Archives of: Carnegie Mellon Logic Seminar

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.

Anton Bernshteyn: From finite combinatorics to descriptive set theory and back

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

Room:     Wean Hall 8220

Speaker:         Anton Bernshteyn
Department of Mathematical Sciences
CMU

Title:     From finite combinatorics to descriptive set theory and back

Abstract:

Many results in finite combinatorics can be extended to infinite structures via compactness – but this transfer is powered by the Axiom of Choice and leads, in general, to highly “pathological” objects. It is natural to ask, which combinatorial constructions can be performed in a “well-behaved” fashion, say, in a Borel or measurable way? This question is addressed in a young branch of descriptive set theory called descriptive combinatorics. We will discuss a class of coloring problems with the requirement that the desired coloring be Baire measurable (i.e., “topologically well-behaved”). The central result of this talk is that the existence of a Baire measurable coloring is equivalent to a purely combinatorial statement, analogs of which have for a long time been studied in finite graph theory with no relation to descriptive set theory.

David Fernández-Bretón: Variations and analogs of Hindman’s theorem

Mathematical logic seminar – May 22 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         David Fernández-Bretón
Department of Mathematics
University of Michigan

Title:     Variations and analogs of Hindman’s theorem

Abstract:

Hindman’s theorem is a Ramsey-theoretic result asserting that, whenever one colours the set of natural numbers with finitely many colours, there will be an infinite set such that all numbers that can be obtained by adding finitely many elements from the set (no repetitions allowed) have the same colour. I will explore generalizations and extensions of this theorem: replacing “natural numbers” with “abelian group” and varying the number of colours, as well as the size of the desired monochromatic set, yields a plethora of very interesting results.

François Le Maître: Ample generics and full groups

Mathematical logic seminar – May 15 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         François Le Maître
Institut de Mathématiques de Jussieu-PRG
Université Paris Diderot

Title:     Ample generics and full groups

Abstract:

In this talk, I will explain why the full group of the action of the rationals on the real line is an example of a connected topologically simple Polish group with ample generics, providing a natural answer to a question of A. Kechris and C. Rosendal. If time permits, I will discuss the following open question: is there a Polish group with ample generics which is not quasi non-archimedean ? The talk is based on a joint work with Adriane Kaïchouh.

Joseph Zielinski: Roelcke precompact sets in Polish groups II

Mathematical logic seminar – May 8 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Joseph Zielinski
Department of Mathematical Sciences
CMU

Title:     Roelcke precompact sets in Polish groups II

Abstract:

In these talks we first recall the uniform structures associated to a topological group. We then present J. Roe’s notion of a coarse space, and consider compatible coarse structures on groups with emphasis on the ‘left-coarse structure’ of a topological group introduced by C. Rosendal. Associated to this notion are the ‘locally bounded Polish groups’: those for which the left-coarse structure is the bounded coarse structure of some compatible, left-invariant metric.

Next, we introduce the Roelcke precompact subsets of a Polish group, which admit equivalent natural definitions both in terms of the lower uniformity on the group and as a subideal of the bounded sets in the left-coarse structure. Through this we define the ‘locally Roelcke precompact Polish groups’ — a subfamily of the locally bounded Polish groups — and present various examples, applications, and several characterizations of these groups.

Joseph Zielinski: Roelcke precompact sets in Polish groups

Mathematical logic seminar – May 1 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Joseph Zielinski
Department of Mathematical Sciences
CMU

Title:     Roelcke precompact sets in Polish groups

Abstract:

In these talks we first recall the uniform structures associated to a topological group. We then present J. Roe’s notion of a coarse space, and consider compatible coarse structures on groups with emphasis on the ‘left-coarse structure’ of a topological group introduced by C. Rosendal. Associated to this notion are the ‘locally bounded Polish groups’: those for which the left-coarse structure is the bounded coarse structure of some compatible, left-invariant metric.

Next, we introduce the Roelcke precompact subsets of a Polish group, which admit equivalent natural definitions both in terms of the lower uniformity on the group and as a subideal of the bounded sets in the left-coarse structure. Through this we define the ‘locally Roelcke precompact Polish groups’ — a subfamily of the locally bounded Polish groups — and present various examples, applications, and several characterizations of these groups.

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)