Events

Jing Zhang, University of Toronto

We describe an organizing framework to study higher dimensional infinitary combinatorics based on \v{C}ech cohomology, originating from works by Barry Mitchell, Barbara Osofsky and others. A central combinatorial notion is $n$-dimensional coherence sequences, generalizing the 1-dimensional ones studied extensively by Todorcevic using the method of minimal walks. We will discuss ZFC results suggesting $\aleph_n$ is not "compact for $(n+1)$-dimensional combinatorics" and consistency results that any regular cardinal greater or equal to $\aleph_{\omega+1}$ can be "compact for $n$-dimensional combinatorics for all $n$". The talk will be purely combinatorial. Joint work with Jeffrey Bergfalk and Chris Lambie-Hanson.

MC 5479

Christine Eagles, Department of Pure Mathematics, University of Waterloo

In stable theories we may understand types of finite rank in terms of a finite collection of minimal types. One such method is a domination decomposition. This talk will serve as an exposition on domination in stable theories, particularly in how it relates to images of a type under a function. In particular, we show that when the fibers of a map f from a type p to another type are almost internal to a minimal type r, then we have that p is domination equivalent to a Morley product of the image of f and some copies of r. This is joint work with Léo Jimenez.

MC 5479

Leo Jimenez, Ohio State University

An ordinary algebraic differential equation is said to be internal to the constants if its general solution is obtained as a rational function of finitely many of its solutions and finitely many constant terms. Any such equation has an algebraic group acting as its Galois group. In this talk, I will use decomposition theorems for algebraic groups to show that some internal equations (do not) split into a product of internal equations. The methods are model-theoretic and could be applied to other contexts. This is a joint work in progress with Christine Eagles.

MC 5479

Andy Zucker, Department of Pure Mathematics, University of Waterloo

Given a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X. This talk will discuss a notion of ultracoproduct for G-flows, which arise from considering ultraproducts of commutative G-C*-algebras by Gelfand duality. We apply the construction to develop an understanding of the properties of various classes of subflows of a flow, i.e. minimal, topologically transitive, etc. For groups which are locally Roelcke precompact, ultracoproducts of G-flows lead to a well-behaved notion of weak containment for a wide class of G-flows, and in particular for all G-flows when G is locally compact. In joint work with Gianluca Basso, we apply ultracoproducts of G-flows to achieve a new characterization of those Polish groups G with the property that every minimal flow has a comeager orbit.

MC 5479

Jason Bell, Department of Pure Mathematics, University of Waterloo

We look at the first-order theory of the real numbers augmented by a predicate X that is in some natural sense self-similar with respect to a positive integer base. We show that there is a dichotomy: either we can define a Cantor set in our structure or our expansion of the reals is interdefinable with the real numbers augmented by a set of the form {1/r, 1/r^2, 1/r^3, …} for some integer r>=2.  In the latter case, this is equivalent to the structure having NIP and NTP_2.  This is joint work with Alexi Block Gorman.

MC 5479

Elliot Kaplan, McMaster University

Let T be a model complete o-minimal theory that extends the theory of real closed ordered fields (RCF). We introduce T-derivations: derivations on models of T which cooperate with T-definable functions. The theory of models of T expanded by a T-derivation has a model completion, in which the derivation acts "generically." If T = RCF, then this model completion is the theory of closed ordered differential fields (CODF) as introduced by Singer. We can recover many of the known facts about CODF (open core, distality) in our setting. We can also describe thorn-rank for models of T with a generic T-derivation. This is joint work with Antongiulio Fornasiero.

MC 5479

Ross Willard, University of Waterloo

An equational theory T is said to be residually finite if every model of the theory can be embedded in a product of finite models of the theory.  Equivalently, T is residually finite if and only if its irreducible models (those that cannot be embedded in products of “simpler” models) are all finite.  In practice, it seems that whenever a theory is both “interesting” and residually finite, then there is a finite upper bound to the sizes of its irreducible models.  In other words, we see a sort of compactness principle for “interesting” equational theories: if such a theory has arbitrarily large finite irreducible models, then it must have an infinite irreducible model.  Whether or not this observation holds generally has been open for almost 50 years.  In this talk I will discuss some recent progress with collaborators Keith Kearnes and Agnes Szendrei.

MC 5479

Nicolas Chavarria, Department of Pure Mathematics, University of Waterloo

We discuss joint work with G. Conant and A. Pillay regarding a version of the Malliaris-Shelah stable regularity lemma realized in the context of continuous logic, which allows us to speak about the structure of stable functions of the form $f:V\times W\to [0,1]$, where we think of $V$ and $W$ as the parts of a "weighted'' bipartite graph. In the process, we will also mention some results about the structure of local Keisler measures in this setting.

MC 5479

Jananan Arulseelan, McMaster University

By the recent MIP*=RE result, the QWEP conjecture is known to be false. Consequently, the universal theory of the hyperfinite II_1 factor is not computable. We will explain these results and their context and then discuss the uncomputability of the universal theories of other Powers factors and the lack of an effective axiomatization of QWEP C^∗ algebras. As an application we show that there is a ultraproduct of non-QWEP algebras with QWEP. This is joint work with Isaac Goldbring and Bradd Hart. 

MC 5479

Christine Eagles, Department of Pure Mathematics, University of Waterloo

An ordinary algebraic differential equation is said to be internal to the constants if its general solution is obtained as a rational function of finitely many of its solutions and finitely many constant terms. Such equations give rise to algebraic groups behaving as Galois groups. In this talk I give a characterisation of when the pullback of the differential logarithm of an equation is internal to the constants when the Galois group is unipotent or a torus. This is joint work in progress with Leo Jimenez.

MC 5479