CUNY Graduate Center
Virtual (email Victoria Gitman for meeting id)
Fridays 11:00am-12:30pm
Organized by Gunter Fuchs and Victoria Gitman
Calendar
September 12
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Rahman Mohammadpour
Institute of Mathematics of Polish Academy of Sciences
Specializing Triples
Abstract
I will talk about weak embeddability and the universality number of the class of Aronszajn trees, with a focus on the role of specializing triples.
The notion of a specializing triple was introduced by Džamonja and Shelah in their strong negative solution to an old problem on the existence of a universal (with respect to weak embeddability) wide Aronszajn tree under Martin's axiom. Their proof has two stages: first, they reprove a theorem of Todorčević showing that under ${\rm MA}_{\omega_1}$ there is no universal Aronszajn tree, and then they show that every wide Aronszajn tree weakly embeds into an Aronszajn tree. The second stage involves a rather complicated ccc forcing. However, already in the first stage, they introduce a new technique: the notion of a specializing triple, and prove that for each Aronszajn tree $T$, there is a ccc forcing adding another Aronszajn tree $T^*$ together with a specializing function on $T^*\otimes T$ such that $(T^*, T, c)$ is a specializing triple. In particular, this shows that $T^*$ does not weakly embed into $T$.
I will explain how a slight but careful modification of this definition makes it possible to accommodate wide trees directly, yielding a more streamlined proof of Džamonja and Shelah’s result. More precisely, for every $\kappa$-wide Aronszajn tree $T$, there is a ccc forcing adding an Aronszajn tree $T^*$ and a function $c$ such that $(T^*, T, c)$ is what I call a left specializing triple. From this, one quickly recovers Džamonja-Shelah’s theorem: under Martin’s axiom, every class of trees of height $\omega_1$ and size less than the continuum but with no cofinal branches either is not universal for Aronszajn trees, or has universality number equal to the continuum.
Finally, I will indicate how the modified definition can also be used to show that this consequence of Martin’s axiom is consistent with the existence of a nonspecial Aronszajn tree.
Video
October 3
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Eyal Kaplan
University of California, Berkeley
The number of normal measures, revisited
Abstract
A central question in the theory of large cardinals was whether the existence of a model of ZFC with exactly two normal measures follows from the consistency of ZFC with a measurable cardinal. This was answered positively by a landmark theorem of Friedman and Magidor, whose proof masterfully combined advanced techniques in the theory of large cardinals, including generalized Sacks forcing, forcing over canonical inner models, coding posets, and nonstationary support iterations.
In this talk, we present a new and simpler proof of the Friedman-Magidor theorem. A notable feature of our approach is that it avoids any use of inner model theory, making it applicable in the presence of very large cardinals that are beyond the current reach of the inner model program. If time permits, we will also discuss additional applications of the technique: the construction of ZFC models with several normal measures but a single normal ultrapower; a nontrivial model of the weak Ultrapower Axiom from the optimal large cardinal assumption; and a generalization of the Friedman–Magidor theorem to extenders.
Video
October 10
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Dan Hathaway
University of Vermont
TBA
Abstract
October 17
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Calliope Ryan-Smith
University of Leeds
TBA
Abstract
October 24
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Bartosz Wcisło
University of Gdańsk
TBA
Abstract
October 31
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Corey Switzer
University of Vienna
TBA
Abstract
November 7
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Emma Palmer
University of Oxford
TBA
Abstract
November 14
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Andrew Brooke-Taylor
University of Leeds
TBA
Abstract
November 21
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Bokai Yao
Peking University
TBA
Abstract
December 5
Virtual (email Victoria Gitman for meeting id)
11:00am NY time
Philip Welch
University of Bristol
TBA
Abstract
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