Archives of: Carnegie Mellon Logic Seminar

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.

Jing Zhang: A Ramsey theorem for (repeated) sums

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

Room: Wean Hall 8220

Speaker: Jing Zhang
Department of Mathematical Sciences
CMU

Title: A Ramsey theorem for (repeated) sums

Abstract:

The motivation is the question: for any finite coloring f: R -> r
does there exist an infinite X such that X + X is monochromatic under f?
Hindman, Leader and Strauss showed the answer is negative if CH holds.
Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky showed the
positive answer is consistent relative to the existence of a certain large
cardinal. I will demonstrate how to eliminate the use of large cardinals.
Other variations of the statement will also be discussed, including some
ZFC results.

Jing Zhang: A Ramsey theorem for (repeated) sums

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

Room:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     A Ramsey theorem for (repeated) sums

Abstract:

The motivation is the question: for any finite coloring f: R -> r , does there exist an infinite X such that X + X is monochromatic under f? Hindman, Leader and Strauss showed the answer is negative if CH holds. Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky showed the positive answer is consistent relative to the existence of a certain large cardinal. I will demonstrate how to eliminate the use of large cardinals. Other variations of the statement will also be discussed, including some ZFC results.

Anush Tserunyan: Hyperfinite subequivalence relations of treed equivalence relations

Mathematical logic seminar – Oct 16 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Anush Tserunyan
Department of Mathematics
University of Illinois at Urbana-Champaign

Title:     Hyperfinite subequivalence relations of treed equivalence relations

Abstract:

A large part of measured group theory studies structural properties of countable groups that hold “on average”. This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on probability spaces. In this vein, the cyclic (more generally, amenable) groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed equivalence relation, deriving some of analogues of structural properties of cyclic subgroups of a free group. In particular, just like any cyclic subgroup is contained in a unique maximal one, we show that any hyperfinite subequivalence relation is contained in a unique maximal one.

Clinton Conley: Realizing abstract systems of congruence II

Mathematical logic seminar – Oct 9 2018
Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Clinton Conley
Department of Mathematical Sciences
CMU

Title: Realizing abstract systems of congruence II

Abstract:

An abstract system of congruence (ASC) is simply an equivalence relation
on the power set of a finite set F satisfying some nondegeneracy
conditions. Given such an ASC and an action of a group G on a set X, a
realization of the ASC is a partition of X into pieces indexed by F such
that whenever two subsets A, B are asc-equivalent, the corresponding
subsets XA and XB of X can be translated to one another in the action.
Familiar notions like paradoxical decompositions can be easily formalized
and refined by the ASC language. Wagon, upon isolating this notion,
characterized those ASCs which can be realized by rotations of the sphere.
He asks whether there is an analogous characterization for realizing ASCs
using partitions with the property of Baire. We provide such a
characterization. This is joint work with Andrew Marks and Spencer Unger.

Nigel Pynn-Coates: Asymptotic valued differential fields and differential-henselianity

Seminar will meet at 4:30pm Monday next week.

Mathematical logic seminar – Oct 1 2018 – NOTE UNUSUAL DAY AND TIME!
Time: 4:30pm – 5:30 pm

Room: Wean Hall 8220

Speaker: Nigel Pynn-Coates
Department of Mathematics
UIUC

Title: Asymptotic valued differential fields and differential-henselianity

Abstract:

It is well known that henselianity plays a fundamental role in the algebra and model theory of valued fields. The notion of differential-henselianity, introduced by Scanlon and developed in further generality by Aschenbrenner, van den Dries, and van der Hoeven, is a natural generalization to the setting of valued differential fields. I will present three related theorems justifying the position that differential-henselianity plays a similarly fundamental role in the algebra of asymptotic valued differential fields, a class arising naturally from the study of Hardy fields and transseries, and that have potential applications to the model theory of such fields.

Clinton Conley: Realizing abstract systems of congruence

Mathematical logic seminar – Sep 25 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Realizing abstract systems of congruence

Abstract:

An abstract system of congruence (ASC) is simply an equivalence relation on the power set of a finite set F satisfying some nondegeneracy conditions. Given such an ASC and an action of a group G on a set X, a realization of the ASC is a partition of X into pieces indexed by F such that whenever two subsets A, B are asc-equivalent, the corresponding subsets XA and XB of X can be translated to one another in the action. Familiar notions like paradoxical decompositions can be easily formalized and refined by the ASC language. Wagon, upon isolating this notion, characterized those ASCs which can be realized by rotations of the sphere. He asks whether there is an analogous characterization for realizing ASCs using partitions with the property of Baire. We provide such a characterization. This is joint work with Andrew Marks and Spencer Unger.

Jing Zhang: How to get the brightest rainbows from the darkest colorings

Mathematical logic seminar – Sep 18 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     How to get the brightest rainbows from the darkest colorings

Abstract:

We will discuss the rainbow (sometimes called polychromatic) variation of the Ramsey theorem on uncountable cardinals. It roughly says if a given coloring on n-tuples satisfies that each color is not used too many times, we can always find a rainbow subset, that is a set in which no two n-tuples from the set get the same color. We will use problems in different settings (inaccessible cardinals, successors of singular cardinals, small uncountable cardinals etc) to demonstrate that the rainbow variation is a “strict combinatorial weakening” of Ramsey theory.