BEST 2016 slides

The 23rd BEST conference was held June 15–16 in San Diego, CA.

Shehzad Ahmed – Jonsson cardinals and pcf theory
Liljana Babinkostova – A weakening of the closure operator
Kyle Beserra – On the conjugacy problem for automorphisms of countable regular trees
Erin Carmody – Killing them softly
William Chan – Every analytic equivalence relation with all Borel classes is Borel somewhere
John Clemens – Relative primeness of equivalence relations
Paul Corazza – The axiom of infinity, quantum field theory, and large cardinals
Cody Dance – Indiscernibles for $L[T_2,x]$
Natasha Dobrinen – Ramsey spaces coding universal triangle-free graphs and applications to Ramsey degrees
Paul Ellis – A Borel amalgamation property
Monroe Eskew – Rigid ideals
Daniel Hathaway – Disjoint Borel functions
Jared Holshouser – Partition properties for non-ordinal sets under the axiom of determinacy
Paul McKenney – Automorphisms of $\mathcal P(\lambda)/\mathcal I_\kappa
Kaethe Minden – Subcomplete forcing and trees
Daniel Soukup – Orientations of graphs with uncountable chromatic number
Simon Thomas – The isomorphism and bi-embeddability relations for finitely generated groups
Douglas Ulrich – A new notion of cardinality for countable first order theories
Kameryn Williams – Minimal models of Kelley-Morse set theory
Martin Zeman – Master conditions from huge embeddings

Ilijas Farah: A consistent failure Glimm’s theorem for nonseparable C*-algebras

Place: Fields Institute (Room 210)

Date: June 24th, 2016 (13:30-15:00)

Speaker: Ilijas Farah

Title: A consistent failure Glimm’s theorem for nonseparable C*-algebras

Abstract: A remarkable 1960 result of J. Glimm provides a sharp dichotomy for
the representation theory of separable C*-algebras. One of its
consequences is that a separable C*-algebra either has a unique
(up to the unitary equivalence) irreducible representation or continuum
many inequivalent irreducible representations. Using some ideas of
Akemann and Weaver, I’ll prove that this conclusion is consistently false
for nonseparable C*-algebras. Many open problems remain.

David Fernández Bretón: Ultrafilters on the rationals generated by perfect sets

Place: Fields Institute (Room 210)

Date: June 17th, 2016 (13:30-15:00)

Speaker: David Fernandez Bretón

Title: Ultrafilters on the rationals generated by perfect sets

Abstract:  In a 1992 paper, van Douwen defined what he calls a
“gruff ultrafilter”: an ultrafilter on the rational numbers which
is generated by perfect (this is, closed and crowded) sets; and
asked whether these ultrafilters exist, providing in the same
paper a proof that they do if cov(M)=c. The question of whether
the existence of gruff ultrafilters can be proved in ZFC alone
remains open, but further progress has been made in the
way of consistently positive answers. In this talk I will
present a proof that gruff ultrafilters exist in the Random
model, as well as in any model satisfying d=c. Joint work
with Michael Hrusak.

Dana Bartosova: Algebra in the Samuel compactification

Place: Fields Institute (Room 210)

Date: June 10, 2016 (13:30-15:00)

Speaker: Dana Bartosova

Title: Algebra in the Samuel compactification

Abstract: The Samuel compactification, or the greatest ambit, is an important compactification of a topological group for its dynamics. In the case of discrete groups, the Samuel compactification coincides with the Cech-Stone compactification and its algebra and combinatorics have been extensively studied. We remind the Samuel compactification for automorphism groups in the ultrafilter language and point out some differences and similarities with the discrete case. We will then apply algebra and combinatorics to answer a problem of Ellis for the group of permutations of the integers. This is a joint work in progress with Andy Zucker (Carnegie Mellon University).

​Nadav Meir: ​​Infinite products of ultrahomogeneous structures

BGU Seminar in Logic, Set Theory and Topology.

Tomorrow we will continue our seminar in Logic, Set Theory and Topology.

Time: Tuesday, June 7th, 12:30-13:45.

Place: Seminar room -101, Math building 58.

Speaker: Nadav Meir (BGU).

Title: Infinite products of ultrahomogeneous structures

Abstract:
We will define the “lexicographic product” of two structures and show that if both structures admit quantifier elimination, then so does their product. As a corollary we get that nice (model theoretic) properties such as (ultra)homogeneity, stability, NIP and more are preserved under taking products.

It is clear how to iterate the product finitely many times, but we will introduce a new infinite product construction which, while not preserving quantifier elimination, does preserve (ultra)homogeneity. As time allows, we will use this to give a negative answer to the last open question from a paper by A. Hasson, M. Kojman and A. Onshuus who asked “Is there a rigid elementarily indivisible* structure?”

As time allows, we will introduce an approach for using the lexicographic product to generalize a result by Lachlan and Shelah to the following: given a finite relational language L, denote by H(L) the class of countable ultrahomogeneous stable L-structures. For M in H(L), define the rank of M to be the maximum value of CR(p,2) where p is a complete 1-type and CR(p,2) is the Shelah’s complete rank. There is a uniform finite bound on the rank of M, where M ranges over H(L). The result was proven by Lachlan and Shelah for L binary and proven in general by Lachlan using the Classification Theorem for finite simple groups.

* A structure M is said to be elementarily indivisible structure if for every colouring of its universe in two colours, there is a monochromatic elementary substructure N of M such that N is isomorphic to M.

Alessandro Vignati: CH and homeomorphisms of Stone-Cech remainders

Place: Fields Institute (Room 210)

Date: June 3rd, 2016 (13:30-15:00)

Speaker: Alessandro Vignati

Title: CH and homeomorphisms of Stone-Cech remainders

Abstract:

If X is locally compact and Polish, it makes sense to ask how many homeomorphisms does X*, the Stone Cech remainder of X, have. It is known that, if X is 0-dimensional, under the Continuum Hypothesis X* has $2^{2^{\aleph_0}}$ many homeomorphisms (Rudin+Parovicenko). The same is true if $X=[0,1)$ (Yu, Dow-KP Hart), or if X is the disjoint union of countably many compact spaces (Coskey-Farah). But the question remains open for, for example, $X=\mathbb{R}^2$. We prove that for a large class of spaces (including $\mathbb{R}^n$, for all n) CH provides $2^{2^{\aleph_0}}$ many homeomorphisms of X*.

Assaf Rinot: Reduced powers of Souslin trees

BGU Seminar in Logic, Set Theory and Topology.

Time: Tuesday, May 31st, 12:30-13:45.

Place: Seminar room -101, Math building 58.
Speaker: Assaf Rinot (BIU)

Title: Reduced powers of Souslin trees

Abstract:

What is the relationship between a Souslin tree and its reduced powers? and is there any difference between, say, the reduced $\omega$-power and the reduced $\omega_1$-power of the same tree?
In this talk, we shall present tools recently developed to answer these sort of questions. For instance, these tools allow to construct an $\omega_6$-Souslin tree whose reduced $\omega_N$-power is Aronszajn iff $n$ is not a prime number.
This is joint work with Ari Brodsky.

Fulgencio Lopez: Banach spaces from Construction Schemes

Place: Fields Institute (Room 210)

Date: May 27th, 2016 (13:30-15:00)

Speaker: Fulgencio Lopez

Title: Banach spaces from Construction Schemes

Abstract: In 2011 J. Lopez-Abad and S. Todorcevic used forcing to construct a
Banach space with an $\varepsilon$-biorthogonal system that didn’t have $\delta$-biorthogonal systems for every $1\leq\delta<\varepsilon$. We show that there is a Banach space with the same property provided there is a capturing Construction Scheme.

Set theory workshop at UIC, October 20-23, 2016

Set theory workshop

The workshop will be held at the University of Illinois at Chicago on October 20-23. Topic will cover forcing, large cardinals, applications of set theory. We will have three tutorials from leading experts and several talks by younger researchers.

The invited speakers are:
Tutorials:

Talks:

Travel support is available. Requests for such should be directed to Dima Sinapova at sinapova@math.uic.edu. Such requests will be handled on a case-by-case basis within the limits of the budget. Graduate students, young researchers, female mathematicians and members of underrepresented groups are particularly encouraged to apply.

Wiesław Kubiś: Abstract Banach-Mazur game

Tuesday, May 31, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Wiesław Kubiś (Czech Academy of Sciences, KSW University)

Title: Abstract Banach-Mazur game

Abstract:

We will discuss an infinite game in which two players alternately choose some objects (structures) from a given class. The only rule is that at each move the structure chosen by the player should extend the one chosen in the previous move by the opponent. One of the players wins if the limit of the chain of structures resulting from the play is isomorphic to some concrete (fixed in advance) object. We will show some basic results and relevant examples concerning the existence of winning strategies.