## Vahagn Aslanyan: Geometry of strongly minimal sets in differentially closed fields

Mathematical logic seminar – Apr 17 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Vahagn Aslanyan
Department of Mathematical Sciences
CMU

Title:     Geometry of strongly minimal sets in differentially closed fields

Abstract:

I will discuss Zilber’s Trichotomy conjecture and some structures (theories) where it holds (the conjecture in its general form was refuted by Hrushovski). In particular, by a result of Hrushovski and Sokolovic differentially closed fields satisfy Zilber’s trichotomy. However, understanding whether a given definable set is strongly minimal or, given a strongly minimal set, understanding the nature of its geometry is not an easy task. I will show how one can use the Ax-Schanuel theorem for the j-function to deduce strong minimality and geometric triviality of the differential equation of the j-function (I will also explain why it is an important example). This result was first proven by Freitag and Scanlon using the analytic properties of the j-function. My approach is completely abstract, I actually prove that once there is an Ax-Schanuel type statement of a certain form for a differential equation E(x,y) then some fibres of E are strongly minimal and geometrically trivial.

## Garrett Ervin: Decomposing the real line into two everywhere isomorphic

Mathematical logic seminar – Apr 10 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Garrett Ervin
Department of Mathematical Sciences
CMU

Title:     Decomposing the real line into two everywhere isomorphic pieces

Abstract:

A dense linear order is said to be homogeneous if it is isomorphic to its restriction to any of its open intervals. The set of rationals ℚ is an example of a homogeneous order, as is the set of irrationals ℝ ∖ ℚ. In general, if X is a homogeneous suborder of the real line ℝ, then ℝ ∖ X is also homogeneous, and there are many examples where both X and ℝ ∖ X are of size continuum. However, it turns out that a homogeneous X can never be isomorphic to ℝ ∖ X. In fact, if ℝ = A ∪ B is any decomposition of ℝ into two disjoint pieces, there is an open interval I such that A restricted to I is not isomorphic to B restricted to I. We will prove this theorem and discuss some related results.

## Andy Zucker: Maximal equivariant compactifications of categorical metric structures

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

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Maximal equivariant compactifications of categorical metric structures

Abstract:

Any completely regular space embeds into a compact space. But suppose G is a topological group and X is a completely regular G-space. There is a largest G-map αX: X → Y where Y is compact and αX has dense image, but αX need not be an embedding. Recently, Pestov has constructed an example of a topological group G and non-trivial flow X for which αX is the map to a singleton.

In this talk, we consider automorphism groups of categorical metric structures, which include the Urysohn sphere, the unit sphere of the Banach lattice Lp, and the unit sphere of the Hilbert space L2. We show that if G is the group of automorphisms of a categorical metric structure X, then αX is the embedding of X into the space of 1-types over X.

(Joint work with Itai Ben Yaacov)

## Andy Zucker: Maximal equivariant compactifications of categorical metric structures

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

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Maximal equivariant compactifications of categorical metric structures

Abstract:

Any completely regular space embeds into a compact space. But suppose G is a topological group and X is a completely regular G-space. There is a largest G-map αX: X → Y where Y is compact and αX has dense image, but αX need not be an embedding. Recently, Pestov has constructed an example of a topological group G and non-trivial flow X for which αX is the map to a singleton.

In this talk, we consider automorphism groups of categorical metric structures, which include the Urysohn sphere, the unit sphere of the Banach lattice Lp, and the unit sphere of the Hilbert space L2. We show that if G is the group of automorphisms of a categorical metric structure X, then αX is the embedding of X into the space of 1-types over X.

(Joint work with Itai Ben Yaacov)

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

Mathematical logic seminar – Feb 27 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 II

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).

## James Cummings: Some strong chain conditions

Mathematical logic seminar – Feb 20 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Some strong chain conditions

Abstract:

One of the basic facts in forcing is that a finite support iteration of ccc forcing is ccc. This underlies (for example) the consistency proof for Martin’s Axiom. In general an iteration of κ-closed κ+-cc forcing with <κ-support fails to be κ+-cc, and we need strngthened forms of the chain condition. I will discuss some of these strong chain conditions and the corresponding iteration theorems.

## 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).

## 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.

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

Mathematical logic seminar – Jan 23 2018
Time:     3:30pm – 4:30 pmRoom:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     Rado’s Conjecture and its Baire Version

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.

## Clinton Conley: Unfriendly colorings of measure-preserving graphs of finite cost

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

Room:     Wean Hall 8220

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Unfriendly colorings of measure-preserving graphs of finite cost

Abstract:

We show that any measure-preserving Borel graph on a standard probability space with finite average degree admits a Borel unfriendly coloring on a conull set. This generalizes the results for group actions discussed last semester, and is joint work with Omer Tamuz.