Archives of: Toronto Set Theory Seminar

Lionel Nguyen Van The: Structural Ramsey theory and topological dynamics for automorphism groups of homogeneous structures.

18/October/2013, 13:30–15:00
Fields institute, Room 210

Speaker:  Lionel Nguyen Van The

Title:  Structural Ramsey theory and topological dynamics for automorphism groups of homogeneous structures.

Abstract: In 2005, Kechris, Pestov, and Todorcevic established a striking connection between structural Ramsey theory and the topological dynamics certain automorphism groups. The purpose of this talk will be to present this connection, together with recent related results.

Eduardo Calderon: Asymptotic models and plegma families

11/October/2013, 13:30–15:00
Fields institute, Room 210

Speaker:  Eduardo Calderon (UNAM)

Title:  Asymptotic models and plegma families

Abstract: We will discuss one of the usual ways in which Ramsey’s theorem is applied to the study of Banach space geometry and then, by means of techniques closely following ones first developed by S. Argyros, V. Kanellopoulos, K. Tyros, we will introduce the concept of an asymptotic model of higher order of a Banach space and establish a relationship between these and higher order spreading models that extends their result of the impossibility of always finding a finite chain of spreading models reaching an $l_p$ space to the context of weakly generated asymptotic models.

David Fernandez: Strongly Productive Ultrafilters

04/October/2013, 13:30–15:00
Fields institute, Room 210

Speaker:  David Fernandez (York University)

Title:  Strongly Productive Ultrafilters

Abstract: The concept of a Strongly Productive Ultrafilter on a semigroup (known as a “strongly summable ultrafilter” when the semigroup is additively denoted) constitute an important concept ever since Hindman defined it, while trying to prove the theorem that now bears his name. In a 1998 paper of Hindman, Protasov and Strauss, it shown that strongly productive ultrafilters on abelian groups are always idempotent, but no further generalization of this fact had been made afterwards. In this talk I will show (at least the main ideas, anyway) the proof that this result holds on a large class of semigroups, which includes all solvable groups and the free semigroup, among others. After that, I’ll discuss a special class of strongly productive ultrafilters on the free semigroup (dubbed “very strongly productive ultrafilters” by N. Hindman and L. Jones), and show that they have the “trivial products property”. This means that (thinking of the free semigroup S as a subset of the free group G) if p is a very strongly productive ultrafilter on S, and q,r are nonprincipal ultrafilters on G such that $qr=p$, then there must be an element x of G such that $q=px$ and $r=x^{-1}p$. This answers a question of Hindman and Jones. Joint work with Martino Lupini.

Stevo Todorcevic: A construction scheme on $\omega_{1}$

27/September/2013, 13:30–15:00
Fields institute, Room 210

Speaker:  Stevo Todorcevic (University of Toronto)

Title:   A construction scheme on $\omega_{1}$

Abstract:  We describe a simple and general construction scheme for describing mathematical structures on domain $\omega_{1}$. Natural requirements on this scheme will reduce the nonseparable structural properties of the resulting mathematical object to some finite-dimensional problems that are easy to state and frequently also easy to solve. The construction scheme is in fact quite easy to use and we illustrate this by some application mainly towards compact convex spaces and normed spaces.

 

Rodrigo Hernandez: Countable dense homogeneous spaces

20/September/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Rodrigo Hernandez (York)

Title:  Countable dense homogeneous spaces

Abstract: A separable space X is countable dense homogeneous (CDH) if every time D and E are countable dense subsets of X, there exists a homeomorphism $h:X\to X$ such that $h[D]=E$. The first examples of CDH spaces were Polish spaces. So the natural open question was whether there exists a CDH metrizable space that is not Polish. By a characterization result by Hrusak and Zamora-Aviles, such a space must be non Borel. In this talk, we will focus on recent progress in this direction. In fact, we only know about two types of CDH non-Borel spaces: non-meager P-filters (with the Cantor set topology) and $\lambda$-sets. Moreover, by arguments similar to those used for the CDH $\lambda$-set, it has also been possible to construct a compact CDH space of uncountable weight.

Daniel Soukup: Monochromatic partitions of edge-colored infinite graphs

13/September/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Daniel Soukup

Title:  Monochromatic partitions of edge-colored infinite graphs

Abstract: Our goal is to find well behaved partitions of edge-colored infinite graphs following a long standing trend in finite combinatorics started by several authors including P. Erdős and R. Rado; in particular, we are interested in partitioning the vertices of complete or nearly complete graphs into monochromatic paths and powers of paths. One of our main results is that for every 2-edge-coloring of the complete graph on $\omega_1$ one can partition the vertices into two monochromatic paths of different colors. Our plan for the talk is to review some results from the literature (both on finite and infinite), sketch some of our results and the ideas involved and finally present the great deal of open problems we facing at the moment. This is a joint work with M. Elekes, L. Soukup and Z. Szentmiklóssy.

 

Connor Meehan: Infinite Games and Analytic Sets

28/Aug/2013, 15:00–16:00
BA6180

Speaker: Connor Meehan

Title: Infinite Games and Analytic Sets

Abstract: In the context of set theory, infinite games have been studied since the mid-20th century and have created an interesting web of connections, such as with measurable cardinals. Upon specifying a subset A of sequences of natural numbers, an infinite game G(A) involves two players alternately choosing natural numbers, with player 1 winning in the event that the resulting sequence x is in A. We will give proofs of Gale and Stewart’s classic results that any open subset A of Baire space leads to the game G(A) being determined (i.e. one of the players has a winning strategy) and that the Axiom of Determinacy (stating that all games are determined) contradicts the Axiom of Choice. With the former we recreate Blackwell’s groundbreaking proof of a classical result about co-analytic sets. A family U of subsets of Baire space is said to have the reduction property if for any B and C in U, there are respective disjoint subsets B* of B and C* of C in U with the same union as B and C; Blackwell proves that the co-analytic sets have the reduction property. Blackwell’s new proof technique with this old result revitalized this area of descriptive set theory and began the development for a slew of new results.

Jack Wright: Nonstandard Analysis and an Application to Combinatorial Number Theory

23/Aug/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Jack Wright

Title: Nonstandard Analysis and an Application to Combinatorial Number Theory

Abstract: Since nonstandard analysis was first formalized in the 60′s it has given mathematicians a framework in which to do rigorous analysis with infinitesimals rather than epsilons and deltas. More importantly, it has also allowed for the application of powerful techniques from logic and model theory to analysis (and other areas of mathematics). This brief presentation will outline some of those tools and discuss one particular application of them.
I will briefly state the key techniques: the transfer principle, the internal definition principle, and the overflow principle. I will then give an indication of the usefulness of these techniques by showing how they have been used to garner some technical results that might be able to help solve the Erd\H{o}s’ famous Conjecture on Arithmetic progressions.

Miguel Angel Mota: Instantiations of Club Guessing. Part I

09/August/2013, 13:30-15:00
Fields Institute, Room 210

Speaker:  Miguel Angel Mota

Title: Instantiations of Club Guessing. Part I

Abstract:

We build a model where Weak Club Guessing fails, $\mho$ holds and the continuum is larger than the second uncountable cardinal. The dual of this result  will be discussed in a future talk.

 

Carlos Uzcategui: Uniform Ramsey theoretic properties

02/August/2013, 13:30-15:00
Fields Institute, Room 210

Speaker:  Carlos Uzcategui

Title: Uniform Ramsey theoretic properties

Abstract:

The classical Ramsey theorem holds uniformly in the following sense. There is a Borel map that for a given coloring of pairs and an infinite set A, it selects an infinite homogeneous subset of A. This fact sugests that the notions of a selective, Frechet, p+ and q+ ideal could also hold uniformly. We will discuss about some of those uniform Ramsey theoretic properties.