AAAS, Pacific Division
2015 SAN FRANCISCO
Symposium Abstracts
02 Boise Extravaganza on Set Theory (BEST)
FAC and WQO Orders and Their Cardinal Invariants, MIRNA DZAMONJA (School of Mathematics,
University of East Anglia, Norwich, NR4 7TJ United Kingdom; [email protected]).
Posets with finite antichain condition (FAC) are amenable to a study versus various rank functions, for
example the rank of the tree of antichains, called the width or other similar cardinal invariants. Some of
them are better understood than the others, with the width probably the least understood until now. The
work of Abraham, Bonnet, Cummings, Dz. and Thompson (2013) gave the characterization of all FAC
orders as the closure of the class of WQO orders and linear orders under lexicographic sums, reversal and
augmentation. This implies that the building blocks for understanding the cardinal invariants of the FAC
orders are the invariants of the WQO and linear orders. In a joint work with Schmitz and Schnoebelen we
study the width function, show it is equivalent to the antichain rank and prove various structural theorems.
Cardinal Invariants of Generalized Continua, ERNEST SCHIMMERLING and LUKE SERAFIN*
(Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213;
[email protected]).
Cardinal invariants such as the bounding, dominating, and splitting numbers of the continuum have
proven very useful for studying aspects of the combinatorial structure and topological properties of the
Baire space ωω and related Polish spaces. This has led several researchers, including Cummings and
Zapletal, to generalize these invariants to function spaces of higher cardinality, in particular to κκ for κ an
uncountable cardinal. We give an overview of research in this area and some results concerning the relative
sizes of generalized cardinal invariants which appear to be new. This is joint work with E. Scimmerling
PRESENTATION yet to be determined
The Universal Minimal Flow of the Homeomorphism Group of the Lelek Fan, ALEKSANDRA
KWIATKOWSKA (Department of Mathematics, University of California, Los Angeles, CA 90095-1555;
[email protected])
The Lelek fan L is a compact and connected space with many symmetries, which can be constructed
from a projective Fraisse limit, and hence it has a very rich homeomorphism group H(L). I will show a
number of properties of H(L) - it is totally disconnected, generated by every neighbourhood of the identity,
has a dense conjugacy class, and is simple. Then I will focus on the dynamics of H(L) and using the
Graham-Rothschild theorem, Kechris-Pestov-Todorcevic correspondence, as well as some new ideas, I will
describe the universal minimal flow of H(L).
The Conjugacy Problem for Automorphism Groups of Countable Homogeneous Structures, SAMUEL
COSKEY1 and PAUL ELLIS*2, (1Department of Mathematics, Boise State University, Boise, ID 83725;
2
Department of Mathematics and Computer Science, Manhattanville College, Purchase, NY 10577;
[email protected]).
Building on past joint work with Scott Schneider, we continue to study the conjugacy problem for the
automorphism groups of a number of countable homogeneous structures. In this talk, I will introduce the
notion of Borel reducibility and give an brief overview of the countable homogeneous linear orders,
undirected graphs, and digraphs. I will explain which of the corresponding conjugacy problems we have
fully classified (up to Borel reducibility), as well as those whose Borel complexity we only still suspect. I
will prove a couple of the easier cases to give the flavor of the quest.
The Density Topology, ANDRES CAICEDO and STUART NYGARD* (Department of Mathematics,
Boise State University, Boise, ID 83725; [email protected]).
The usual topology on the real line uses a basis of open intervals. Instead of open intervals, we take as
our basis sets which have locally full Lebesgue measure. The resulting topology is not only a refinement of
the original topology, but it occurs "naturally." That is, the topology arises in many spaces. We consider the
density topology on the space of continuous functions.
PRESENTATION yet to be determined
Covering Properties of Derived Models, TREVOR M. WILSON (Department of Mathematics, University
of California, Irvine, Irvine, CA 92697-3875; [email protected]).
Derived models are models of the axiom of determinacy (AD) that are obtained from large cardinals
via forcing. They are, in a sense, maximal canonical models of AD, whereas core models are maximal
canonical models of ZFC + large cardinals. We discuss covering results for derived models at inaccessible
limits of Woodin cardinals and at weakly compact limits of Woodin cardinals. By "covering" we mean that
these derived models are close to V in terms of certain cardinals and cofinalities. These results are parallel
to inner-model-theoretic covering theorems of Jensen and Kunen respectively, although the method of
proof is quite different. We also discuss an open problem regarding covering for derived models at singular
limits of Woodin cardinals.
Joint Laver Diamonds, MIHA E HABIC (Graduate Center, CUNY, New York, NY 10016;
[email protected]).
A joint Laver diamond sequence for a given large cardinal is a sequence of Laver functions for that
large cardinal which can simultaneously anticipate any given sequence of target sets of the same length via
a single large cardinal embedding. In particular, we consider joint Laver diamonds for supercompact and
strong cardinals, as well as their level-by-level stratifications. We show that, in most of these cases, the
existence of joint Laver diamond sequences is equiconsistent with the large cardinal notion itself. In the
supercompact case we also give separation results, showing that the existence of a joint Laver diamond
sequence of some given length does not imply the existence of one of any greater length.
PRESENTATION to be determined
Productively Countably Tight Spaces and Selective Games, LEANDRO F. AURICHI1, ANGELO
BELLA2 and RODRIGO R. DIAS*3 (1Institute of Mathematical Sciences and Computation, University of
Sao Paulo in Sao Carlos, Brazil; 2Department of Mathematics, University of Catania, Catania, Italy;
3
Institute of Mathematics and Statistics, University of Sao Paolo, Sao Paulo, Brazil; [email protected]).
The connections between topological games and productivity of countable tightness were first
established by Gruenhage in 1976. In a recent work, Aurichi and Bella investigated such connections in the
scope of selection principles, showing that productivity of countable tightness is a property that lies strictly
between countable strong fan tightness and the existence of a winning strategy for the second player in the
selective game naturally associated to it. We start by showing that the nonexistence of a winning strategy
for the first player in this game is not related to productivity of countable tightness, which answers a
question from Aurichi and Bella's work. This leads to the study of variations of these properties; among
other things, we show that, although countable strong fan tightness is equivalent to the selection principle
S2(Ωp, Ωp), the selective games associated to these properties are different.
Applications of Almost Compatible Functions for Limited Information Strategies in Infinite Length Games,
STEVEN CLONTZ (Department of Mathematics, Auburn University, Auburn, AL 36849;
[email protected]).
Scheepers introduced the statement S(κ) which asserts the existence of pairwise almost-compatible
functions (agreeing on all but finitely many ordinals) for each countable subset of κ. S(ω1) is a theorem of
ZFC, and S(κ) is false for κ greater than the continuum; otherwise this statement is independent of ZFC.
The presenter will demonstrate several games for which S(κ) implies the existence of a winning strategy
relying on at most two moves of the opponent (and perhaps the round number), including a special case of
the topological game characterizing Menger's covering property.
The Selective Strong Screenability Game, LILJANA BABINKOSTOVA* and MARION SCHEEPERS
(Department
of
Mathematics,
Boise
State
University,
Boise,
ID
83725;
[email protected]).
Selective screenability, introduced in a 1978 paper, and selective strong screenability, coincide in a
wide class of topological spaces. The two corresponding games, however, are vastly different from each
other. In this talk we describe results that illustrate this remark, and indicate that a deeper investigation of
the selective strong screenability game is necessary.
Baire Measurable Paradoxical Decompositions via Matchings, ANDREW MARKS1 and SPENCER
UNGER*2 (1Department of Mathematics, California Institute of Technology, Pasadena, CA 91125;
2
Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095-1555;
[email protected]).
We show that every locally finite Borel bipartite graph satisfying a strengthening on Hall's condition
has a Borel perfect matching on some comeager invariant set. We apply this to show that every Borel
action of a countable nonamenable group on a Polish space has a Baire measurable paradoxical
decomposition. This is joint work with Andrew Marks.
Finite Forms of Gowers’ FINk Theorem, DIANA OJEDA-ARISTIZABAL (Department of Mathematics,
University of Toronto, Toronto, Ontario, Canada M5S 2E4; [email protected]).
Hindman’s Theorem states that given any finite coloring of FIN, the collection of finite sets of natural
numbers, there exists an infinite sequence of elements of FIN all of whose finite unions get the same color.
Gowers formulated and proved a generalization of Hindman’s Theorem to obtain the oscillation stability of
c0, the space of real sequences converging to 0 endowed with the supremum norm. We present a
combinatorial proof of the finite version of Gowers’ generalization of Hindman’s Theorem. For a fixed
natural number k, the objects that are being colored are finitely supported functions defined on the natural
numbers, that take values -k, -(k - 1), ... , -1, 0, 1, ..., k. We first obtain the result for functions that take only
non-negative values and then describe how to reduce the general result to this special case. Our approach to
obtain the general result from the special case for non-negative functions also works for the infinite version
of Gowers’ Theorem.
Products of Topological Ramsey Spaces, TIMOTHY TRUJILLO (Department of Mathematics, Texas
State University, San Marcos, TX 78666-4616; [email protected]).
Two of the most well known topological Ramsey spaces are the Millliken space and the Ellentuck
space. We will introduce a method for constructing products of these spaces with one another and ask some
questions about its applications and if the construction can be extended to arbitrary topological Ramsey
spaces.