Set Theory Talks

Speaker:     David Jaz Myers, NYU Abu Dhabi.

Date and Time:     Wednesday November 6, 2024, SPECIAL TIME: 2:00 PM NYC TIME (contact N Yanofsky noson@sci.brooklyn.cuny.edu for zoom link)

Title:     Contextads: Para and Kleisli constructions as wreath products.

Abstract: Given a comonad D on a category C, we can produce a double category whose tight maps are those of C and whose loose maps are Kleisli maps for D --- this is the Kleisli double category kl(D). Given a monoidal right action & : C x M --> C, we can produce a double category Para(&) whose tight maps are those of C and whose loose maps A -|-> B are pairs (P, f : A & P --> B) of a parameter space P in M and a parameterised map f.

In this talk, we'll see both these as special cases of a general construction: the Ctx construction which takes a *contextad* on a (double) category and produces a new double category. We'll see that this construction is "just" the wreath product of pseudo-monads in Span(Cat). We'll then exploit this observation to find 2-algebraic structure on the Ctx constructions of suitably structured contextads; vastly generalizing the old observation that a colax monoidal comonad has a monoidal Kleisli category.

This is joint work with Matteo Capucci.

Geoff Galgon,
Distributivity and Base trees for P(κ)/<κ

For κ a regular uncountable cardinal, we show that distributivity and base trees for P(κ)/<κ of intermediate height in the cardinal interval [ω,κ) exist in certain models. We also show that base trees of height κ can exist as well as base trees of various heights ≥κ+ depending on the spectrum of cardinalities of towers in P(κ)/<κ. These constructions answer questions of V. Fischer, M. Koelbing, and W. Wohofsky in certain models.

Philipp Schlicht Kurt Gödel Research Center

Russell Miller, CUNY
Computable reductions on groups and fields

Hjorth and Thomas established that the complexity of the isomorphism problem for torsion-free abelian groups of finite rank grows dramatically higher as the rank increases: for each r, there is no Borel function F that maps each rank-(r+1) group G to a rank-r group F(G) in such a way that G0≅G1⟺F(G0)≅F(G1). We say that there is no Borel reduction from isomorphism on TFAbr+1 to isomorphism on TFAbr. (From lower to higher rank, in contrast, such a reduction is readily seen.) Fields of transcendence degree r over Q have very similar computability properties to groups in TFAbr. This being so, we extend their investigations to include the isomorphism relations on the classes FDr of such fields. We show that there do exist reductions (not merely Borel, but actually computable, and moreover functorial) from each TFAbr to the corresponding FDr, and also from each FDr to FDr+1 (which proves more challenging than it was for the groups!). It remains open whether a theorem analogous to that of Hjorth-Thomas holds for the fields, but we use the notion of countable reductions to show that the fundamental obstacle to a reduction from TFAbr+1 to TFAbr is the uncountability of these spaces. This is joint work with Meng-Che 'Turbo' Ho and Julia Knight.

Sun Mengzhou, National University of Singapore
The Kaufmann–Clote question on end extensions of models of arithmetic and the weak regularity principle

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each n∈N and any countable model of BΣn+2, we construct a proper Σn+2-elementary end extension satisfying BΣn+1, which answers a question by Clote positively. We also give a characterization of countable models of IΣn+2 in terms of their end extendibility similar to the case of BΣn+2. Along the proof, we will introduce a new type of regularity principles in arithmetic called the weak regularity principle, which serves as a bridge between the model's end extendibility and the amount of induction or collection it satisfies.
The talk is based on this paper from arxiv:2409.03527.

Speaker:     David Jaz Myers, NYU Abu Dhabi.

Date and Time:     Wednesday November 6, 2024, ZOOM TALK. TIME TBA (contact N Yanofsky noson@sci.brooklyn.cuny.edu for zoom link)

Title:     Contextads: Para and Kleisli constructions as wreath products.

Abstract: Given a comonad D on a category C, we can produce a double category whose tight maps are those of C and whose loose maps are Kleisli maps for D --- this is the Kleisli double category kl(D). Given a monoidal right action & : C x M --> C, we can produce a double category Para(&) whose tight maps are those of C and whose loose maps A -|-> B are pairs (P, f : A & P --> B) of a parameter space P in M and a parameterised map f.

In this talk, we'll see both these as special cases of a general construction: the Ctx construction which takes a *contextad* on a (double) category and produces a new double category. We'll see that this construction is "just" the wreath product of pseudo-monads in Span(Cat). We'll then exploit this observation to find 2-algebraic structure on the Ctx constructions of suitably structured contextads; vastly generalizing the old observation that a colax monoidal comonad has a monoidal Kleisli category.

This is joint work with Matteo Capucci.

Geoff Galgon,
Distributivity and Base trees for P(κ)/<κ

For κ a regular uncountable cardinal, we show that distributivity and base trees for P(κ)/<κ of intermediate height in the cardinal interval [ω,κ) exist in certain models. We also show that base trees of height κ can exist as well as base trees of various heights ≥κ+ depending on the spectrum of cardinalities of towers in P(κ)/<κ. These constructions answer questions of V. Fischer, M. Koelbing, and W. Wohofsky in certain models.

More Borel chromatic numbers

Borel chromatic numbers of definable graphs on Polish spaces have been studied for 25 years, starting with the seminal paper by Kechris, Solecky and Todorcevic. I will talk about some recent results about the consistent separation of uncountable Borel chromatic numbers of some particular graphs and about the Borel chromatic number of graphs related to Turing reducibility.

Sun Mengzhou, National University of Singapore
The Kaufmann–Clote question on end extensions of models of arithmetic and the weak regularity principle

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each n∈N and any countable model of BΣn+2, we construct a proper Σn+2-elementary end extension satisfying BΣn+1, which answers a question by Clote positively. We also give a characterization of countable models of IΣn+2 in terms of their end extendibility similar to the case of BΣn+2. Along the proof, we will introduce a new type of regularity principles in arithmetic called the weak regularity principle, which serves as a bridge between the model's end extendibility and the amount of induction or collection it satisfies.
The talk is based on this paper from arxiv:2409.03527.

Title:     A formal category theory for oo-T-multicategories.

Abstract: We will explore a framework for oo-T-multicategories. To begin, we build a schema for multicategories out of the simplex schema and the monoid schema. The multicategory schema, D_m, inherits the structure of a monad from the +1 monad on the monoid schema. Simplicial T-multicategories are monad preserving functors out of the multicategory schema, [D_m, T], into another monad T. The framework is larger than just [D_m,T]. A larger structure describes notions of yoneda lemma and fibration. Inner fibrant, simplicial T-multicategories are oo-T-multicategories. oo-T-multicategories generalize oo-categories and oo-operads: oo-operads are fm-multicategories, oo-categories are Id-multicategories.

We use this framework to study oo-fc-multicategories, or "oo - virtual double categories". In general, under various assumptions on T (which hold for fc), the collection of oo-T-multicategories [D_m, T] has other useful structure. One such structure is a join operation. This join operation points towards a synthetic definition of op/cartesian cells, which we hope will model oo-virtual equipments. If there is time, I will explain the motivation for this study as it relates to ontologies, meta-theories and type theories.

Takashi Yamazoe, Kobe University
Cichoń's maximum with the uniformity and the covering of the σ-ideal E generated by closed null sets

Let E denote the σ-ideal generated by closed null sets on R. We show that the uniformity and the covering of E can be added to Cichoń's maximum with distinct values, more specifically, it is consistent that ℵ1<add(N)<cov(N)<b<non(E)<non(M)<cov(M)<cov(E)<d<non(N)<cof(N)<2ℵ0 holds.

Victoria Gitman, CUNY
Baby measurable cardinals

Measurable cardinals and other large cardinals on the larger side of things are characterized by the existence of elementary embeddings j:V→M from the universe V of sets into a transitive submodel M. The clear pattern the large cardinals in that region follow is that the closer the submodel M is to V the stronger the large cardinal notion. Smaller large cardinals, such as weakly compact or Ramsey cardinals, are known chiefly for their combinatorial properties, such as the existence of large homogeneous sets for colorings. But, it turns out that they too have elementary embeddings characterizations with embeddings on the correspondingly small models M of (a fragment) of set theory (usually ZFC−, the theory ZFC with powerset axiom removed). Elementary embeddings of V are often by-definable with the existence of certain ultrafilters or systems of ultrafilters. The classical example is that κ is measurable if and only if there is a κ-complete ultrafilter on κ. The model M is then the transitive collapse of the ultrapower of V by U. The connection between elementary embedding and ultrafilters also exists in the case of the small elementary embeddings. A typical elementary embedding characterization of a small large cardinal κ follows the following template: for every A⊆κ, there is a (technical condition) model M, with A∈M, for which there is an M-ultrafilter U on κ with (technical properties). A subset U⊆P(κ)∩M is an M-ultrafilter if the structure ⟨M,∈,U⟩, with a predicate for U, satisfies that U is a κ-complete ultrafilter on κ, meaning that U measures all the sets in M and its completeness applies to sequences that are elements of M. The reason we need to add a predicate for U is that in most interesting case, and in contrast to the situation with measurable cardinals, U is not an element of M (indeed in most cases, P(κ) does not exist in M). While the structure M usually satisfies some large fragment of ZFC, once, we add a predicate for the M-ultrafilter U, the structure ⟨M,∈,U⟩ can fail to satisfy even Σ0-separation. In this talk, I will discuss how smaller large cardinals follow the pattern that the more set theory the structure ⟨M,∈,U⟩ satisfies the stronger the resulting large cardinal notion. I will use these observations to introduce a new hierarchy of large cardinals between Ramsey and measurable cardinals. This is joint work with Philipp Schlicht, based on earlier work by Bovykin and McKenzie.

Title:     A formal category theory for oo-T-multicategories.

Abstract: We will explore a framework for oo-T-multicategories. To begin, we build a schema for multicategories out of the simplex schema and the monoid schema. The multicategory schema, D_m, inherits the structure of a monad from the +1 monad on the monoid schema. Simplicial T-multicategories are monad preserving functors out of the multicategory schema, [D_m, T], into another monad T. The framework is larger than just [D_m,T]. A larger structure describes notions of yoneda lemma and fibration. Inner fibrant, simplicial T-multicategories are oo-T-multicategories. oo-T-multicategories generalize oo-categories and oo-operads: oo-operads are fm-multicategories, oo-categories are Id-multicategories.

We use this framework to study oo-fc-multicategories, or "oo - virtual double categories". In general, under various assumptions on T (which hold for fc), the collection of oo-T-multicategories [D_m, T] has other useful structure. One such structure is a join operation. This join operation points towards a synthetic definition of op/cartesian cells, which we hope will model oo-virtual equipments. If there is time, I will explain the motivation for this study as it relates to ontologies, meta-theories and type theories.

David Marker, University of Illinois at Chicago
Rigid real closed fields

Shelah showed that it is consistent that there are uncountable rigid non-archimedean real closed fields and, later, he and Mekler proved this in ZFC. Answering a question of Enayat, Charlie Steinhorn and I show that there are countable rigid non-archimedean real closed fields by constructing one of transcendence degree two.

David Marker, University of Illinois at Chicago
Rigid real closed fields

Shelah showed that it is consistent that there are uncountable rigid non-archimedean real closed fields and, later, he and Mekler proved this in ZFC. Answering a question of Enayat, Charlie Steinhorn and I show that there are countable rigid non-archimedean real closed fields by constructing one of transcendence degree two.

David Marker, University of Illinois at Chicago
Rigid real closed fields

Shelah showed that it is consistent that there are uncountable rigid non-archimedean real closed fields and, later, he and Mekler proved this in ZFC. Answering a question of Enayat, Charlie Steinhorn and I show that there are countable rigid non-archimedean real closed fields by constructing one of transcendence degree two.

Hello everyone,

This week our weekly Nankai Logic Colloquium will be in the afternoon.

Our speaker this week will be Rizos Sklinos from the Chinese Academy of Sciences. This talk will take place this Friday,  June 7th,  from 4pm to 5pm (UTC+8, Beijing time). 

This is going to be an online/offline hybrid event. Follow the link below to join the Zoom meeting. Please use your real name to join the meeting.

Time ：16:00pm, Jun. 7, 2024(Beijing Time)

Zoom Number ： 436 658 8683

Passcode ：477893

Best wishes,

Ming Xiao

Speaker:     Emilio Minichiello , The CUNY Graduate Center.

Date and Time:     Wednesday May 22, 2024, 7:00 - 8:30 PM. IN PERSON TALK!

Title:     Presenting Profunctors.

Abstract: In categorical database theory, profunctors are ubiquitous. For example, they are used to define schemas in the algebraic data model. However, they can also be used to query and migrate data. In this talk, we will discuss an interesting phenomenon that arises when trying to model profunctors in a computer. We will introduce two notions of profunctor presentations: the UnCurried and Curried presentations. They are modeled on thinking of profunctors as functors P: C^op x D -> Set and as functors P: C^op -> Set^D, respectively. Semantically of course, these are equivalent, but their syntactic properties are quite different. The UnCurried presentations are more intuitive and easier to work with, but they carry a fatal flaw: there does not exist a semantics-preserving composition operation of UnCurried presentations that also preserves finiteness. Therefore we introduce the Curried presentations and show that they remedy this flaw. In the process, we characterize which UnCurried Presentations can be made Curried, and discuss some applications. This talk will be based off of this recent preprint which is joint work with Gabriel Goren Roig and Joshua Meyers.

Hello everyone,

This week our weekly Nankai Logic Colloquium is going to be in the afternoon, but at an irregular time, as we have two speakers this week.

_____________________________________________________________________

The records of past talks can be accessed at https://space.bilibili.com/253421893. 

Best wishes,

Ming Xiao

Speaker:     Raymond Puzio.

Date and Time:     Wednesday May 15, 2024, 7:00 - 8:30 PM. IN-PERSON!

Title:     Uniqueness of Classical Retrodiction.

Abstract: In previous talks at this Category seminar and at the Topology, Geometry and Physics seminar, Arthur Parzygnat showed how Bayesian inversion and its generalization to quantum mechanics may be interpreted as a functor on a suitable category of states which satisfies certain axioms. Such a functor is called a retrodiction and Parzygnat and collaborators conjectured that retrodiction is unique. In this talk, I will present a proof of this conjecture for the special case of classical probability theory on finite state spaces.

In this special case, the category in question has non-degenerate probability distributions on finite sets as its objects and stochastic matrices as its morphisms. After preliminary definitions and lemmas, the proof proceeds in three main steps.

In the first step, we focus on certain groups of automorphisms of certain objects. As a consequence of the axioms, it follows that these groups are preserved under any retrodiction functor and that the restriction of the functor to such a group is a certain kind of group automorphism. Since this group is isomorphic to a Lie group, it is easy to prove that the restriction of a retrodiction to such a group must equal Bayesian inversion if we assume continuity. If we do not make that assumption, we need to work harder and derive continuity "from scratch" starting from the positivity condition in the definition of stochastic matrix.

In the second step, we broaden our attention to the full automorphism groups of objects of our category corresponding to uniform distributions. We show that these groups are generated by the union of the subgroup consisting of permutation matrices and the subgroup considered in the first step. From this fact, it follows that the restriction of a retrodiction to this larger group must equal Bayesian inversion.

In the third step, we finally consider all the objects and morphisms of our category. As a consequence of what we have shown in the first two steps and some preliminary lemmas, it follows that retrodiction is given by matrix conjugation. Furthermore, Bayesian inversion is the special case where the conjugating matrices are diagonal matrices. Because the hom sets of our category are convex polytopes and a retrodiction functor is a continuous bijection of such sets, a retodiction must map polytope faces to faces. By an algebraic argument, this fact implies that the conjugating matrices are diagonal, answering the conjecture in the affirmative.

Paper.

Speaker:     Emilio Minichiello , The CUNY Graduate Center.

Date and Time:     Wednesday May 22, 2024, 7:00 - 8:30 PM. IN PERSON TALK!

Title:     Presenting Profunctors.

Abstract: In categorical database theory, profunctors are ubiquitous. For example, they are used to define schemas in the algebraic data model. However, they can also be used to query and migrate data. In this talk, we will discuss an interesting phenomenon that arises when trying to model profunctors in a computer. We will introduce two notions of profunctor presentations: the UnCurried and Curried presentations. They are modeled on thinking of profunctors as functors P: C^op x D -> Set and as functors P: C^op -> Set^D, respectively. Semantically of course, these are equivalent, but their syntactic properties are quite different. The UnCurried presentations are more intuitive and easier to work with, but they carry a fatal flaw: there does not exist a semantics-preserving composition operation of UnCurried presentations that also preserves finiteness. Therefore we introduce the Curried presentations and show that they remedy this flaw. In the process, we characterize which UnCurried Presentations can be made Curried, and discuss some applications. This talk will be based off of this recent preprint which is joint work with Gabriel Goren Roig and Joshua Meyers.

Ali Enayat, University of Gothenburg
Tarski's undefinability of truth theorem strikes again

Tarski's undefinability of truth theorem has two versions, the first one deals with truth itself, takes some effort to prove, and is a descendant of the Epimenides (liar) paradox. The second one deals with the related concept of satisfaction, has a one-line proof, and is a descendent of Russell's paradox. This talk is about the first one, which appeared in the 1953 monograph 'Undecidable Theories' by Tarski, Mostowski, and Robinson; it was employed there to show the essential undecidability of consistent theories that can represent all recursive functions (a strong form of the Gödel-Rosser incompleteness theorem). I will present Tarski's original 1953 formulation (which differs from the common formulation in modern expositions) and will explain how it was used in my recent work with Albert Visser to show that no consistent completion of a sequential theory whose signature is finite is axiomatizable by a collection of sentences of bounded quantifier-alternation-depth. A variant of this result was proved independently by Emil Jeřábek, as I will explain. Our proof method has a pedagogical dividend since it allows one to replace the cryptic Gödel-Carnap fixed point lemma with the perspicuous undefinability of truth theorem in the proof of the Gödel-Rosser incompleteness theorem.

Speaker:     Juan Orendain, Case Western Univeristy.

Date and Time:     Wednesday May 8, 2024, 7:00 - 8:30 PM. ZOOM TALK.

Title:     Canonical squares in fully faithful and absolutely dense equipments.

Abstract: Equipments are categorical structures of dimension 2 having two separate types of 1-arrows -vertical and horizontal- and supporting restriction and extension of horizontal arrows along vertical ones. Equipments were defined by Wood in [W] as 2-functors satisfying certain conditions, but can also be understood as double categories satisfying a fibrancy condition as in [Sh]. In the zoo of 2-dimensional categorical structures, equipments nicely fit in between 2-categories and double categories, and are generally considered as the 2-dimensional categorical structures where synthetic category theory is done, and in some cases, where monoidal bicategories are more naturally defined.

In a previous talk in the seminar, I discussed the problem of lifting a 2-category into a double category along a given category of vertical arrows, and how this problem allows us to define a notion of length on double categories. The length of a double category is a number that roughly measures the amount of work one needs to do to reconstruct the double category from a bicategory along its set of vertical arrows.

In this talk I will review the length of double categories, and I will discuss two recent developments in the theory: In the paper [OM] a method for constructing different double categories from a given bicategory is presented. I will explain how this construction works. One of the main ingredients of the construction are so-called canonical squares. In the preprint [O] it is proven that in certain classes of equipments -fully faithful and absolutely dense- every square that can be canonical is indeed canonical. I will explain how from this, it can be concluded that fully faithful and absolutely dense equipments are of length 1, and so they can be 'easily' reconstructed from their horizontal bicategories.

References:
[O] Length of fully faithful framed bicategories. arXiv:2402.16296.
[OM] J. Orendain, R. Maldonado-Herrera, Internalizations of decorated bicategories via π-indexings. To appear in Applied Categorical Structures. arXiv:2310.18673.
[W] R. K. Wood, Abstract Proarrows I, Cahiers de topologie et géométrie différentielle 23 3 (1982) 279-290.
[Sh] M. Shulman, Framed bicategories and monoidal fibrations. Theory and Applications of Categories, Vol. 20, No. 18, 2008, pp. 650–738.

Alf Dolich, CUNY
The decidability of the rings Z/mZ

In this expository talk I will discuss recent work of Derakhshan and Macintyre on the decidability of the common theory of the rings Z/mZ as m varies through the natural numbers m>1.

Roman Kossak, CUNY
The lattice problem for models of arithmetic

The lattice problem for models of PA is to determine which lattices can be represented either as lattices of elementary substructures of a model of PA or, more generally, which can be represented as lattices of elementary substructures of a model N that contain a given elementary substructure M of N.

Since the 1970's, the problem generated much research with highly nontrivial results with proofs combining specific methods in the model theory of arithmetic with lattice theory and various combinatorial theorems. The problem has a definite answer in the case of distributive lattices, and, despite much effort, there are still many open questions in the nondistributive case. I will briefly survey some early results and present a few proofs that illustrate the difference between the distributive and nondistributive cases.

Spencer Unger, University of Toronto
Iterated ultrapower methods in analysis of Prikry type forcing

We survey some old and new results in singular cardinal combinatorics whose proofs can be phrased in terms of iterated ultrapowers and ask a few questions.

Christian Wolf, CUNY
Computability of entropy and pressure on compact symbolic spaces beyond finite type

In this talk we discuss the computability of the entropy Htop(X) and topological pressure Ptop(ϕ) on compact shift spaces X and continuous potentials ϕ:X→R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the entropy pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and Beta shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further show that the generalized pressure function (X,ϕ)↦Ptop(X,ϕ|X) is not computable for a large set of shift spaces X and potentials ϕ. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability. The topic of the talk is joint work with Michael Burr (Clemson U.), Shuddho Das (Texas Tech) and Yun Yang (Virginia Tech).

Speaker:     Juan Orendain, Case Western Univeristy.

Date and Time:     Wednesday May 8, 2024, 7:00 - 8:30 PM. ZOOM TALK.

Title:     Canonical squares in regularly framed bicategories.

Roman Kossak, CUNY
The lattice problem for models of arithmetic

The lattice problem for models of PA is to determine which lattices can be represented either as lattices of elementary substructures of a model of PA or, more generally, which can be represented as lattices of elementary substructures of a model N that contain a given elementary substructure M of N.

Since the 1970's, the problem generated much research with highly nontrivial results with proofs combining specific methods in the model theory of arithmetic with lattice theory and various combinatorial theorems. The problem has a definite answer in the case of distributive lattices, and, despite much effort, there are still many open questions in the nondistributive case. I will briefly survey some early results and present a few proofs that illustrate the difference between the distributive and nondistributive cases.

Spencer Unger, University of Toronto
Iterated ultrapower methods in analysis of Prikry type forcing

We survey some old and new results in singular cardinal combinatorics whose proofs can be phrased in terms of iterated ultrapowers and ask a few questions.

Christian Wolf, CUNY
Computability of entropy and pressure on compact symbolic spaces beyond finite type

In this talk we discuss the computability of the entropy Htop(X) and topological pressure Ptop(ϕ) on compact shift spaces X and continuous potentials ϕ:X→R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the entropy pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and Beta shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further show that the generalized pressure function (X,ϕ)↦Ptop(X,ϕ|X) is not computable for a large set of shift spaces X and potentials ϕ. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability. The topic of the talk is joint work with Michael Burr (Clemson U.), Shuddho Das (Texas Tech) and Yun Yang (Virginia Tech).

Speaker:     Juan Orendain, Case Western Univeristy.

Date and Time:     Wednesday May 8, 2024, 7:00 - 8:30 PM. ZOOM TALK.

Title:     Canonical squares in regularly framed bicategories.

Roman Kossak, CUNY
The lattice problem for models of arithmetic

The lattice problem for models of PA is to determine which lattices can be represented either as lattices of elementary substructures of a model of PA or, more generally, which can be represented as lattices of elementary substructures of a model N that contain a given elementary substructure M of N.

Since the 1970's, the problem generated much research with highly nontrivial results with proofs combining specific methods in the model theory of arithmetic with lattice theory and various combinatorial theorems. The problem has a definite answer in the case of distributive lattices, and, despite much effort, there are still many open questions in the nondistributive case. I will briefly survey some early results and present a few proofs that illustrate the difference between the distributive and nondistributive cases.

Spencer Unger, University of Toronto
Iterated ultrapower methods in analysis of Prikry type forcing

We survey some old and new results in singular cardinal combinatorics whose proofs can be phrased in terms of iterated ultrapowers and ask a few questions.

Christian Wolf, CUNY
Computability of entropy and pressure on compact symbolic spaces beyond finite type

In this talk we discuss the computability of the entropy Htop(X) and topological pressure Ptop(ϕ) on compact shift spaces X and continuous potentials ϕ:X→R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the entropy pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and Beta shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further show that the generalized pressure function (X,ϕ)↦Ptop(X,ϕ|X) is not computable for a large set of shift spaces X and potentials ϕ. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability. The topic of the talk is joint work with Michael Burr (Clemson U.), Shuddho Das (Texas Tech) and Yun Yang (Virginia Tech).

Hans Schoutens, CUNY
Geometric tools for the decidability of the existential theory of Fp[[t]]

I will give a brief survey how tools from algebraic geometry can be used in finding solutions to Diophantine equations over Fp[[t]] and similar rings. These tools include Artin approximation, arc spaces, motives and resolution of singularities. This approach yields the definability of the existential theory of Fp[[t]] (in the ring language with a constant for t) contingent upon the validity of resolution of singularities (Denef-Schoutens). Anscombe-Fehm proved a weaker result using model-theoretic tools and together with Dittmann, they gave a proof assuming only the weaker 'local uniformization conjecture.'