Events

Salma Shaheen, Department of Pure Mathematics, University of Waterloo

In 1976, S. Eilenberg and M.-P. Schützenberger posed the following Diabolical question: if A is a finite algebraic structure, Σ is the set of all identities true in A, and there exists a finite subset F of Σ such that F and Σ have exactly the same finite models, must there also exist a finite subset F' of Σ such that F' and Σ have exactly the same finite and infinite models? (That is, must the identities of A be "finitely based"?). It is known that any counter example to their question must be inherently nonfinitely based (INFB) but not inherently nonfinitely based in the finite sense (INFBfin). In this talk, I will show that the algebras constructed by Lawrence and Willard from group action do not provide a counter example to this question. If time permits, I will give the first known examples of inherently nonfinitely based "automatic algebras" constructed from group actions.

MC 5479

George Domat, Rice University

I'll give a broad introduction to big mapping class groups and why they may be of interest to set theorists. These are a rich class of Polish groups coming from the worlds of low-dimensional topology and geometric group theory. Although they are not defined as such, they all arise as automorphism groups of countable structures. Various tools and properties coming from the fields of descriptive set theory and topological dynamics such as automatic continuity and Rosendal's coarse geometry machinery have proved useful in understanding these groups.

MC 5479

Russell Miller, City University of New York

Fix a computable presentation $\overline{\mathbb Q}$ of the algebraic closure of the rational numbers. The absolute Galois group of the rational numbers, which is precisely the automorphism group of the field $\overline{\mathbb Q}$, may then be viewed as a collection of paths through a finite-branching tree. Each individual automorphism has a Turing degree. We will use known results in computability to try to build natural countable elementary subgroups of the absolute Galois group. Several intriguing questions in number theory will appear as we measure the extent to which we succeed in doing so.

MC 5403

David Chodounsky, Institute of Mathematics of the Czech Academy of Sciences

We define games which characterize countable coloring numbers of analytic graphs on Polish spaces. These games can provide simple verification of the countable chromatic number of certain graphs. (Eg. the rational distance graph on the Euclidean plane.) Joint work with Jindrich Zapletal.

MC 5403

Omar Leon Sanchez, University of Manchester

While the model theory of differential fields in characteristic zero has been vastly studied since the late 60s, its positive characteristic counterpart has received little attention (or rather exploration). As far as I am aware, the only work on this direction appears in a series of papers of Carol Wood from the early 70's, where she investigates the class of differentially closed fields in characteristic p>0. This theory DCF_p is complete and stable; and one can think of it as the differential analogue of ACF_p. The question that I will address in this talk is: what is the differential analogue of the theory of separably closed fields SCF_p? Somewhat surprisingly, to my knowledge, this has not been dealt with elsewhere. This is joint work with Kai Ino.

MC 5403

Valentina Harizanov, George Washington University

Cohesive powers of computable structures are effective analogs of ultrapowers, where the role of an ultrafilter is played by a cohesive set of natural numbers. A set is cohesive if it is infinite and cannot be split into two infinite pieces by any computably enumerable set. The inspiration for cohesive powers goes back to Skolem’s 1934 construction of a countable nonstandard model of arithmetic. The elements of a cohesive power are equivalence classes of partial computable functions, so the power is at most countable structure. We will show how cohesive powers could give us nonstandard models with interesting properties.

MC 5501

Nicolas Chavarria Gomez, Department of Pure Mathematics, University of Waterloo

We explore some extensions of the classical result, in the model theory of modules, of positive primitive elimination. We first consider the case of length functions. Then we move to the case of adding homomorphisms to abelian structures. The latter requires us to consider a different formalism of continuous logic which is appropriate for the correct formulation of the result. This is joint work with Anand Pillay.

MC 5479

Chris Schulz, Department of Pure Mathematics, University of Waterloo

The k-automatic sets are those subsets of N^d whose base-k representations form a regular language. Building on theorems of Büchi and Bès, we aim to characterize the partial preorder among k-automatic sets of definability over (N, +). We give a conjecture—that this preorder contains exactly three equivalence classes—and discuss our progress toward proving this conjecture. This talk is based on joint work with Alexi Block Gorman (OSU) and Jason Bell (Waterloo).

MC 5479

Gianluca Basso, University of Turin

We generalize a theorem of Gutman, Tsankov and Zucker on the non-existence of generic chains of subcontinua in manifolds of dimension at least 3 to a large class of spaces. Among them is the Menger curve, a 1-dimensional planar continuum. Using Bing's partition theorem, we reduce the problem to a combinatorial statement about walks on finite graphs. The theorem has dynamical consequences which can be interpreted as non-rigidity results for the homeomorphism groups of the spaces involved. This is joint work with A. Codenotti and A. Vaccaro.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Natasha Dobrinen, University of Notre Dame

(joint work with Tom Benhamou)  The Tukey structure of ultrafilters on $\omega$ has been studied extensively in the last two decades with various works of Blass, Dobrinen, Kuzeljevic,  Mijares, Milovich, Raghavan, Shelah, Todorcevic, Trujillo, and Verner.  Research on the Galvin property for ultrafilters over uncountable cardinals, in particular on measurable cardinals, has gained recent momentum, due to applications in infinite combinatorics, cardinal arithmetic, and inner models and forcing theory,  with various works of Benhamou, Garti, Gitik, Poveda, and Shelah. Joint work with Tom Benhamou began with the observation that the Galvin property is equivalent to being not Tukey maximal; hence, Tukey types refine various Galvin properties.  We initiate the development of the Tukey theory of ultrafilters on measurable cardinals, allowing the flow of results from the countable to the uncountable and vice versa.  The situation for ultrafilters on measurable cardinals turns out to be quite different from that on $\omega$, sometimes greatly simplifying the situation on $\omega$ and sometimes posing new obstacles.  The structure of the Tukey classes also turns out to be sensitive to different large cardinal hypotheses.  We will present results from our preprint arXiv:2304.07214  and ongoing work. 

MC 5479