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SUMMARY:Keegan Dasilva Barbosa (University of Toronto)
DTSTART;VALUE=DATE-TIME:20200410T173000Z
DTEND;VALUE=DATE-TIME:20200410T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/1
DESCRIPTION:Title: A Decomposition Theorem for Aronszajn Lines\nby Keegan Dasilva
Barbosa (University of Toronto) as part of Toronto set theory seminar\n\n
\nAbstract\nWe will prove that under the proper forcing axiom\, the class
of all Aronszajn lines behave like $\\sigma$-scattered orders under the em
beddability relation. In particular\, we show that the class of better qua
si order labeled fragmented Aronszajn lines is itself a better quasi order
. Moreover\, we show that every better quasi order labeled Aronszajn line
can be expressed as a finite sum of labeled types which are algebraically
indecomposable. By encoding lines with finite labeled trees\, we are also
able to deduce a decomposition result\, that for every Aronszajn line $L$\
, there is an $n\\in \\omega$ such that for any finite colouring of $L$\,
there is a subset $L\\prime$ of $L$ isomorphic to $L$ which uses no more t
han $n$ colours. \n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/1/
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SUMMARY:Matteo Viale (University of Torino)
DTSTART;VALUE=DATE-TIME:20200417T173000Z
DTEND;VALUE=DATE-TIME:20200417T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/2
DESCRIPTION:Title: Tameness for Set Theory\nby Matteo Viale (University of Torino
) as part of Toronto set theory seminar\n\n\nAbstract\nWe show that (assum
ing large cardinals) set theory is a tractable (and we dare to say tame) f
irst order theory when formalized in a first order signature with natural
predicate symbols for the basic definable concepts of second and third ord
er arithmetic\, and appealing to the model-theoretic notions of model comp
leteness and model companionship.\n\nSpecifically we develop a general fra
mework linking generic absoluteness results to model companionship and sho
w that (with the required care in details) a $\\Pi_2$-property formalized
in an appropriate language for second or third order number theory is forc
ible from some T extending ZFC + large cardinals if and only if it is cons
istent with the universal fragment of T if and only if it is realized in t
he model companion of T.\n\nPart (but not all) of our results are conditio
nal to the proof of Schindler and Asperò that Woodin's axiom (*) can be f
orced by a stationary set preserving forcing.\n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/2/
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BEGIN:VEVENT
SUMMARY:Todd Eisworth (Ohio State University)
DTSTART;VALUE=DATE-TIME:20200424T173000Z
DTEND;VALUE=DATE-TIME:20200424T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/3
DESCRIPTION:Title: Representability and pseudopowers\nby Todd Eisworth (Ohio Stat
e University) as part of Toronto set theory seminar\n\n\nAbstract\nWe will
prove some basic facts about Shelah's pseudopower function\, and derive s
ome new (?) ZFC results in cardinal arithmetic using basic topological ide
as. This talk is designed to be an introduction to this part of pcf theory
.\n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/3/
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SUMMARY:Paul Szeptycki (York University)
DTSTART;VALUE=DATE-TIME:20200501T173000Z
DTEND;VALUE=DATE-TIME:20200501T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/4
DESCRIPTION:Title: Strong convergence properties and an example from a $\\Box$ sequen
ce\nby Paul Szeptycki (York University) as part of Toronto set theory
seminar\n\n\nAbstract\nWe present an example of a space constructed from $
\\Box(\\kappa)$ answering some questions of Arhangel'skii.\n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/4/
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BEGIN:VEVENT
SUMMARY:Dima Sinapova (UIC)
DTSTART;VALUE=DATE-TIME:20200508T173000Z
DTEND;VALUE=DATE-TIME:20200508T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/5
DESCRIPTION:Title: Iteration\, reflection\, and Prikry forcing.\nby Dima Sinapova
(UIC) as part of Toronto set theory seminar\n\n\nAbstract\nThere is an in
herent tension between stationary reflection and the failure of SCH. The f
ormer is a compactness type principle that follows from large cardinals. T
he latter is an instance of incompactness\, and usually obtained using Pri
kry forcing. We describe a Prikry style iteration\, and use it to force st
ationary reflection in the presence of not SCH. Then we discuss the situat
ion at smaller cardinals. This is joint work with Alejandro Poveda and Ass
af Rinot.\n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/5/
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SUMMARY:Vinicius de Oliveira Rodrigues (University of São Paulo and Unive
rsity of São Paulo\, Institute of Mathematics and Statistics)
DTSTART;VALUE=DATE-TIME:20200522T173000Z
DTEND;VALUE=DATE-TIME:20200522T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/6
DESCRIPTION:Title: Pseudocompact hyperspaces of Isbell-Mrówka spaces\nby Viniciu
s de Oliveira Rodrigues (University of São Paulo and University of São P
aulo\, Institute of Mathematics and Statistics) as part of Toronto set the
ory seminar\n\n\nAbstract\nJ. Ginsburg has asked what is the relation betw
een the pseudocompactness of the $\\omega$-th power of a topological space
$X$ and the pseudocompactness of its Vietoris Hyperspace\, $\\exp(X)$. M.
Hrusak\, I. Martínez-Ruiz and F. Hernandez-Hernandez studied this questi
on restricted to Isbell-Mrówka spaces\, that is\, spaces of the form $\\P
si(A)$ where A is an almost disjoint family. Regarding these spaces\, if $
\\exp(X)$ is pseudocompact\, then $X^\\omega$ is also pseudocompact\, and
$X^\\omega$ is pseudocompact iff $A$ is a MAD family. They showed that if
the cardinal characteristic $\\mathfrak{p}$ is $\\mathfrak{c}$\, then for
every MAD family $A$\, $\\exp(\\Psi(A))$ is pseudocompact\, and if the car
dinal characteristic $\\mathfrak{h}$ is less than $\\mathfrak{c}$\, there
exists a MAD family $A$ such that $\\exp(\\Psi(A))$ is not pseudocompact.
They asked if there exists a MAD family $A$ (in ZFC) such that $\\exp(\\Ps
i(A))$ is pseudocompact.\n\nIn this talk\, we present some new results on
the (consistent) existence of MAD families whose hyperspaces of their Isbe
ll-Mrówka spaces are (or are not) pseudocompact by constructing new examp
les. Moreover\, we give some combinatorial equivalences for every Isbell-M
rówka space from a MAD family having pseudocompact hyperspace. This is a
joint work with\, O. Guzman\, M. Hrusak\, S. Todorcevic and A. Tomita.\n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/6/
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SUMMARY:Jamal Kawach (University of Toronto)
DTSTART;VALUE=DATE-TIME:20200612T173000Z
DTEND;VALUE=DATE-TIME:20200612T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/7
DESCRIPTION:Title: Dual Ramsey theory for countable ordinals\nby Jamal Kawach (Un
iversity of Toronto) as part of Toronto set theory seminar\n\n\nAbstract\n
Using techniques from the theory of topological Ramsey spaces\, we prove a
dual Ramsey theorem for countable ordinals. Specifically\, for each count
able ordinal $\\alpha$ we define a topological Ramsey space of equivalence
relations on $\\omega$ which code equivalence relations on $\\alpha$\, up
to a necessary restriction on the set of minimal representatives of the e
quivalence classes. This extends the classical dual Ramsey theorem of Carl
son and Simpson. This is joint work with Stevo Todorcevic.\n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/7/
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SUMMARY:Will Brian (University of North Carolina at Charlotte)
DTSTART;VALUE=DATE-TIME:20200626T173000Z
DTEND;VALUE=DATE-TIME:20200626T183000Z
DTSTAMP;VALUE=DATE-TIME:20240614T084131Z
UID:FieldsSetTheory/8
DESCRIPTION:Title: Limited-information strategies in Banach-Mazur games\nby Will
Brian (University of North Carolina at Charlotte) as part of Toronto set t
heory seminar\n\n\nAbstract\nThe Banach-Mazur game is an infinite-length g
ame played on a topological space X\, in which two players take turns choo
sing members of an infinite decreasing sequence of open sets\, the first p
layer trying to ensure that the intersection of this sequence is empty\, a
nd the second that it is not. A limited-information strategy for one of th
e players is a game plan that\, on any given move\, depends on only a smal
l part of the game's history. In this talk we will discuss Telgársky's co
njecture\, which asserts roughly that there must be topological spaces whe
re winning strategies for the Banach-Mazur game cannot be too limited\, bu
t must rely on large parts of the game's history in a significant way. Rec
ently\, it was shown that this conjecture fails in models of set theory sa
tisfying GCH + $\\Box$. In such models it is always possible for one playe
r to code all information concerning a game's history into a small piece o
f it. We will discuss these so-called coding strategies\, why assuming GCH
+ $\\Box$ makes them work so well\, and what can go wrong in other models
of set theory.\n
LOCATION:https://researchseminars.org/talk/FieldsSetTheory/8/
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