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SCIENTIFIC PROGRAMS AND ACTIVITIES |
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| November 24, 2024 |
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Abstracts
Applications of Ramsey theory in topological dynamics We show that the group of linear isometries of the Gurarij space is extremely
amenable and compute the universal minimal flows of the group of affine homeomorphisms
of the Poulsen simplex and the group of homeomorphisms of the Lelek fan. This
is a joint work with Jordi Lopez-Abad and Brice Mbombo, and Aleksandra Kwiatkowska.
Subsymmetric sequences in large Banach spaces
We present a method of constructing Banach spaces of large densities without subsymmetric basic sequences, based on the existence of certain sequences of compact, hereditary and large families of finite sets. We also give an idea of how to construct those families for every cardinal smaller than the first inaccessible cardinal, improving results by S. Argyros and P. Motakis. This is a joint work with J. Lopez-Abad and S. Todorcevic.
Quotients of strongly proper posets, and related topics I will discuss when quotients of strongly proper posets have the $\omega_1$
approximation property. As an application we prove the conjecture of Viale
and Weiss, that $ISP(\omega_2)$ is consistent with arbitrarily large continuum.
This is joint work with John Krueger.
Higher dimensional Ellentuck spaces Abstract TBA.
Alan Dow
Parametrized $\diamondsuit$-principles and canonical models We will review recent results concerning definable versions of the weak diamond.
Mitchell's theorem revisited Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of $\omega_2 \cap \textrm{cof}(\omega_1)$ in the approachability ideal $I[\omega_2]$. In this talk I will give an overview of a new proof of Mitchell's theorem, which is derived from an abstract framework of side condition methods.
Approximate Ramsey properties of finite dimensional normed spaces. We present the following result: For every finite dimensional normed spaces $F$ and $G$, every integer $r$ and every numbers $\theta\geq 1$ and $\varepsilon>0$ there exists a finite dimensional space $H$ containing a linear isometric copy of $G$ and such that every $r$-coloring of the set of linear $\theta$-isometric embeddings $\mathrm{Emb}_\theta(F,H)$ has an $\varepsilon$-monochromatic set of the form $\gamma \circ \mathrm{Emb}(F,G)$, for some $\gamma\in \mathrm{Emb}(G,H)$. We will discuss several applications, as the extremely amenability of the group of linear surjective isometries of the Gurarij space or the fact that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is the Poulsen simplex itself. The proof of the main result is of combinatorial nature, as it uses the dual Ramsey theorem.We will also mention other approximate Ramsey results for $\ell_p^n$'s, this time using the concentration of measure.
Open questions on ultrafilters arising from p, t and model theory The talk will be about a range of open problems, and some related theorems, at the intersection of set theory, model theory, and general topology, mainly around construction of ultrafilters.
Blass--Shelah forcing revisited We force with $\sigma$-centred subforcings of forgetful versions of Blass--Shelah
forcings. The pure parts of the conditions are taken from suitable centred
sets $C$ in a space of sequences of normed subsets of powersets of finite
sets. We call the resulting subforcing ${\mathbb BS}(C)$. We sketch a proof
of the following: A $P$-point ${\mathcal U}$ in the ground model is preserved
by forcing with ${\mathbb BS}(C)$ iff the projection of $C$ to ${\mathcal
P}(\omega)$ is not below ${\mathcal U}$ in the Rudin--Blass order.
Coherent adequate forcing and preserving CH In the last years there has been a second boom of the technique of forcing
with side conditions (see for instance the recent works of Asper\'{o}-Mota,
Krueger and Neeman describing three different perspectives of this technique).
The first boom took place in the 1980s when Todorcevic discovered a method
of forcing in which elementary substructures are included in the conditions
of a forcing poset to ensure that the forcing poset preserves cardinals. More
than twenty years later, Friedman and Mitchell independently took the first
step in generalizing the method from adding small (of size at most the first
uncountable cardinal) generic objects to adding larger objects by defining
forcing posets with finite conditions for adding a club subset on the second
uncountable cardinal. However, neither of these results show how to force
(with side conditions together with another finite set of objects) the existence
of such a large object together with the continuum being small. In this talk
we will discuss new results in this area. This is joint work with John Krueger.
A Lindelof topolotical group with non-Lindelof square This is a joint work with Liuzhen. We generalize Moore's construction for an L space to get an L group. We also prove that its square is not Lindelof. This answers a question of Arhangel'skii. We also apply the method to higher finite powers.
A microscopic approach to Souslin trees constructions We present an approach to construct $\kappa$-Souslin trees that is insensitive to the identity of the cardinal $\kappa$, thereby, allowing to transform constructions from successor of regulars to successor of singulars and to inaccessible. This is obtained by redirecting all constructions through a parametrized proxy principle. The construction is carried as a transfinite sequence of microscopic steps that is indifferent of the "big picture". Indeed, the features of the outcome tree are determined by the parameters of the proxy principle that one starts with. This is joint work with Ari. M. Brodsky.
The topological conjugacy relation of free minimal G-subshifts During this talk I will discuss the descriptive set-theoretic complexity of the topological conjugacy relation for free minimal G-subshifts for various countable groups G. For residually finite countable groups G we will see that there exists a probability measure on the set of free minimal G-subshifts, which is invariant under a natural action of G and such that the stabilizers of points in this action are a.e. amenable. As a consequence, we will get that if G is a countable residually finite non-amenable group, then the relation of topological conjugacy on free minimal G-subshifts is not amenable. On the other hand, for the group G=Z, we will look at the class of subshifts with separated holes and see that the conjugacy relation is an amenable equivalence relation there. This is joint work (in progress) with Todor Tsankov.
Prikry forcing and square properties
Prikry type forcing is the standard way of constructing models of the failure of SCH. On the other hand not SCH is at odds with the failure of the weaker square principles. I will go over some consistency results about not SCH and failure of squares. Then I will present a dichotomy theorem characterizing what type of Prikry posets add weak square. This is joint work with Spencer Unger.
Universal functions and universal graphs
A function of two variables F(x,y) was defined by Sierpinski to be universal if for every other function G(x,y) there exist functions h(x) and k(y) such that G(x,y)=F(h(x),k(y)). Various aspects of this question were examined in a paper LMSW (Larson, Miller, Steprans and Weiss). While the universality of Sierpinski seems similar to model theoretic universality, there is a key difference in the role played by the range of the function in the two cases. This was the motivation for the following question asked in LMSW: Does the existence of a universal graph of cardinality aleph_1 imply the existence of a universal colouring of the complete graph on omega_1 with countably many colours? I will discuss joint work with S. Shelah providing a negative answer to this question.
A disjoint union theorem for trees In this talk we will present an infinitary disjoint union theorem for level
products of trees, which can be viewed as a dual form of the Halpern-Läuchli
theorem. A consequence of the dual Ramsey theorem due to T.J. Carlson and
S.G. Simpson is that for every Suslin measurable finite coloring of the powerset
of the natural numbers, there exists a sequence (Xn)n2N of pairwise disjoint
non-empty subsets of N such that the set
The tree property We survey recent partial progress towards a positive answer to a question
of Magidor "Is it consistent that every regular cardinal greater than
aleph_1 has the tree property?" We also discuss some obstructions to
better results.
Universally Baire subsets of 2^\kappa We generalize the notion of universally Baire set of reals and define the universally Baire subsets of 2^\kappa for \kappa an arbitrary infinite cardinal. We show that the standard theory of universally Baire sets of reals can be naturally generalized to this setting and that the properties of universally Baire subsets of 2^{\omega_1} are inestricably intertwined with forcing axioms. This is joint work with Daisuke Ikegami.
Interpreter for Topologists Given a transitive model M of set theory, we find an interpretation functor between topological spaces of M and V which commutes with very many topological operations. The theory is developed from scratch all the way to computations in functional analysis.
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