL meeting
at Waterloo* 
May 6 (*Monday*) 
Marion Scheepers 
May 3
Stewart Library
*please note non standard location

Martino Lupini
Borel complexity of equivalence relations from operator
algebras
I will give an overview of the study from the point of
view of descriptive set theory of the Borel complexity of
equivalence relatons arising within the theory of C*algebras.
No previous knowledge of operator algebras will be assumed.

Apr 26 
Dana Bartosova
Filter dynamical systems II
This time, we will focus on general topological groups
and show that many dynamical notions naturally translate
into the language of filters. We will construct disjoint
large subsets of nonprecompact topological groups showing
that the greatest ambit has infinitely many disjoint minimal
left ideals. This result was inspired by a problem of Ellis
asking for which groups homomorphisms from the greatest
ambit into the universal minimal flow separate points. We
will finish up with a sketch of a simplified proof that
groups of isometries of generalized Urysohn spaces are extremely
amenable.

Apr 19 
Konstantinos Tyros
A discussion on Density Ramsey Theory
We will present some recent results in Density Ramsey Theory.
In particular, we will present a density version of a result
due to Carlson and Simpson concerning left variable words,
which consists a common extension of the Density HalesJewett
Theorem and the Density HalpernLäuchli Theorem. If
time permits we will also present some applications.

Apr 12 
Daniel Soukup, University of Toronto
Partitioning bases of topological spaces
The purpose of this talk is to investigate whether an arbitrary
base for a dense in itself topological space can be partitioned
into two bases; these spaces will be called base resolvable.
First, we review positive results, i.e. that several classes
of spaces are base resolvable: metric spaces and leftor
right separated spaces. Furthermore, every T_3 (locally)
Lindelöf space is base resolvable. Second, we aim to
outline the construction of a non base resolvable space;
this is done by isolating a new partition property of partially
ordered sets. Our strongest result in this direction is
that, consistently, there is a 0dimensional, 1st countable
Hausdorff space of weight ? 1 and size continuum which is
non base resolvable.

Apr 5 
David Chodounsky
Gaps and Towers in $P(\omega)/fin$
We study the structure of $\subset$ relation on towers
($\subset^*$chains) and gaps in $P(\omega)/fin$. We define
Suslin towers and Hausdorff towers and discuss their existence
in various models of set theory. Then some of the results
and methods are used to provide examples of indestructible
gaps not equivalent to a Hausdorff gap.

Mar 29 
Good Friday, no seminar 
Mar 22 
Dana Bartosova
Filter dynamical systems
We show how we can view the universal minimal flow of a
topological group G as a space of filters on G with the
structure of right topological semigroup. This approach
allows us to translate a variety of dynamical properties
into the language of filters and use set theoretic and combinatorial
methods to understand dynamics of G.

Mar 15 
Miguel Angel Mota
On a question of Abraham and Cummings
The technique of ensuring properness of a given forcing
notion by incorporating elementary substructures of some
large enough model into its definition as side conditions
may be traced back to Todorcevic. The more specific approach
of considering symmetric systems of countable structures
as side conditions in the context in which one starts with
a model of CH and wants to obtain a forcing notion which
is proper and does not collapse cardinals is quite natural.
In fact, this approach (also created by Todorcevic) has
already shown up in several places in the literature. The
main novelty of the method created by Asperó and
Mota is that it incorporates the use of symmetric systems
of structures as side conditions affecting all iterands
of a given forcing iteration rather than a single forcing
as in the above references. As an interesting application
of this method, we answer a question of Abraham and Cummings
by showing that a negative polychromatic Ramsey relation
is consistent together with MA and a large continuum. This
is joint work with Asperó

Mar 8 
Chris Eagle
Omitting types in infinitary [0,1]valued logic (presentation)
In firstorder logic many interesting nonelementary classes
of mathematical structures can be classified by the types
that they realize or omit. The classical Omitting Types
Theorem characterizes those types which can be omitted in
models of a fixed theory $T$ as the ones which are not generated
over $T$ by a single formula. The Omitting Types Theorem
has close connections to the Baire Category Theorem, which
we will use to give a topological proof of an Omitting Types
Theorem for a logic for metric structures which is analogous
to $\mathcal{L}_{\omega_1, \omega}$.

Mar 1

Natasha May
A new class of spaces all finite powers Lindelof
We consider a new class of open covers and classes of spaces
defined from them, called ? spaces (“iota spaces”).
We explore their relationship with ? spaces (that is, spaces
having all finite powers Lindelof) An example of a hereditarily
? space whose square is not hereditarily Lindel of is provided.
Time permitting, we will also explore a potential duality
between the ? covering property of X and convergence properties
of C p (X) .

Feb 22

Reading Week, no seminar 
Feb 15 
Mike Pawliuk
Using Tsequences to create a robust family of topological
groups
Using the methods of Protasov and Zelenyuk (“Topologies
on Abelian Groups”, 1991) I will describe a method
for constructing topological groups. We will focus on creating
topologies on the integers (with the usual group operation)
where nontrivial sequence converge to 0. Not all sequences
in the Integers admit a T_2 group topology on the integers;
those that do are called Tsequences.
We will examine three different aspects of Tsequences.
First we will see that there are as many (different) Tsequences
as possible, and that only certain types of chains of topologies
given by Tsequences are possible. Then we will see that
group topologies given by a (nontrivial) Tsequence are
all examples of sequential spaces that are not FrechetUrysohn.
Finally, we will give a nice diagonalization technique that
produces many topological groups on the integers that do
not admit characters.

Feb 8 
Fields closed due to weather conditions

Feb 1 
Saeed Ghasemi
Automorphisms of Borel quotients of FDDalgebras
Assume there exists a measurable cardinal. Using generalized
groupwise Silver forcing I build a model of set theory in
which every automorphism of a Borel quotient of a FDDalgebra
(finite dimensional decomposition) has a *homomorphism
representation (lifting).

Jan 25 
Frank Tall
More topological consequences of PFA(S)[S] (part 2 of
talk) 
Jan 18 
Frank Tall
More topological consequences of PFA(S)[S] (part 1 of
talk)
Abstract:
http://settheory.mathtalks.org/franktallmoretopologicalconsequencesofpfasspart1/ 
Jan 11 
Assaf Rinot
Chromatic number of graphs  large gaps
We shall present a construction of graphs of large size
and large chromatic number whose any smaller subgraphs are
countably chromatic. The construction builds on our notion
of Ostaszewski square. It follows that if the weak covering
lemma holds, and kappa is the successor of a strong limit
singular cardinal, then there exists a graph of size and
chromatic number kappa, whose all smaller subgraphs are
countably chromatic.


Affiliated
talks: York University (Mondays 3:304:30 pm)
 December 3, Alan Dow
 November 14, Grigor Sargsyan
 November 5, Menachem Magidor 
201213

Speaker and Talk
Title

Dec. 14
11:00 a.m.
Room 230 
Trevor Wilson
Wellbehaved measures and weak covering for derived
models
For an inner model $M$ containing all the reals
and satisfying the Axiom of Determinacy, we show that countably
complete measures over $M$ on ordinals less than $\Theta^M$
are “wellbehaved.” In particular every such measure
is ordinaldefinable from $M$, generalizing a theorem of Kunen
that says “AD implies that every measure on an ordinal
less than Theta is ordinaldefinable.” This generalization
is useful in constructing weak homogeneity systems consisting
of measures over $M$. As an application, we get a kind of
weak covering result that applies to weakly compact cardinal
whose successor is not computed correctly in HOD. Namely,
if $\delta$ is a weakly compact limit of Woodin cardinals,
and $(\delta^+)^{\text{HOD}} < \delta^+$, then the derived
model at $\delta$ satisfies “every set is Suslin.”
The necessary facts about weak homogeneity systems, Suslin
sets, and derived models will all be covered in the talk.

Dec. 14
1:30 p.m.
Room 230 
Alex Rennet
Axiomatizability, Ultraproducts and OMinimality
Suppose a theory T is given as the set of sentences true
in all structures in a fixed language which share some nonfirst
order property P. For instance, if P is ‘stable’
or ‘ominimal’ or ‘finite’, we get the
Ltheory of stability, ominimality or finiteness respectively.
A classic modeltheoretic result describes the models of
a theory constructed in this way as exactly those with the
given property, or their ultraproducts (up to elementary
equivalence).
The main result I’ll focus on in my talk is a general
failure of recursive axiomatizability for certain theories
of this kind. I’ll explain why the question of whether
such a theory has a recursive axiomatization is natural,
and give examples which do have such an axiomatization.
In the case of ominimality (a modeltheoretic property
related to nonstandard analysis) this answers negatively
a suggestion from the recent literature. I’ll go through
the proof of this result, which pleasantly involves minimal
modeltheoretic detail.

Dec. 7
1:30 p.m.
Room 230 
Bill Mitchell
The Chang Model — Again
A few years ago, I gave several talks — with varying
degrees of tentativeness — describing a weak version
of Woodin’s sharp for the Chang model. I will discuss
what the optimal result, that is, the actual sharp, might
look like, and how the picture I had of this is wrong in
a major way. I will then discuss the proof of the result
I did claim.

Dec. 7
11:15 a.m.
Room 230 
Sean Cox
Antichain catching at $\omega_1$ versus antichain catching
at $\omega_2$ (slides
of the talk)
I’ll discuss a property of normal ideals, called
projective antichain catching, which lies (implicationwise)
between saturation and precipitousness. For ideals on $\omega_1$,
projective antichain catching is equivalent to precipitousness;
in fact it gives a nice characterization of the statement
“$NS_{\omega_1}$ is precipitous” in terms of
FengJech’s notion of projective stationarity
(this is due essentially to Schindler).
For ideals on $\omega_2$, however, projective antichain
catching is strictly between saturation and precipitousness
(and much stronger than precipitousness, in consistency
strength). Proving that projective antichain catching does
not imply saturation—in fact does not imply even strongness
of the ideal—involves a modification of the KunenMagidor
constructions of saturated ideals to work in the context
of supercompact towers which are not almost huge. This is
joint work with Martin
Zeman.

Nov.30, 2012
11:00 a.m.
Room 230 
Christina Brech
The Banach space $\ell_\infty/c_0$ in the Cohen model
We will present some results concerning nonexistence of
isomorphic copies of certain Banach spaces inside $\ell_\infty/c_0$
in the Cohen model. As opposed to results obtained under
CH, we conclude that in the Cohen model the space $\ell_\infty/c_0$
cannot contain a copy of all Banach spaces of density continuum
and cannot be written as an $\ell_\infty$sum of any given
Banach space X.
These are joint results
with Piotr
Koszmider.

Nov. 30, 2012
1:30 p.m.
Room 230 
Alan Dow
Nontrivial copies of N* in N*
It is a rather old problem of E. van Douwen to determine
if there is (in ZFC) a nontrivial copy of N* in N*. It
is folklore that for any countable discrete subset D of
N*, the subspace consisting of the limit points of D is
itself a copy of N*. These are known as the trivial copies
of N*. CH implies there are many nontrivial copies of N*
but, following Shelah’s work in on the consistency
of the nonexistence of autohomeomorphisms of N*, negative
partial results were obtained by W. Just and I. Farah. In
particular, it follows from the PFA that the dual ideal
generated by any possible nontrivial copy of N* would have
to satisfy the ccc over fin property (Farah).

Nov. 28
11:00 a.m.
Room 230

Lynn Scow (UIC)
$I$indexed Indiscernible Sets and Trees
Fix any $L’$structure ${I}$ on an underlying set
$I$. An ${I}$indexed indiscernible set is a set of parameters
$A = \{a_i : i
\in I\}$ where the $a_i$ are samelength finite tuples from
some structure $M$ and $A$ satisfies a homogeneity condition:
$\textrm{tp}(a_{i_1}, \ldots, a_{i_n};M)=\textrm{tp}(a_{j_1},
\ldots,a_{j_n};M)$ provided that $\textrm{qftp}(i_1,\ldots,i_n;{I})=\textrm{qftp}(j_1,\ldots,j_n;{I})$,
where $\textrm{qftp}$ denotes the quantifierfree type.
${I}$indexed indiscernible sets were introduced by Shelah
in the 70′s and have important applications in model
theory.
In this talk, I will dicuss wellknown examples of trees
${I}$ for which ${I}$indexed indiscernible sets are particularly
wellbehaved.
In particular, we will look at the structure ${I}_t =
(\omega^{<\omega},\unlhd,\le,\wedge)$ where $\unlhd$
is the partial order on the tree, $\wedge$ is the meet in
this order, and $\le$ is the lexicographical order. Takeuchi
and Tsuboi proved that ${I}_t$indexed indiscernibles have
a certain technical property, the modeling property.
By a dictionary theorem that I will present in this talk,
we may conclude that age(${I}_t$) is a Ramsey class.

Nov. 23
11:00 a.m. 
Todd Eisworth (Ohio
University)
A proof of Shelah’s “Cov vs. pp” theorem
We give a relatively easy proof of one of the core results
of Shelah’s “Cardinal Arithmetic”. The intent
is to present enough details so that the statement of the
theorem and the ideas underlying the proof are clear, even
if we don’t have enough time to prove every lemma completely.
We assume only a minimal knowledge of pcf theory: the basics
as outlined in Abraham & Magidor’s chapter of the
Handbook of Set Theory are more than enough.

Nov. 23
1:30 p.m. 
Jose Iovino (University of Texas at Arlington)
Model theory for structures of analysis, and omitting uncountable
types
Over the years, a number of different frameworks have been
proposed to study the model theory of structures of functional
analysis. All of these frameworks have turned out to be
equivalent. I will state a recent result that, among other
things, explains this equivalence. The result characterizes
these modeltheoretic frameworks in terms of a version for
uncountable languages of the classical omitting types theorem.
This result is joint with X. Caicedo.

Nov. 28
11:00 a.m. 
Lynn Scow (UIC)
$I$indexed Indiscernible Sets and Trees
http://settheory.mathtalks.org/lynnscow/

Nov 9
1:30
Room 230 
Michael Hrusak (UNAM)
Malykhin’s Problem
We prove that consitently every separable Frechet group
is metrizable.

Nov 2
11:00 a.m.
Room 230 
A.R.D. Mathias
The truth predicate and the forcing theorem in weak
subsystems of ZF
Devlin in his book "Constructibility"; sought
a theory true in every limit Goedel fragment $L_{\omega\nu}$
and every Jensen fragment $J_\nu$ (where $\nu\ge 1$) and
strong enough to define the truth predicate for $\Delta_0$
formulae.
For some years I sought to identify the weakest fragment
of ZF that would support a recognisable theory of set forcing,
and in particular the definition of $p\Vdash \phi$ for $\phi$
a $\Delta_0$ formula.</p>
These two quests turn out to have common ground and have
resulted in the theory of rudimentary recursion and provident
sets, which will be described in the talk.

Nov 2
1:30
Room 230 
Antonio Avilés
$P(\omega)/fin$ and its close relatives
We shall discuss uncountable Fraisse limits and iterated
pushout constructions. This is related to the problem of
finding structural characterizations of the Boolean algebra
$P(\omega)/fin$ like Parovichenko’s theorem under CH,
and DowHart‘s characterization in the model obtained
by adding $\aleph_2$ Cohen reals to a model of CH. We shall
explore some connections with Banach spaces as well.

Oct 19
Room 230 
István Juhász (Rényi Institute)
Resolvability
Let $\kappa$ be a (finite or infinite) cardinal number.
A topological space $X$ is said to be $\kappa$resolvable
(resp. almost $\kappa$resolvable) if there are $\kappa$
dense sets in $X$ that are pairwise disjoint (resp. almost
disjoint w.r.t. the nowhere dense ideal on $X$). The space
$X$ is maximally resolvable iff it is $\Delta(X)$resolvable,
where $$\Delta(X)=\min\{G : G\neq\emptyset\text{ open}\}.$$
In the first part of this talk we deal with the separation
of various resolvability and almost resolvability properties.
In the second part we describe results that deduce resolvability
properties from certain topological properties. In particular,
we present a recent joint result with M.
Magidor that characterizes maximal resolvability of
monotonically normal spaces in terms of maximal decomposability
of ultrafilters. We also report on work in progress, joint
with L.
Soukup and Z.
Szentmiklossy, concerning the problem of Malychin that asks
the following:
How resolvable is a regular Lindelof space in which
every nonempty open set is uncountable?

Oct 12
Room 230 
Dilip Raghavan (National University of Singapore)
More about the closed almostdisjointness number
Abstract: will be available in here: http://settheory.mathtalks.org/speaker/dilipraghavan/

Oct 5
Room 210

Piotr Koszmider (Polish Academy of Sciences)
On RadonNikodym compact spaces
We solve an old problem of Isaac Namioka proving that continuous
images of RadonNikodym compacta do not have to be RadonNikodym.
The construction requires certain combinatorial guessing
principle and is based on topological resolutions. We discuss
some consequences in the Banach space theory. This result
was obtained together with Antonio Avilés.

Sept. 28
11:00 a.m.
Room 2135** (Bahen Centre)

Michal Doucha (Charles University)
Canonization of analytic equivalence relations for the
CarlsonSimpson forcing

Sept. 28

Saharon Shelah (HUJI and Rutgers)
Weak axiom of choice : can the dead be resurrected

Sept. 21

Andreas Blass (University of Michigan)
The next best thing to a Ppoint
I'll present a couple of examples of ultrafilters that
are not Ppoints but have the strongest weak (i.e., squarebracket,
exponent 2) partition property that is possible for nonPpoints.
The two examples differ in other respects, one of which
shows a curious aspect of forcing. I plan to also summarize
preliminary definitions and results, to place these examples
in context.

Sept. 14

NO SEMINAR

Sept. 7
11:00 a.m.

Justin Moore (Cornell University)
Thompson’s group is amenable
I will demonstrate that Thompson’s group F is amenable.
This
will be done by exhibiting an idempotent measure on the
free
nonassociative groupoid on one generator. This in turn can
be used to generalize Hindman’s theorem to the setting
of nonassociative operations.
Talk attendees:
For those of you who attended my talk on Friday, the proof
of the amenability of F has now been checked sufficiently
that I've made the announcement completely public. A preprint
will appear as part of the ArXiv'x listing today at 7pm
EDT. It is also available on my webpage at: http://www.math.cornell.edu/~justin/Ftp/amen_F.pdf

Sept. 7

Marcin Sabok
Extreme amenability of abelian L_0 groups
http://settheory.mathtalks.org/marcinsabok/

Aug 31

Menachem Magidor (HUJI)
On w1strongly compact cardinals

Aug. 24

Lajos Soukup (Alfréd Rényi Institute
of Mathematics)
On properties of ladder systems on $\omega_1$

Aug. 24

Joel David Hamkins (The City University of New York)
Every countable model of set theory embeds into its own
constructible universe.
I shall give an account of my recent theorem showing that
every countable model of set theory $M$, including every
wellfounded model, is isomorphic to a submodel of its own
constructible universe. In other words, there is an embedding
$j:M\to L^M$ that is elementary for quantifierfree assertions.
The proof uses universal digraph combinatorics, including
an acyclic version of the countable random digraph, which
I call the countable random $\mathbb{Q}$graded digraph,
and higher analogues arising as uncountable Fraisse limits,
leading to the hypnagogic digraph, a sethomogeneous, classuniversal,
surrealnumbersgraded acyclic class digraph, closely connected
with the surreal numbers. The proof shows that $L^M$ contains
a submodel that is a universal acyclic digraph of rank $\text{Ord}^M$.
The method of proof also establishes that the countable
models of set theory are linearly preordered by embeddability:
for any two countable models of set
theory, one of them is isomorphic to a submodel of the other.
Indeed, the biembedability classes form a wellordered
chain of length $\omega_1+1$. Specifically, the countable
wellfounded models are ordered by embedability in accordance
with the heights of their ordinals; every shorter model
embeds into every taller model; every model of set theory
$M$ is universal for all countable wellfounded binary relations
of rank at most $\text{Ord}^M$; and every illfounded model
of set theory is universal for all countable acyclic binary
relations. Finally, strengthening a classical theorem of
Ressayre, the same proof method shows that if $M$ is any
nonstandard model of PA, then every countable model of set
theoryin particular, every model of ZFCis isomorphic
to a submodel of the hereditarily finite sets $HF^M$
of $M$. Indeed, $HF^M$ is universal for all countable acyclic
binary relations.

Aug. 17
11:00 a.m.

Teruyuki Yorioka
An application of AsperoMota’s iteration to Todorcevic’s
OCA
It is an open question whether it is consistent that Todorcevic’s
OCA holds together with the continuum larger than ?2. As
many people knows, it is sometimes trouble to force set
theoretic statements together with the continuum larger
than ?2. Aspero and Mota gave a new method of a forcing
iteration which forces that the size of the continuum is
larger than ?2, and they have proved that, for example,
it is consistent that the axiom holds together with the
continuum larger than ?2.
In the middle of 1990's, Ilijas Farah proved that it is
consistent that Todorcevic’s OCA for separable metric
spaces of size ?1 holds togetherwith the continuum larger
than ?2 It is given a different proof of this result using
AsperoMota’s iteration.

Aug. 17

Dima Sinapova
A bad scale and not SCH at ??
Starting from a supercompact, we construct a model in which
SCH fails at ?? and there is a bad scale at ??. The existence
of a bad scale implies the failure of weak square. The construction
uses two Prikry type forcings defined in different ground
models and a suitably defined projection between them.
This is joint work with Spencer Unger.

Aug 10

Tadatoshi Miyamoto (Nanzan)
A limit stage for proper iterated forcing of length omega
Given any notion of forcing P which is proper, we may form
a bigger notion of forcing Q with side conditions in such
a way that Q is proper and projects down to P.
Similary, given any iterated forcing (P_n, Q_n) of length
omega which iterates the proper partial orders Q_n, we may
form a bigger notion of forcing Q with side conditions in
such a way that Q projects down to each P_n.

July 20

David Milovich (Texas A&M)
Davies trees and stratified inverse limits
A Davies tree is a method of performing transfinite recursive
constructions longer than $\omega_1$ yet proceeding one
countable piece at a time. At any given stage, the tree
organizes all previous stages into finitely many nice pieces.
I will discuss how a simpler structure induces a canonical
Davies tree, extending the applicability of the Davies tree
and simplifying its use. I will also survey some proofs
that use Davies trees. The original such proof (Davies,
1963) shows (in ZFC) that the plane is a countable union
of rotations of graphs of functions. Most recently, I have
used a Davies tree in a proof that every zerodimensional
openly generated compactum is a continuous image of a homogeneous
zerodimensional openly generated compactum.

July 13
Bahen Centre, Room 1210

Tony Wong (Caltech)
Diagonal forms for incidence matrices and zerosum Ramsey
theory
Let $H$ be a $t$uniform hypergraph on $k$ vertices, with
$a_i\geq0$ denoting the multiplicity of the $i$th edge,
$1\leq i\leq\binom{k}{t}$. Let ${\textbf{h}}=(a_1,\dotsc,a_{\binom{k}{t}})^\top$,
and $N_t(H)$ the matrix whose columns are the images of
${\textbf{h}}$ under the symmetric group $S_k$. We determine
a diagonal form (Smith normal form) of $N_t(H)$ for a very
general class of $H$. Now, assume $H$ is simple. Let $K^{(t)}_n$
be the complete $t$uniform hypergraph on $n$ vertices,
and $R(H,\mathbb{Z}_p)$ the zerosum (mod $p$) Ramsey number,
which is the minimum $n\in\mathbb{N}$ such that for every
coloring $c:E\big(K^{(t)}_n\big)\to\mathbb{Z}_p$, there
exists a copy $H'$ isomorphic to $H$ inside $K^{(t)}_n$
such that $\sum_{e\in E(H')}c(e)=0$. Through finding a diagonal
form of $N_t(H)$, we reprove a theorem of Y. Caro in Caro
(1994) that gives the value $R(G,\mathbb{Z}_2)$ for any
simple graph $G$. Further, we show that for any $t$, $R(H,\mathbb{Z}_2)$
is almost surely $k$ as $k\to\infty$, where $k$ is the number
of vertices of $H$. Similar techniques can also be applied
to determine the zerosum (mod $2$) bipartite Ramsey numbers,
$B(G,\mathbb{Z}_2)$, introduced in CaroYuster (1998).

July 6

Brent Cody (Fields)
Easton's Theorem for Woodin cardinals
Easton proved that the continuum function $\kappa\mapsto
2^\kappa$ on regular cardinals can be forced to behave in
any way that is consistent with K\"onig's Theorem ($\kappa<\
cf (2^\kappa)$) and monotonicity ($\kappa<\lambda$ implies
$2^\kappa\leq 2^\lambda$). In the presence of large cardinals,
there are additional restrictions on the possible behaviors
of the continuum function on regular cardinals. A natural
question to ask is: given a large cardinal $\kappa$, what
possible behaviors of the continuum function can we force
while preserving the large cardinal property of $\kappa$?
I will give a brief outline of the literature in this area.
I will also sketch a proof of the following result from
my dissertation. Suppose $\delta$ is a Woodin cardinal and
$F$ is any class function from the regular cardinals to
the cardinals such that (1) $\delta$ is a closure point
of $F$, (2) $\kappa< \ cf (F(\kappa))$ for each $\kappa\in
\ REG$, (3) $\kappa<\lambda$ implies $F(\kappa)\leq F(\lambda)$
for $\kappa,\lambda\in \ REG$. Then there is a cofinalitypreserving
forcing extension in which $\delta$ remains Woodin and $2^\gamma=F(\gamma)$
for each regular cardinal $\gamma$. The proof uses the tuning
fork method of Friedman and Thompson as well as some lifting
techniques due to Friedman and Honsik.
