Archives of: Carnegie Mellon Logic Seminar

Jing Zhang: Poset dimension and singular cardinals

Mathematical logic seminar – Mar 19 2019
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
CMU

Title:     Poset dimension and singular cardinals

Abstract:

The dimension of a poset (P, ≤P) is defined as the least cardinal λ such that there exists a λ-sized collection of linear extensions of P realizing P, that is to say a ≤P b if and only a ≤ b in any linear extension in the collection. We will focus on the poset Pα(κ), that is the poset of subsets of κ of size less than α partially ordered by inclusion, and determine completely the dimension of such posets under GCH. Then we will mention a few consistency results when GCH fails. In particular, we point out the connection between the dimension of the poset Pα (2κ) and the density of 2κ under the <α-box product topology, and show it is consistent that they are different.

Hector Alonzo Barriga-Acosta: Some combinatorics on the normality of the countable box product of the convergent sequence

Mathematical logic seminar – Mar 5 2019
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Hector Alonzo Barriga Acosta
Universidad Nacional Autónoma de México

Title: Some combinatorics on the normality of the countable box product
of the convergent sequence

Abstract:

The normality of □ (ω + 1)^ω is a question raised in the 40’s (it is
known
that consistently this space is normal). Through the years many different
tecniques have been developed, but non of them have solved the question in
ZFC. We’ll take a look to a combinatorial point of view, given by Judy
Roitman, of this problem.

Raphaël Carroy: Strongly surjective linear orders

Mathematical logic seminar – Feb 26 2019
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Raphaël Carroy (KGRC, Vienna)

Title: Strongly surjective linear orders

Abstract:

When a linear order has an increasing surjection onto each of its
suborders we say that it is strongly surjective. We prove that countable
strongly surjective orders are the union of an analytic and a coanalytic
set, and that moreover they are complete for this class of sets. If time
allows it, I’ll also discuss the existence of uncountable strongly
surjective orders. This is a joint work with Riccardo Camerlo and Alberto
Marcone.

Assaf Shani: Borel reducibility and symmetric models

Mathematical logic seminar – Feb 19 2019
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Assaf Shani
Department of Mathematics
UCLA

Title:     Borel reducibility and symmetric models

Mathematical logic seminar – Feb 19 2019
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Assaf Shani
Department of Mathematics
UCLA

Title:     Borel reducibility and symmetric models

Abstract:

We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of S∞, and the study of symmetric models of set theory without choice, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence relation ≅*ω+1,0 is strictly below ≅*ω+1 in Borel reducibility. By results of Hjorth-Kechris-Louveau, ≅*ω+1 corresponds to Σ0ω+1 actions of S∞, while ≅*ω+1,0 corresponds to Σ0ω+1 actions of “well behaved” closed subgroups of S∞, for example abelian groups. For these proofs we analyze the models Mn, n<ω, developed by Monro (1973), and extend his construction past ω, through all countable ordinals. This answers a question of Karagila (2016).

Clinton Conley: Ode on a one-ended subforest

Mathematical logic seminar – Feb 12 2019
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Clinton Conley
Department of Mathematical Sciences
CMU

Title: Ode on a one-ended subforest

Abstract:

Many arguments in (finite) graph theory follow this pattern: postpone some
onerous task until the last possible moment, after you’ve arranged things
to make the task as easy as possible. In the descriptive set-theoretic
milieu, the one-ended forest provides a portal to a procrastinator’s
wonderland in which the onerous task instead wanders off to infinity. We
discuss a few instances of this phenomenon and some applications to
coloring and treeing graphs.

James Cummings: More on compactness

Mathematical logic seminar – Feb 5 2019
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     More on compactness

Abstract:

I’ll discuss compactness phenomena at regular cardinals, particularly successors of singular cardinals. In particular I’ll explain why ℵω+1 is in some respects completely different from ℵω2+1.

(This talk is related to the talks about compactness that I gave in the Fall semester, but is logically independent)

Anton Bernshteyn: Independent Sets in Algebraic Hypergraphs

Mathematical logic seminar – Jan 29 2019
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Anton Bernshteyn
Department of Mathematical Sciences
CMU

Title:     Independent Sets in Algebraic Hypergraphs

Abstract:

An active avenue of research in modern combinatorics is extending classical extremal results to the so-called sparse random setting. The basic hope is that certain properties that a given “dense” structure is known to enjoy should be inherited by a randomly chosen “sparse” substructure. One of the powerful general approaches for proving such results is the hypergraph containers method, developed independently by Balogh, Morris, and Samotij and Saxton and Thomason. Another major line of study is establishing combinatorial results for algebraic or, more generally, definable structures. In this talk, we will combine the two directions and address the following problem: Given a “dense” algebraically defined hypergraph, when can we show that the subhypergraph induced by a generic low-dimensional algebraic set of vertices is also fairly “dense”? This is joint work with Michelle Delcourt (University of Waterloo) and Anush Tserunyan (University of Illinois at Urbana–Champaign).

Aristotelis Panagiotopoulos: Higher dimensional obstructions for star-reductions

The last meeting before the break. Happy holidays.
Seminar will resume in the New Year

Mathematical logic seminar – Dec 11 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Aristotelis Panagiotopoulos
Department of Mathematics
Caltech

Title:     Higher dimensional obstructions for star-reductions

Abstract:

In this talk we will consider *-reductions between orbit equivalence relations. These are Baire measurable reductions which preserve generic notions, i.e., preimages of comeager sets are comeager. In short, *-reductions are weaker than Borel reductions in the sense of definability, but as we will see, they are much more sensitive to the dynamics of the orbit equivalence relations in question.

Based on a past joint work with M. Lupini we will introduce a notion of dimension for Polish G-spaces. This dimension is always 0 whenever the group G admits a complete and left invariant metric, but in principle, it can take any value n within 0,1,….∞ For each such n we will produce a free action of S∞ which is generically n-dimensional and we will deduce that the associated orbit equivalence relations are pairwise incomparable with respect to *-reductions.

This is in joint work with A. Kruckman.

James Cummings: Regular cardinals and compactness

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

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Regular cardinals and compactness

Abstract:

This talk is a sequel of sorts to last week’s talk on singular compactness, but is completely independent of it. I will discuss phenomena of compactness and incompactness for regular cardinals, with particular emphasis on stationary reflection and problems about transversals.

James Cummings: Shelah’s singular compactness theorem

Mathematical logic seminar – Nov 27 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Shelah’s singular compactness theorem

Abstract:

Shelah’s singular compactness theorem is a general result showing that a singular cardinal λ has properties reminiscent of those enjoyed by large cardinals: for example

If G is an abelian group of size λ and every subgroup of G with size less than λ is free, then G is free.

If X is a family of size λ of countable sets, and every subfamily of size less than λ has a transversal, then X has a transversal.

I will prove a version of the singular compactness theorem, and discuss some complementary consistency results for λ regular.