SCIENTIFIC PROGRAMS AND ACTIVITIES

Seminars to June 30, 2013 (Seminars in the 2013-14 academic year)
at 1:30 p.m. in the Fields Institute, Room 210

Asger Törnquist
Talking about a theorem he is proving at the moment and is not yet ready to be revealed

A well-known 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.

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 self-contained). 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.

Frank Tall
A provisional solution to Nyikos’ manifold problem

Peter Nyikos observed that, although the Long Line is a non-metrizable, hereditarily normal manifold, it is difficult to find such a manifold of dimension > 1. Indeed the only such examples are constructed with extra set-theorteic 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

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 cofinality-preserving forcing? Could the same graph have different chromatic numbers in different cofinality-preserving 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.

Dilip Raghavan
Combinatorial dichotomies and cardinal invariants

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known 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 $S\not\rightarrow ({\aleph}_{1}, \omega+2)^2$.

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 Graham-Rothschild’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.

In his PhD Thesis Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is K-sigma – 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 K-sigma (since it is countable) but not compactly generated.

Beros showed that for any Polish group G, every K-sigma subgroup of G is compactly generated iff every countable subgroup of G is compactly generated. Beros showed that any K-sigma 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 (n-1)-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 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.

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 non-precompact 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.

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 Hales-Jewett Theorem and the Density Halpern-Läuchli Theorem. If time permits we will also present some applications.

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 left-or 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 0-dimensional, 1st countable Hausdorff space of weight ? 1 and size continuum which is non base resolvable.

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.

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.

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

Chris Eagle
Omitting types in infinitary [0,1]-valued logic (presentation)

In first-order logic many interesting non-elementary 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 $\mathcal{L}_{\omega_1, \omega}$.

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) .

Mike Pawliuk
Using T-sequences 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 non-trivial sequence converge to 0. Not all sequences in the Integers admit a T_2 group topology on the integers; those that do are called T-sequences.

We will examine three different aspects of T-sequences. First we will see that there are as many (different) T-sequences as possible, and that only certain types of chains of topologies given by T-sequences are possible. Then we will see that group topologies given by a (non-trivial) T-sequence are all examples of sequential spaces that are not Frechet-Urysohn. Finally, we will give a nice diagonalization technique that produces many topological groups on the integers that do not admit characters.

Fields closed due to weather conditions

Saeed Ghasemi
Automorphisms of Borel quotients of FDD-algebras

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 FDD-algebra (finite dimensional decomposition) has a *-homomorphism representation (lifting).

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.

Trevor Wilson
Well-behaved 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 “well-behaved.” In particular every such measure is ordinal-definable from $M$, generalizing a theorem of Kunen that says “AD implies that every measure on an ordinal less than Theta is ordinal-definable.” 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.

Alex Rennet
Axiomatizability, Ultraproducts and O-Minimality

Suppose a theory T is given as the set of sentences true in all structures in a fixed language which share some non-first order property P. For instance, if P is ‘stable’ or ‘o-minimal’ or ‘finite’, we get the L-theory of stability, o-minimality or finiteness respectively. A classic model-theoretic 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 o-minimality (a model-theoretic property related to non-standard analysis) this answers negatively a suggestion from the recent literature. I’ll go through the proof of this result, which pleasantly involves minimal model-theoretic detail.

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.

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 (implication-wise) 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 Feng-Jech’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 Kunen-Magidor constructions of saturated ideals to work in the context of supercompact towers which are not almost huge. This is joint work with Martin Zeman.

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.

Alan Dow
Non-trivial copies of N* in N*

It is a rather old problem of E. van Douwen to determine if there is (in ZFC) a non-trivial 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 non-trivial copies of N* but, following Shelah’s work in on the consistency of the non-existence 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 non-trivial copy of N* would have to satisfy the ccc over fin property (Farah).

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 same-length 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 quantifier-free 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 well-known examples of trees ${I}$ for which ${I}$-indexed indiscernible sets are particularly well-behaved.
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.

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.

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 model-theoretic frameworks in terms of a version for uncountable languages of the classical omitting types theorem. This result is joint with X. Caicedo.

Lynn Scow (UIC)
$I$-indexed Indiscernible Sets and Trees

http://settheory.mathtalks.org/lynn-scow/

Michael Hrusak (UNAM)
Malykhin’s Problem

We prove that consitently every separable Frechet group is metrizable.

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.

Antonio Avilés
$P(\omega)/fin$ and its close relatives

We shall discuss uncountable Fraisse limits and iterated push-out 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 Dow-Hart‘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.

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 non-empty open set is uncountable?

Dilip Raghavan (National University of Singapore)
More about the closed almost-disjointness number

Abstract: will be available in here: http://settheory.mathtalks.org/speaker/dilip-raghavan/

Piotr Koszmider (Polish Academy of Sciences)
On Radon-Nikodym compact spaces

We solve an old problem of Isaac Namioka proving that continuous images of Radon-Nikodym compacta do not have to be Radon-Nikodym. 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 Carlson-Simpson forcing

Sept. 28

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

Andreas Blass (University of Michigan)
The next best thing to a P-point

I'll present a couple of examples of ultrafilters that are not P-points but have the strongest weak (i.e., square-bracket, exponent 2) partition property that is possible for non-P-points. 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/marcin-sabok/

Aug 31

Menachem Magidor (HUJI)
On w1-strongly 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 well-founded 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 quantifier-free 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 set-homogeneous, class-universal, surreal-numbers-graded 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 pre-ordered by embeddability: for any two countable models of set
theory, one of them is isomorphic to a submodel of the other. Indeed, the bi-embedability classes form a well-ordered chain of length $\omega_1+1$. Specifically, the countable well-founded 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 well-founded binary relations of rank at most $\text{Ord}^M$; and every ill-founded 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 theory---in particular, every model of ZFC---is 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 Aspero-Mota’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 Aspero-Mota’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 zero-dimensional openly generated compactum is a continuous image of a homogeneous zero-dimensional openly generated compactum.

July 13
Bahen Centre, Room 1210

Tony Wong (Caltech)
Diagonal forms for incidence matrices and zero-sum 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 zero-sum (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 zero-sum (mod $2$) bipartite Ramsey numbers, $B(G,\mathbb{Z}_2)$, introduced in Caro-Yuster (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 cofinality-preserving 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.