# Logic Seminar

## Department of Mathematics, The Ohio State University

Attention: Our meeting time has changed to 2:30–3:25pm.

Hello! Welcome to the website for the Logic Seminar in the Ohio State Department of Mathematics. We normally meet on Tuesdays from 2:30–3:25pm in Enarson Classroom Building 354. To join on Zoom or to be added to our mailing list, please contact Nigel Pynn-Coates at pynn-coates.1@osu.edu.

### Upcoming Seminars (2022–2023)

#### February 7

Gabe Conant (OSU)
Title: Pseudofinite compactifications and additive combinatorics, part II
Abstract: In the second of two talks, I will use the work discussed in the first talk to prove a nonabelian analogue of the Bogolyubov–Ruzsa Lemma, which is a fundamental result in additive combinatorics about large subsets of abelian groups. This will include some slight simplifications to the original proof using the language of continuous logic. (The first talk will be treated as a black box, and thus is not required for understanding the second talk.)

#### February 14

Title: Regular solutions of systems of transexponential-polynomial equations
Abstract: It is unknown whether there are o-minimal fields that are "transexponential", i.e., which define functions that eventually grow faster than any tower of exponential functions. In recent work, I constructed a Hardy field closed under a transexponential function which satisfies E(x+1) = exp E(x). Since the unary definable functions in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by E could be o-minimal. To actually prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach arising from Wilkie’s proof that the real exponential field is o-minimal. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.

#### February 28

Elliot Kaplan (McMaster University)
Title: TBA
Abstract: TBA

#### March 21

Sonia L'Innocente (University of Camerino)
Title: The universal *-regular R-ring
Abstract:
Olivier's construction of the universal commutative (von Neumann) regular ring over a commutative ring is generalized to obtain the universal *-regular ring over a noncommutative ring (R, *) with involution. The construction of a universal *-regular ring proceeds similarly with the Moore–Penrose inverse replacing the role of the group inverse in the construction of universal abelian regular rings.

The involution of (R, *) induces an involution on the modular lattice L(R, 1) of positive primitive formulae in the language of left R-modules. It is shown that *-regular ring coordinatizes the quotient lattice of L(R, 1) modulo the least congruence for which the involution designates an orthogonal complement.

Some explicit examples will be given in the context of some algebras, as the Jacobson algebra. This is joint work with Ivo Herzog.

#### April 4

Alex Kruckman (Wesleyan University)
Title: TBA
Abstract: TBA

#### April 11

Nick Ramsey (Notre Dame)
Title: TBA
Abstract: TBA

#### April 25

Isabella Scott (University of Chicago)
Title: TBA
Abstract: TBA

#### May 2

Scott Mutchnik (Berkeley)
Title: TBA
Abstract: TBA

### Past Seminars (2022–2023)

#### January 31

Gabe Conant (OSU)
Title: Pseudofinite compactifications and additive combinatorics, part I
Abstract: In the first of two talks, I will discuss pseudofinite structures and groups, as well as the notion of a "definable" compactification. Then I will present a new proof of a result of Pillay that any definable compactification of a pseudofinite group has an abelian connected component. Pillay's original proof used a powerhouse theorem of Breuillard, Green, and Tao on the structure of approximate groups. The new proof is much simpler, and uses only classical tools from harmonic analysis and representation theory of compact groups (along with work from one surprise name).

#### January 17 (Joint Geometric Group Theory–Logic Seminar)

Arman Darbinyan (OSU)
Title: Geometric interpretation of Turing degrees and some applications in geometric group theory
Abstract:
Classical theorems of Novikov, Boone, and Higman establish important connections between recursive and recursively enumerable sets on the one hand and the theory of finitely presented groups on the other hand. Despite the groundbreaking nature of these results, not much is known about possible extensions of those results to the realm of higher Turing degrees. My talk will be a contribution in this direction.

In my talk, I will discuss how one can view Turing degrees through invariants coming from geometric group theory. In particular, I will introduce a new quasi-isometric invariant, called (λ, μ)-taut filling spectra, and will show how this invariant can grasp Turing degrees of arbitrary degrees. Through these means, we will also introduce an interesting special class of functions, called Turing saturated maps, which will be useful for our applications.

As a main application, we will discuss a purely geometric group theoretical result that establishes quasi-isometric diversity of f.g. left-orderable simple groups, which is an important class of groups discovered by Hyde and Lodha. Previous attempts to obtain this result through more usual geometric/algebraic tools have not succeeded, which emphasizes the importance of the idea of Turing degrees in this context.

The talk is aimed to have an expository component on concepts from geometric group theory.

#### November 29

Neil Tennant (OSU Philosophy)
Title: Core Proofs as Objects of Search: Preserving Relevance and Epistemic Gains
Abstract: We explain how the model-invariant rules of inference of Core Logic (in natural deduction: introduction and elimination rules; in sequent calculus: Right and Left rules) arise naturally from the model-relative rules of evaluation (verification and falsification) by simple 'morphing'. We then explain how two important features of the rules for core proof provide constraints on 'bottom-up' proof search, without loss of completeness. Relevance of premises to conclusions is always preserved. So too are any epistemic gains made when pursuing solutions to the deductive sub-problems posed in the course of search.

#### November 15

Yayi Fu (Notre Dame), on Zoom
Title: Strong Erdos Hajnal in VC minimal theory
Abstract: We will show that if $T$ is a VC minimal theory (e.g. ACVF) and $M\models T$, then for any $d$ and any definable relation $E(x,y)\subseteq M^2$ of complexity $\leq d$ in Swiss Cheese decomposition, there is $k_d>0$ such that for any disjoint finite $A,B\subseteq M$, there exist $A'\subseteq A$, $B'\subseteq B$ with $|A'|\geq k_d |A|$, $|B'|\geq k_d |B|$ such that $A'\times B' \subseteq E$ or $A'\times B'\subseteq\neg E$.

#### November 8

Nicolas Chavarria Gomez (Notre Dame)
Title: Positive primitive elimination in a continuous setting
Abstract: We show positive primitive elimination for abelian structures with a homomorphism to a compact group, in analogy to the classical result for modules, as in [Ziegler, 1984]. To do this, we first develop an appropriate logic to handle these "enriched" structures and define the correct analogues of positive primitive-formulas. We finish by showing that this elimination result implies stability of the structure. This is joint work with Anand Pillay.

#### October 18

Liling Ko (OSU)
Title: Disorderly gamblers can outperform orderly ones
Abstract: In a game of casino versus gamblers, a casino has infinitely many pennies, each hidden under a cup. The cups are arranged in a line, and some are empty. A gambler picks infinitely many cups, selecting the next cup after checking the contents of the previously selected ones. A gambler wins if the density of the selected cups with pennies is non-zero. Gamblers are allowed to be disorderly, selecting cup i after selecting cup j>i. Are disorderly gamblers more difficult to beat than orderly ones? The statement is true if gamblers are also adaptive, where they may select a different cup depending on the outcomes of the uncovered ones. We show that the claim also holds in the non-adaptive setting. The proof involves infinitary combinatorial arguments. This is joint work with Justin Miller.

#### October 4

Ivo Herzog (OSU)
Title: The model theory of countable abelian p-groups
Abstract:
The countable abelian p-groups that have no divisible summands are determined, up to isomorphism, by their Ulm invariants. This classification can be used to determine the homogeneous countable abelian p-groups. One such abelian p-group turns out to be a universal countable abelian p-group for purity, i.e., every countable abelian p-group admits a pure embedding into it. It is the last step needed to complete the solution to Fuchs' Problem 5.1 below ℵω.

We will start off with some background, including how Ulm's Theorem is used to obtain a Scott sentence as well as some motivating examples. This is joint work with Marcos Mazari Armida.

#### September 20

Nigel Pynn-Coates (OSU)
Title: Monotone T-convex T-differential fields
Abstract: Let T be a suitably nice o-minimal theory extending the theory of real closed fields. A T-convex T-differential field is an expansion of a model of T by a valuation and a derivation, each of which is compatible with the o-minimal structure, the former in the T-convex sense of van den Dries–Lewenberg and the latter in the sense of Fornasiero–Kaplan. When T is the theory of the real field with restricted analytic functions, we can expand an ordered differential Hahn field (a kind of generalized power series field) to a T-convex T-differential field, in which case the derivation is monotone, i.e., weakly contractive with respect to the valuation (monotone differential Hahn fields were studied earlier by Scanlon and Hakobyan). I will describe joint ongoing work with Kaplan on monotone T-convex T-differential fields, achieving among other results an Ax–Kochen/Ershov type theorem for such structures. A key step is isolating an appropriate analogue of henselianity in this setting. I will explain these terms.

#### September 6

Kyle Gannon (UCLA)
Title: Extension domination
Abstract: Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its type setting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion. This is joint work with Jinhe Ye.

#### August 30

Matthew DeVilbiss (OSU)
Title: Towards a general method for showing strong minimality of differential equations
Abstract: In this talk, I will outline a technique for showing that nonlinear algebraic differential equations are strongly minimal. This is used to prove the strong minimality of generic differential equations with sufficiently large degree, answering a question of Poizat (1980). I will also discuss ongoing work in applying this method to differential equations of interest whose coefficients are not generic. This is joint work with James Freitag.