Mid-Atlantic Mathematical Logic Seminar
Fall Fest 2024

I'm going to talk about an old question of van Douwen: Are the shift map and its inverse conjugate in the automorphism group of P(ω)/fin? By the mid 1980's, van Douwen and Shelah proved that it is consistent they are not conjugate. Specifically, any automorphism witnessing their conjugacy would need to be nontrivial (van Douwen), but it is consistent that all automorphisms are trivial (Shelah). In this talk I'm going to discuss the complementary result: it is consistent that the shift map and its inverse are conjugate and, in fact, it follows from CH.

Let \(X\) be a standard Borel space, and let \(f\) be a Borel acyclic function on \(X\). A set \(H\) is hitting for \(f\) if, for each \(x \in X\), there is \(k \in \mathbb{N}\) such that \(f^k(x) \in H\). In this talk, we give some applications of hitting sets to Borel combinatorics. First, we use independent hitting sets to construct witnesses to finite Borel asymptotic dimension on non-dominating subsets of the shift graph on \([\mathbb{N}]^{\mathbb{N}}\). By combining this construction with work of Todorčević and Vidnyánszky, we compute the complexity of the set of Borel subgraphs of the shift graph that have Borel asymptotic dimension at most \(1\). Using a similar argument, we also solve the following decision problem: If \(H\) is a finite directed graph and \(f\) is an acyclic Borel function, when is there a Borel homomorphism from the directed graph generated by \(f\) to \(H\)? This is joint work in progress with Jan Grebík.

A wide variety of games turn out to be intertwined with Challenge and Response games. While often not directly equivalent, known results about one type of game turn out to be translatable into results about a different type of game. This talk discusses a family of games, called normal Welch games where winning strategies of length at least \(\omega+1\) produce normal precipitous ideals that can concentrate on any positive set for the superineffable ideal on a regular cardinal \(\kappa\). The super ineffable ideal is a proper ideal ideal on \(\kappa\) if and only if \(\kappa\) is completely ineffable (in the sense of Abramson, Harrington, Kleinberg and Zwicker). Moreover, player II has a winning tactic in the game just in case \(\kappa\) is completely superineffable. (The latter result is translatable to slightly weaker earlier result of Neilsen and Welch on a different type of game.)

A typical corollary of this is that if \(\kappa\) is a \(\kappa^{+2}\)-supercompact cardinal then \(\kappa\) is a stationary limit of non-measurable cardinals \(\lambda\) that carry normal precipitous ideals that concentrate on cardinals \(\alpha\) that are \(\alpha^+\) supercompact.

The investigation of the superineffable ideal explicates results of Baumgartner about the related hierarchy of \(n\)-ineffability, partition relations such as \[ \kappa\rightarrow [Stat]^n_2\] and a different large cardinal hierarchy that is intertwined in consistency strength.

These results are joint work with Julian Eshkol and Menachem Magidor.

When solving a differential equation, one sometimes finds that solutions can be expressed using a finite number of fixed, particular solutions, and some complex numbers. As an example, the set of solutions of a linear differential equation is a finite-dimensional complex vector space. A model-theoretic incarnation of this phenomenon is internality to the constants in a differentially closed field of characteristic zero. In this talk, I will define what this means, and discuss some recent progress, joint with Christine Eagles, on finding concrete methods to determine whether or not the solution set of a differential equation is internal. A corollary of our method also gives a criteria for solutions to be Liouvillian: I will show a concrete application to Lotka-Volterra systems.

Every Borel graph G on a Polish space X induces an analytic equivalence relation on X by considering two points equivalent if there is a path between them in G. Equivalence relations which arise in this way are called Borel graphable. In recent work, Tyler Arant, Alexander Kechris and I have examined the question of which analytic equivalence relations are Borel graphable. In this talk, I will discuss some of our results, focusing mostly on the case of equivalence relations which are the orbit equivalence relation of some Polish group action. In this setting, we have shown that there are many Polish groups (including both all connected groups and the group of permutations of the natural numbers) whose Borel actions all have Borel graphable orbit equivalence relations. I will also discuss the (open) question of whether these results can be extended to all Polish groups.

A theorem of Elekes and Szabó (2012) recognizes algebraic groups among certain complex algebraic surfaces in \(\mathbb{C}^3\) with “large” intersections with finite grids. Bays and Breuillard (2018) generalized it to any arity and dimension.

In this talk I present a generalization to relations definable in strongly minimal structures with distal expansions (includes algebraically and differentially closed fields of characteristic \(0\)); and also relations definable in o-minimal structures,

Our methods also provide explicit bounds on the power saving exponent in the non-group case.

A universality question asks: given a class equipped with a notion of embedding, does it have a universal object? An example of an interesting class is the class of models (of some fixed size) of some first order theory. Often, when the size of the objects is uncountable, the answer becomes independent of ZFC. In this talk I will concentrate in a natural class which is not a first order elementary class: trees with no cofinal branches.

In the last decade, computability theorists studied almost computable sets based on the notion of asymptotic density, such as generically and coarsely computable sets. We extended this approach by defining and investigating densely and coarsely computable structures. After all, these notions were originally motivated by problems in finitely presented groups. While in generic computability we may not have answers on inputs from a small set, in coarse computability we may have wrong answers on inputs from a small set. These notions are stratified by natural elementarity conditions for large substructures. This is joint work with Wesley Calvert, Douglas Cenzer, and David Gonzalez.