Title: The étale open topology and the stable fields conjecture
Speaker's homepage
For any field $K$, we introduce natural topologies on $K$-points of varieties over $K$, which is defined to be the weakest topology such that étale morphisms are open. This topology turns out to be natural in a lot of settings. For example, when $K$ is algebraically closed, it is easy to see that we have the Zariski topology, and the procedure picks up the valuation topology in many henselian valued fields. Moreover, many topological properties correspond to the algebraic properties of the field. As an application of this correspondence, we will show that large stable fields are separably closed. Joint work with Will Johnson, Chieu-Minh Tran, and Erik Walsberg.
Title: Rigid branchwise-real tree orders
Speaker's homepage
A branchwise-real tree order is a partial order tree in which every branch is isomorphic to a real interval. In this talk, I give several methods of constructing examples of these which are rigid (i.e. without non-trivial automorphisms), subject to increasing uniformity conditions. I show that there is a rigid branchwise-real tree order in which every branching point has the same degree, one in which every point is branching and of the same degree, and finally one in which every point is branching of the same degree and which admits no monotonic function into the reals. Trees are grown iteratively in stages, and a key technique is the construction (in ZFC) of a family of colourings of $(0,\infty)$ which is 'sufficiently generic', using these colourings to determine how to proceed with the construction.
Title: On the universality problem for $\aleph_2$-Aronszajn and wide $\aleph_2$-Aronszajn trees
Speaker's homepage
We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of the possible existence of a universal tree under weak embeddings in the classes of $\aleph_2$-Aronszajn and wide $\aleph_2$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA + CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_2$-Aronszajn tree $T$ without a weak ascent path, there is an $\aleph_2$-cc countably closed forcing forcing which specialises $T$ and adds an $\aleph_2$-Aronszajn tree which does not embed into $T$. One cannot however apply the ShFA to this forcing.
Further, we construct a model à la Laver-Shelah in which there are $\aleph_2$-Aronszajn trees, but none is universal. Work in progress is to obtain an analogue for universal wide $\aleph_2$-Aronszajn trees. We also comment on some negative ZFC results in the case that the embeddings are assumed to have a strong preservation property.
Title: Galvin's problem in higher dimensions
Speaker's homepage
This talk will discuss recent work on Galvin's conjecture in Ramsey theory. I will review the background and discuss previous work on the two dimensional case before focusing on the recent work on dimensions greater than 2. This is joint work with Stevo Todorčević.
Title: Simple automorphism groups
Speaker's homepage
The automorphism groups of many homogeneous structures (Riemannian symmetric spaces, projective spaces, trees, algebraically closed fields, Urysohn space etc) are abstractly simple groups - or at least are simple after taking an obvious quotient.
We present criteria to prove simplicity for a broad range of structures based on the notion of stationary independence.
Title: Set theory without powerset
Speaker's homepage
Many natural set-theoretic structures satisfy the basic axioms of set theory, but not the powerset axiom. These include the collections $H_{\kappa^+}$ of sets whose transitive closure has size at most $\kappa$, forcing extensions of models of ${\rm ZFC}$ by pretame (but not tame) forcing, and first-order models that are morally equivalent to models of the second-order Kelley-Morse set theory (with class choice). It turns out that a reasonable set theory in the absence of the powerset axiom is not simply ${\rm ZFC}$ with the powerset axiom removed. Without the powerset axiom, the Replacement scheme is not equivalent to the Collection scheme, and the various forms of the Axiom of Choice are not equivalent. In this talk, I will give an overview of the properties of a robust set theory without powerset, ${\rm ZFC}^-$, whose axioms are ${\rm ZFC}$ without the powerset axiom, with the Collection scheme instead of the Replacement scheme and the Well-Ordering Principle instead of the Axiom of Choice. While a great deal of standard set theory can be carried out in ${\rm ZFC}^-$, for instance, forcing works mostly as it does in ${\rm ZFC}$, there are several important properties that are known to fail and some which we still don't know whether they hold. For example, the Intermediate Model Theorem fails for ${\rm ZFC}^-$, and so does ground model definability, and it is not known whether ${\rm HOD}$ is definable. I will also discuss a strengthening of ${\rm ZFC}^-$ obtained by adding the Dependent Choice Scheme, and some rather strange ${\rm ZFC}^-$-models.
Title: Twenty Years of Tameness
Speaker's homepage
In the 1970s Saharon Shelah initiated a program to develop classification theory for non-elementary classes, and eventually settled on the setting of abstract elementary classes. For over three decades, limited progress was made, most of which required additional set theoretic axioms. In 2001, Rami Grossberg and I introduced the model theoretic concept of tameness which opened the door for stability results in abstract elementary classes in ZFC. During the following 20 years, tameness along with limit models have been used by several mathematicians to prove categoricity theorems and to develop non-first order analogs to forking calculus and stability theory, solving a very large number of problems posed by Shelah in ZFC. Recently, Marcus Mazari-Armida found applications to Abelian group theory and ring theory. In this presentation I will highlight some of the more surprising results involving tameness and limit models.
Title: Backward ergodic theorem along trees and its consequences
Speaker's homepage
In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), ..., T^n(x)\}$ in the "future" of a point $x$. In joint work with Jenna Zomback, we prove a backward ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over arbitrary trees of possible "pasts" of $x$. Somewhat unexpectedly, this theorem yields ergodic theorems for actions of free groups, where the averages are taken along arbitrary subtrees of the standard Cayley graph rooted at the identity. This strengthens results of Grigorchuk (1987), Nevo (1994), and Bufetov (2000).
