Category Archives: Seminars

Will Brian: Autohomeomorphisms of ω∗ : the quotient relation

Place: Fields Institute (Room 210)

Date: February 16, 2018 (13:30-15:00)

Speaker: Will Brian

Title: Autohomeomorphisms of ω∗ : the quotient relation

Abstract: Given two autohomeomorphisms f and g of N*, we say that f is a quotient of g when there is a continuous surjection Q from N* to N* such that Qg = fQ. In other words, f is a quotient of g if it is the “continuous image” of g, in the appropriate sense.

I have been investigating this relation, and will present some of the results of that investigation in my talk. For example, under CH: there are many universal autohomeomorphisms (an autohomeomorphism is universal if everything else is a quotient of it); the quotient relation has uncountable chains and antichains; there is an exact description of the quotients of a given trivial map. Under OCA+MA the picture is still murky: for example, there is a jointly universal pair of autohomeomorphisms (meaning everything else is a quotient of one or the other), but I do not know if there is a single universal automorphism. I will sketch some of these results and include several open questions.

Rick Statman: Completeness of BCD for an operational semantics; forcing for proof theorists

Mathematical logic seminar – Feb 13 2018
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Rick Statman
Department of Mathematical Sciences
CMU

Title: Completeness of BCD for an operational semantics; forcing for proof theorists

Abstract:

Intersection types provide a type discipline for untyped λ-calculus. The formal theory for assigning intersection types to lambda terms is BCD (Barendregt, Coppo, and Dezani). We show that BCD is complete for a natural operational semantics. The proof uses a primitive forcing construction based on Beth models (similar to Kripke models).

Zach Norwood: Coding along trees and remarkable cardinals

Time: Mon, 02/12/2018 – 4:00pm – 5:30pm
Location: RH 440R

Speaker: Zach Norwood (UCLA)

Title: Coding along trees and remarkable cardinals

Abstract. A major project in set theory aims to explore the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler’s proof does not resemble Kunen’s, however, using almost-disjoint coding instead of Kunen’s innovative method of coding along branchless trees. We show how to reconcile these two proofs, giving a new proof of Schindler’s theorem that generalizes Kunen’s methods and suggests further investigation of non-thin trees.

David J. Fernández Bretón: Models of set theory with union ultrafilters and small covering of meagre

Thursday, February 15, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Models of set theory with union ultrafilters and small covering of meagre

Abstract:

Union ultrafilters are ultrafilters that arise naturally from Hindman’s finite unions theorem, in much the same way that selective ultrafilters arise from Ramsey’s theorem, and they are very important objects from the perspective of algebra in the Cech–Stone compactification. The existence of union ultrafilters is known to be independent from the ZFC axioms (due to Hindman and Blass–Hindman), and is known to follow from a number of set-theoretic hypothesis, of which the weakest one is that the covering of meagre equals the continuum (this is due to Eisworth). I will show that such hypothesis is not a necessary condition, by exhibiting a number of different models of ZFC that have a covering of meagre strictly less than the continuum, while at the same time satisfying the existence of union ultrafilters.

Wang Wei: Combinatorics and Probability in First and Second Order Arithmetic

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 14 February 2018, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Wang Wei

Title: Combinatorics and Probability in First and Second Order Arithmetic

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:
Recent years see emergence of connections between the reverse mathematics
of Ramsey theory and computable measure theory or algorithmic randomness.
Here we consider two simple propositions in measure theory which have
interesting connections to the reverse mathematics of Ramsey theory. The
first is that every set X in Cantor space of positive Lebesgue measure is
non-empty. If X is assumed to be effectively closed then this is the
well-known axiom WWKL-0. However, if X is allowed to be a
little wilder and the proposition is twisted a bit, then it could help in
understanding the first order theory of some Ramseyan theorems. The second
is that every set X in Cantor space of positive measure has a perfect
subset. This proposition is somehow related to a tree version of Ramsey's
theorem. But unlike the first one, it is not familiar to people either in
algorithmic randomness or reverse mathematics.

Frank Tall: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Place: Fields Institute (Room 210)

Date: February 9, 2018 (13:30-15:00)

Speaker: Frank Tall

Title: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Abstract: I will not assume knowledge from my previous talks on this subject. We define co-K-analytic spaces and provide evidence that this is the “correct generalization” of ‘co-analytic’ to non-metrizable spaces. As before, we view the classic work of Rogers and Jayne on analytic sets through the lens of

Arhangel’skii’s work on generalized metric spaces, while we investigate the question of whether definable Menger spaces are sigma-compact.

David Belanger: Randomness versus induction

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 07 February 2018, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: David Belanger

Title: Randomness versus induction

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

We look at some recent work towards finding the axiomatic strength of the
statement: There is a Martin-Loef random set of natural numbers.

Stevo Todorcevic: P-ideal dichotomy and versions of Souslin Hypothesis, continued

Place: Fields Institute (Room 210)

Date: February 2, 2018 (13:30-15:00)

Speaker: Stevo Todorcevic

Title: P-ideal dichotomy and versions of Souslin Hypothesis, continued

Abstract: This is a joint work with B. kuzeljevic. This talk will be about the relationship of PID with various forms of SH such as, for example, the statement that all Aronszajn trees are Q-embeddable.

Two talks: Aurichi and de Rancourt

The seminar meets on Wednesday February 7th at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Time permitting, we will have two talks.

  1. Leandro Aurichi — Some problems with products of Lindelöf spaces
    Some problems about the preservation of the Lindelöf property on
    products will be discussed.
  2. Noé de Rancourt — Ramsey theory with and without pigeonhole principle
    I will present an abstract infinite-dimensional Ramsey principle that doesn’t need any pigeonhole principle, and then I will compare the cases where the pigeonhole principle holds and the cases where it doesn’t, in a metamathematical way.

 

Jing Zhang: Rado’s Conjecture and its Baire Version II

Mathematical logic seminar – Jan 30 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     Rado’s Conjecture and its Baire Version II

Abstract:

Rado’s Conjecture is a reflection/compactness principle formulated by Todorčević, who also showed its consistency relative to the existence of strongly compact cardinals. One of its equivalent forms asserts that any nonspecial tree of height ω1 has a nonspecial subtree of size less or equal to ℵ1. Although it is incompatible with Martin’s Axiom, Rado’s Conjecture turns out to imply a lot of consequences of forcing axioms, for example Strong Chang’s Conjecture, failure of square principles, the semi-stationary reflection principle, the Singular Cardinal Hypothesis etcetera. In fact, almost all known consequences of Rado’s Conjecture are consequences of a weaker statement, the Baire version of it which asserts any Baire tree of height ω1 has a nonspecial subtree of size less or equal to ℵ1.

We will show that in the forcing extension by countable support iteration of Sacks forcing of strongly compact length, the Baire version of Rado’s Conjecture holds. Using a classical Mitchell style model, we show Rado’s conjecture along with not-CH does not imply ω2 has the super tree property, answering a question by Torres-Pérez and Wu. We will also see that in general the Baire version of Rado’s Conjecture does not imply Rado’s Conjecture.