Failure of GCH on a measurable with the Ultrapower Axiom

Speaker: 

Eyal Kaplan

Institution: 

UC Berkeley

Time: 

Monday, December 2, 2024 - 4:00pm to 5:40pm

Location: 

RH 440 R

The Ultrapower Axiom (UA) roughly states that any pair of ultrapowers can be compared by internal ultrapowers. The Axiom was extensively studied by Gabriel Goldberg, leading to a series of striking results.

Goldberg asked whether UA is consistent with a measurable cardinal that violates GCH. The main challenge is that UA is not easily preserved under forcing constructions, especially ones that achieve violation of GCH on a measurable from large cardinal assumptions. For example, such forcings might create normal measures which are incomparable in the Mitchell order – a property that negates UA.
In this talk, we sketch the proof that the failure of GCH on the least measurable cardinal can indeed be forced while preserving UA, starting from the minimal canonical inner model carrying a (\kappa, \kappa^{++})-extender. We will present the forcing construction and sketch the main proof ideas. This is a joint work with Omer Ben-Neria.

Shuffling Posets and new failures of Squares

Speaker: 

Omer Ben-Neria

Institution: 

Hebrew University of Jerusalem

Time: 

Tuesday, October 15, 2024 - 3:00pm to 4:30pm

Location: 

RH 440R

The purpose of this talk is to introduce a new forcing method from a joint project with Daniel Iosub called "Shuffling", and explain how it is used to obtain new consistency results involving different square principles.  Given a poset P, the shuffling method aims to form a related poset that captures certain essential generic sets added by P while avoiding other undesirable ones. After introducing the method, I will explain its connection with square principles and how it is used to answer questions by Jensen, Cummings and Friedman about the failure of the global square principle, and questions about the points at which squares fail. 

Topological Erdős Similarity Conjecture and Strong Measure Zero Sets

Speaker: 

Yeonwook Jung

Institution: 

UC Irvine

Time: 

Monday, October 7, 2024 - 4:00pm to 5:30pm

Location: 

RH 440 R

 It is well-known that a finite set is universal, that is, each Lebesgue measurable set with positive measure contains an affine copy of a finite set. The Erdős similarity conjecture, which remains open, states that there is no infinite universal set. In 2022, Gallagher, Lai, and Weber considered a topological version of this conjecture, defining a set to be topologically universal if each dense G-delta set contains an affine copy of the set. They conjectured that there are no such uncountable sets. In this thesis, we give a full classification of topologically universal sets as a special subfamily of measure zero sets. As a corollary, we prove that the topological Erdős similarity conjecture is independent of ZFC. We generalize this result to arbitrary locally compact Polish groups, and use the measure-category duality to pose and investigate the full-measure Erdős similarity conjecture.

The Arithmetic of Linear Orders

Speaker: 

Garrett Irvin

Institution: 

California Institute of Technology

Time: 

Monday, April 22, 2024 - 4:00pm to 5:20pm

Host: 

Location: 

RH 440R

There are two natural arithmetic operations on the class of linear orders: the sum + and lexicographic product x. These operations generalize the sum and product of ordinals. 

The arithmetic laws obeyed by the sum were uncovered in the pre-forcing days of set theory and are surprisingly nice. For example, while the left cancellation law A + X \cong B + X => A \cong B is not true in general, its failure can be completely characterized: a linear order X fails to cancel in some such isomorphism if and only if there is a non-empty order R such that R + X \cong X. Right cancellation is symmetrically characterized. 

Tarski and Aronszajn characterized the commuting pairs of linear orders, i.e. the pairs X and Y such that X + Y \cong Y + X.

Lindenbaum showed that X + X \cong Y + Y implies X \cong Y for linear orders X and Y. More generally, the finite cancellation law nX \cong nY => X \cong Y holds. Lindenbaum showed that the sum even satisfies the Euclidean algorithm! 

On the other hand, the arithmetic of the lexicographic product is much less well understood. The lone totally general classical result is due to Morel, who characterized when the right cancellation law A x X \cong B x X => A \cong B holds. Morel showed that an order X fails to cancel in some such isomorphism if and only if there is a non-singleton order R such that R x X \cong X, in analogy with the additive case. 

In this talk we focus on the question of whether Morel’s cancellation theorem is true on the left. We’ll show that, while the literal left-sided version of Morel’s theorem is false, an appropriately reformulated version is true. Our results suggest that a complete characterization of left cancellation in lexicographic products is possible. We’ll also discuss how our work might help in proving multiplicative versions of Tarski’s, Aronszajn’s, and Lindenbaum’s additive laws.

This is joint work with Eric Paul.  

Extensions of the Axiom of Determinacy and the ABCD Conjecture

Speaker: 

Nam Trang

Institution: 

University of North Texas

Time: 

Monday, April 15, 2024 - 4:00pm to 5:20pm

Host: 

Location: 

RH 440 R

The axiom AD^+, a structural strengthening of the Axiom of Determinacy (AD), was introduced by Hugh Woodin in the 1980's. AD^+ resolves many basic structural questions unsettled by AD. However, there are still many basic questions not answered by AD^+. One such class of questions concerns comparing cardinalities of sets under AD^+: given any two sets X and Y, how can we compare |X| and |Y|? One concrete instance of this is the following conjecture. 

 

Conjecture (the ABCD conjecture): suppose \alpha,\beta,\gamma,\delta are infinite cardinals such that \beta \leq \alpha and \delta\leq \gamma. Then |\alpha^\beta| \leq |\gamma^\delta| if and only if \alpha\leq \gamma and \beta \leq \delta.

 

The ABCD Conjecture is false under ZFC. It is open whether AD^+ implies the conjecture holds, but many instances of the conjecture have been established (by work of Woodin, Chan-Jackson-Trang etc). We introduce a structural strengthening of the axiom AD^+, called AD^{++}. AD^{++} implies the ABCD Conjecture and appears to have other interesting consequences not known to follow from AD^+. We do not know if AD^+ implies AD^{++} but some special cases have been proved. We will define these notions and discuss some of the partial results mentioned above. This is ongoing joint work with W. Chan and S. Jackson.

Lossless expansion and measure hyperfiniteness

Speaker: 

Jan Grebik

Institution: 

UCLA

Time: 

Monday, April 1, 2024 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440 R

Abstract: The notions of measure hyperfiniteness and measure reducibility of countable Borel equivalence relations are variants of the usual notions of hyperfiniteness and Borel reducibility. Conley and Miller proved that every basis for the countable Borel equivalence relations strictly above E_0 under measure reducibility is uncountable and asked whether there is a "measure successor of E_0"—i.e. a countable Borel equivalence relation E such that E is not measure reducible to E_0 and any F which is measure reducible to E is either equivalent to E or measure reducible to E_0. In an ongoing work with Patrick Lutz, we have isolated a combinatorial condition on a Borel group action (a strong form of expansion that we call "lossless expansion" after a similar property which is studied in computer science and finite combinatorics) which implies that the associated orbit equivalence relation is a measure successor of E_0. We have also found several examples of group actions which are plausible candidates for satisfying this condition. In this talk, I will explain the context for Conley and Miller's question, the condition that we have isolated and discuss some of the candidate examples we have identified.

All of this is joint work with Patrick Lutz.

Combinatorics of Very Large Cardinals

Speaker: 

Julian Eshkol

Institution: 

UC Irvine

Time: 

Monday, March 11, 2024 - 4:00pm to 5:20pm

Host: 

Location: 

RH 340 R

At and above the level of measurability, large cardinal notions are typically characterized by the existence of certain elementary embeddings of the universe into an inner model. We may contrast this with smaller large cardinal notions, whose characterizations tend to be strictly combinatorial. In this series of talks, we survey results from Magidor's thesis, in which he shows that the large notion of supercompactness can also be viewed combinatorially, and in this light supercompactness is seen to be a natural strengthening of ineffability. We will also survey modern results which show how these strong combinatorial principles can be forced to hold at small successor cardinals.

The combinatorics of Large Cardinals

Speaker: 

Julian Eshkol

Institution: 

UC Irvine

Time: 

Monday, March 4, 2024 - 4:00pm to 5:50pm

Host: 

Location: 

RH 340N

At and above the level of measurability, large cardinal notions are typically characterized by the existence of certain elementary embeddings of the universe into an inner model. We may contrast this with smaller large cardinal notions, whose characterizations tend to be strictly combinatorial. In this series of talks, we survey results from Magidor's thesis, in which he shows that the large notion of supercompactness can also be viewed combinatorially, and in this light supercompactness is seen to be a natural strengthening of ineffability. We will also survey modern results which show how these strong combinatorial principles can be forced to hold at small successor cardinals.

Ineffability

Speaker: 

Julian Eshkol

Institution: 

UC Irvine

Time: 

Monday, February 26, 2024 - 4:00pm to 5:20pm

Host: 

Location: 

RH 340

Ineffability

This is the first of series of seminars that surveys the results of ineffability and its use in forcing extensions.

The first talk will be about the results in Magidor's Thesis where the fundamental notions were introduced. 

 

Algorithmic Randomness

Speaker: 

Michael Hehman

Institution: 

UC Irvine

Time: 

Tuesday, February 20, 2024 - 11:00am to 12:30pm

Host: 

Location: 

440R

NOTE: Tuesday meeting

This is the last lecture in an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.

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