The Logic seminar generally meets on Fridays, 2–3:30p.m., at Caltech or UCLA. Contact Alexander Kechris or Itay Neeman if you wish to give a talk.

Schedule of talks, going back to Fall
2017, in ** reverse chronological order**:

Friday May 24 2019 | ||||

14:00-15:30 (MS 6221) | Aristotelis Panagiotopoulos (Caltech) | Higher dimensional obstructions for star reductions | ||

Abstract. In this talk we will consider \(*\)-reductions between orbit equivalence relations. These are Baire measurable reductions which preserve generic notions, i.e., preimages of comeager sets are comeager. In short, \(*\)-reductions are weaker than Borel reductions in the sense of definability, but much more sensitive to the dynamics of the orbit equivalence relations in question.
Based on a past joint work with M. Lupini we will introduce a notion of dimension for Polish \(G\)-spaces. This dimension is always \(0\) whenever the group \(G\) admits a complete and left invariant metric, but in principle, it can take any value $n$ within \(\{0,1,\ldots\}\cup\{\infty\}\). For each such \(n\) we will produce a free action of \(S_{\infty}\) which is generically \(n\)-dimensional and we will deduce that the associated orbit equivalence relations are pairwise incomparable with respect to \(*\)-reductions. This is a joint work with A. Kruckman. | ||||

Friday May 17 2019 | ||||

14:00-15:30 (MS 6221) | Nam Trang (UNT) | The strength of Sealing | ||

Abstract. Sealing is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. generic-LSA is the statement that the Largest Suslin Axiom (LSA) holds in all generic extensions. Under a mild large cardinal hypothesis, we show that Sealing is equiconsistent with generic-LSA. As a consequence, Sealing is weaker than the theory "ZFC+there is a Woodin cardinal which is a limit of Woodin cardinals". Sealing's consistency being weak represents an obstruction to the current program of descriptive inner model theory. Going beyond this bound in core model induction applications seems challenging and requires us to construct third order objects (subsets of the universally Baire sets). We will state the precise theorems and explain their impact on current developments of inner model theory. Time allowed, we'll say a bit about how to overcome the obstructions imposed by Sealing. This is joint work with G. Sargsyan. | ||||

Friday May 03 2019 | ||||

14:00-15:30 (MS 6221) | Michal Tomasz Godziszewski (University of Warsaw) | \(\Pi^0_1\)-computable quotient presentations of nonstandard models of arithmetic | ||

Abstract. A computable quotient presentation of a mathematical structure \(\mathcal A\) consists of a computable structure on the natural numbers \(\langle \mathbb{N},\star,\ast,\dots \rangle\), meaning that the operations and relations of the structure are computable, and an equivalence relation \(E\) on \(\mathbb{N}\), not necessarily computable but which is a congruence with respect to this structure, such that the quotient \(\langle \mathbb{N},\star,\ast,\dots \rangle\) is isomorphic to the given structure \(\mathcal{A}\). Thus, one may consider computable quotient presentations of graphs, groups, orders, rings and so on.
A natural question asked by B. Khoussainov in 2016, is if the Tennenbaum Thoerem extends to the context of computable presentations of nonstandard models of arithmetic. In a joint work with J.D. Hamkins we have proved that no nonstandard model of arithmetic admits a computable quotient presentation by a computably enumerable equivalence relation on the natural numbers.
However, as it happens, there exists a nonstandard model of arithmetic admitting a computable quotient presentation by a co-c.e. equivalence relation. Actually, there are infinitely many of those. The idea of the proof consists in simulating the Henkin construction via finite injury priority argument. What is quite surprising, the construction works (i.e. injury lemma holds) by Hilbert's Basis Theorem.
During the talk I'll present ideas of the proof of the latter result, which is joint work with T. Slaman and L. Harington. | ||||

Friday Apr 19 2019 | ||||

14:00-15:30 (MS 6221) | Matthew Foreman (UC Irvine) | Games on weakly compact cardinals | ||

Abstract. Weakly compact cardinals are equivalent to the statement that every \(\kappa\)-complete filter on a Boolean algebra \({\mathcal B}\) of size \(\kappa\) can be extended to a \(\kappa\)-complete ultrafilter on \({\mathcal B}\). One can continue this finitely many times. Can it be continued transfinitely?
Fix a cardinal \(\kappa\) and consider the following game \(\mathcal G_\gamma\) of ordinal length \(\gamma\): Player I plays a a sequence of collections \(\mathcal S_\alpha\subseteq P(\kappa)\) of size \(\kappa\) and player II plays an increasing sequence of \(\kappa\)-complete ultrafilters \(U_\alpha\) on \(\bigcup_{\beta\le \alpha}\mathcal S_\beta\). Player II wins if she can continue playing until stage \(\gamma\). Clearly if \(\kappa\) is measurable then II wins the game of any length. Welch asked whether the property that " II has a winning strategy in \(\mathcal G_\gamma\)" can hold at a non-measurable cardinal. The main result in this talk is that if II wins \(\mathcal G_{\omega_1}\) then there is a precipitous ideal on \(\kappa\) whose quotient has a countably closed dense subset. Hence the answer to Welch's question, at least for \(\gamma\ge \omega_1\), is no. In joint work with Magidor, we prove that it is consistent at a non-measurable cardinal for II to have a winning strategy in \(\mathcal G_{\omega_1}\), hence the theorem is not vacuous. | ||||

Friday Feb 22 2019 | ||||

14:00-15:30 (MS 6221) | Thomas Gilton (UCLA) | The Abraham-Rubin-Shelah Open Coloring Axiom with a Large Continuum | ||

Abstract. In their 1985 paper, Abraham, Rubin, and Shelah studied a number of combinatorial principles about \(\aleph_1\)-sized objects. One such axiom is the so-called "ARS Open Coloring Axiom" (hereafter \(\mathsf{OCA}_{ARS}\)), which concerns decompositions of \(\aleph_1\)-sized sets of reals into homogeneous sets for clopen colorings. One of the main open questions from their paper is whether or not \(\mathsf{OCA}_{ARS}\) is consistent with a value of the continuum greater than \(\aleph_2\) (it implies that the continuum is at least \(\aleph_2\)).
There are two additional theorems which complicate the situation. First, Moore has shown that Todorcevic's Open Coloring Axiom (hereafter \(\mathsf{OCA}_T\)) together with \(\mathsf{OCA}_{ARS}\) decides the value of the continuum to be exactly \(\aleph_2\), but second, Farah has shown that (a restricted version of) \(\mathsf{OCA}_T\) is consistent with an arbitrarily large value of the continuum. It is therefore of interest whether or not \(\mathsf{OCA}_{ARS}\) on its own decides the value of \(2^{\aleph_0}\). Recently Gilton and Itay Neeman have answered this question, showing that \(\mathsf{OCA}_{ARS}\) is in fact consistent with \(2^{\aleph_0}=\aleph_3\). As in the original ARS paper, we need to build so-called preassignments of colors in order to add the requisite homogeneous sets. However, these can only be built over models satisfying the \(\mathsf{CH}\). To get around this difficulty, we build preassignments with very strong symmetry conditions, which allow us to combine them in many different ways, using a new type of poset called a partition product. In this talk, we will motivate and define partition products, sketch the construction of these preassignments, and show how, as a result, we can obtain a model of \(\mathsf{OCA}_{ARS}\) with a large value of the continuum. | ||||

Friday Nov 30 2018 | ||||

14:00-15:30 (MS 6221) | Asgar Jamneshan (UCLA) | Measure-theoretic sheaves and kernel structures | ||

Abstract. A measure-theoretic sheaf is constructed over an arbitrary sigma-finite measure space. The collection of such sheaves forms a topos, and is an instance of a Boolean-valued model of ZFC. In this talk, adopting a variant of the sheaf-theoretic approach, we present semantics of some fundamental structures in this model such as vector spaces, topological spaces, and measure spaces. Interestingly, it turns out that these Boolean-valued structures allow for a meaningful interpretation in a standard model as well which often can be characterized by kernel type objects. This opens a way for applications of Boolean-valued techniques to analysis and its applications. We present details on two of these applications in probability and ergodic theory. | ||||

Friday Nov 16 2018 | ||||

14:00-15:30 (MS 6221) | Kyle Gannon (Notre Dame) | Local Keisler Measures | ||

Abstract. The connection between finitely additive probability measures and NIP theories was first noticed by Keisler. Around 20 years later, the work of Hrushovski, Peterzil, Pillay, and Simon greatly expanded this connection. Out of this research came the concept of generically stable measures. In the context of NIP theories, these particular measures exhibit stable behavior. In particular, Hrushovski, Pillay, and Simon demonstrated that generically stable measures admit a natural finite approximation. In this talk, we will discuss generically stable measures in the local setting. We will describe connections between these measures and concepts in functional analysis as well as show that this interpretation allows us to derive an approximation theorem. | ||||

Friday Nov 09 2018 | ||||

14:00-15:30 (MS 6221) | Anush Tserunyan (UIUC) | Independent sets in finite and algebraic hypergraphs | ||

Abstract. An active line of research in modern combinatorics is extending classical results from the dense setting (e.g., Szemeredi's theorem) to the sparse random setting. These results state that a property of a given "dense" structure is inherited by a randomly chosen "sparse" substructure. A recent breakthrough tool for proving such statements is the Balogh-Morris-Samotij and Saxton-Thomason hypergraph containers method, which bounds the number of independent sets in finite hypergraphs. In a joint work with A. Bernshteyn, M. Delcourt, and H. Towsner, we give a new — elementary and nonalgorithmic — proof of the containers theorem for finite hypergraphs. Our proof is inspired by considering hyperfinite hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. Applying this intuition in another setting with a notion of dimension, namely, algebraically closed fields, A. Bernshteyn, M. Delcourt, and I prove an analogous theorem for "dense" algebraically definable hypergraphs: any Zariski-generic low-dimensional subset of such hypergraphs is itself "dense" (in particular, not independent). | ||||

Friday Nov 02 2018 | ||||

14:00-15:30 (MS 6221) | Antonio Montalban (UC Berkeley) | The uniform Martin conjecture for the many-one degrees | ||

Abstract. We will discuss a variant of the uniform Martin's conjecture for the many-one degrees. | ||||

Friday Oct 19 2018 | ||||

14:00-15:30 (MS 6221) | John Susice (UCLA) | $\Box_{\kappa, 2}$ and the Special Aronszajn Tree Property at $\kappa^+$. | ||

Abstract. We show the consistency of $\square_{\kappa, 2}$ plus $SATP(\kappa^+)$ for regular $\kappa$ assuming a weakly compact. Using methods of Golshani-Hayut we also establish a global consistency result for successors of regulars from class many supercompacts. | ||||

Friday Jun 08 2018 | ||||

14:00-15:30 (MS 6221) | Travis Nell (UIUC) | Distal Types and Large Dimension in Dense Pairs | ||

Abstract. Let T be an o-minimal theory extending that of ordered abelian groups. In joint work with Hieronymi, we showed that the theory of dense pairs of models of T is non-distal. I will now discuss the connection between distal and non-distal types in this theory to the notion of Large Dimension, which arises from a suitable pre-geometry. | ||||

Friday Jun 01 2018 | ||||

14:00-15:30 (MS 6221) | Allen Gehret (UCLA) | Towards a model theory of logarithmic transseries | ||

Abstract. In this talk I will first define and describe the mathematical object $\mathbb{T}_{\log}$: the ordered valued differential field of logarithmic transseries. I will then discuss a strategy I have developed for proving $\mathbb{T}_{\log}$ is model complete in a certain language that I will introduce. I reduce the problem of model completeness down to a few precise conjectures about the nature of logarithmic derivatives, logarithms, solutions of linear differential equations, and differential-transcendence. Recent progress made in the last year will also be mentioned. | ||||

Friday May 25 2018 | ||||

14:00-15:30 (MS 6221) | Assaf Shani (UCLA) | Borel equivalence relations and symmetric models | ||

Abstract. We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models of set theory without choice, and apply it to solve questions of Hjorth-Kechris-Louveau.
E.g., we show that $\Sigma^0_{\omega+1}$ equivalence relations induced by abelian group actions are strictly simpler than general $\Sigma^0_{\omega+1}$ orbit equivalence relations.
The proof goes through studying symmetric models generated by generic invariants of these equivalence relations.
We will use models which were constructed by G. Monro in 1973 to separate the ``generalized Kinna-Wagner principles'', $\mathrm{KWP}^n$.
We show that these models correspond to the irreducibilities along the finite Friedman-Stanley jumps, $=^{+n}<_B =^{+(n+1)}$.
We extend Monro's results through limit stages, thus showing the consistency of $\mathrm{KWP}^{\omega+1}\wedge\neg\mathrm{KWP}^\omega$, answering a question of Karagila. | ||||

Friday Apr 27 2018 | ||||

14:00-15:30 (MS 6221) | Lewis Bowen (UT Austin) | Weak Pinsker entropy | ||

Abstract. The Weak Pinsker entropy is an easy-to-define measure-conjugacy invariant of measure-preserving actions of countable groups inspired by the classical Weak Pinsker conjecture, recently proven by Tim Austin for amenable groups. I'll discuss two recent results: (1) (joint with Robin Tucker-Drob) if the group is Bernoulli cocycle-superrigid, then WP-entropy is an orbit-equivalence invariant, (2) WP entropy is bounded by sofic entropy and can be strictly less. | ||||

Friday Apr 20 2018 | ||||

14:00-15:30 (MS 6221) | Zach Norwood (UCLA) | Coding along trees and remarkable cardinals | ||

Abstract. It is a major goal of modern set theory to understand the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler's proof does not resemble Kunen's, however, using almost-disjoint coding instead of Kunen's innovative method of coding along branchless trees. We show how to reconcile this gap, giving a new proof of Schindler's theorem that generalizes Kunen's methods and suggests further investigation of non-thin trees. This work is joint with Itay Neeman. | ||||

Friday Mar 16 2018 | ||||

14:00-15:30 (MS 6221) | Alex Kruckman (Indiana U.) | Independence in generic expansions and fusions | ||

Abstract. The word "generic" is often applied to a theory $T^*$ when it arises as a model companion of a base theory $T$, augmented with extra structure (e.g. a generic automorphism, a generic predicate, a generic order, etc.). Generic theories exhibit lots of "random" behavior, so they are rarely stable or NIP, but they can sometimes be shown to be simple by characterizing a well-behaved notion of independence in $T^*$ (namely non-forking independence) in terms of independence in $T$. Recently, there has been increased interest in the property NSOP1, a generalization of simplicity, spurred by the work of Chernikov, Kaplan, and Ramsey, who showed that NSOP1 theories can also be characterized by the existence of a well-behaved notion of independence (namely Kim independence). In this talk, I will present a number of preservation results for simplicity and NSOP1 under generic constructions, and characterizations of notions of independence in the resulting theories. In joint work with Nicholas Ramsey, generic expansion and generic Skolemization: add new symbols to the language, interpreted arbitrarily or as Skolem functions, and take the model companion. And in very recent results towards a joint project with Minh Chieu Tran and Erik Walsberg, interpolative fusion: given an $L_1$-theory $T_1$ and and $L_2$-theory $T_2$, which intersect in an $L_0$-theory $T_0$, take the model companion of the union of $T_1$ and $T_2$. | ||||

Friday Mar 02 2018 | ||||

14:00-15:30 (MS 6221) | Vadim Kulikov (University of Helsinki) | Towards non-classifiability of open 3-manifolds | ||

Abstract. It has been shown by Goldman (1971) that the open 2-manifolds can be completely classified by algebraic structures. It is known that the open 3-manifolds are more complicated than the open 2-manifolds as witnessed e.g. by the Whitehead manifolds. This presents us with a question: do open 3-manifolds admit classification by countable structures? The conjecture is "no" and the attempt is to use the theory of turbulence developed by Hjorth, Kechris and others to attack this problem. In this talk I will present results surrounding this conjecture and put them in a broader context of descriptive set theory and Borel-reducibility. Finally I will present some new ideas on how to tackle the conjecture of non-classifiability of open 3-manifolds. | ||||

Friday Feb 16 2018 | ||||

14:00-15:30 (MS 6221) | Matthew Foreman (UCI) | Classifying diffeomorphisms of the torus (part 2) | ||

Abstract. In 1932 von Neumann proposed classifying the statistical behavior of diffeomorphisms of compact manifolds. Notable progress was made by Halmos and von Neumann using spectral invariants. Later, Bernoulli shifts were classified by Ornstein using Kolmogorov's notion of entropy.
In these talks we show that the general von Neumann problem is intractable in a rigorous sense: the collection of pairs of diffeomorphisms of the torus that are isomorphic is complete analytic — in particular not Borel. It follows that there is no inherently countable procedure for determining whether a given pair (S,T) is isomorphic. A problem closely related to the isomorphism problem is the Realization Problem: which abstract measure preserving transformations can be realized as diffeomorphisms of a compact manifold. The only known restriction is that the transformations must have finite entropy. As a byproduct of the anti-classification result stated above, we show that a large class of previously unknown examples can be realized smoothly. For example there are measure distal diffeomorphisms of the 2-torus of arbitrary countable ordinal height, and diffeomorphisms of the torus whose simplex of invariant measures is affinely homeomorphic to an arbitrary compact Choquet simplex. This latter results are proved by establishing a Global Structure Theorem that says that there is a categorical isomorphism between the collection of ergodic transformations with odometer factors and diffeomorphisms realized by the Anosov-Katok method off conjugacy. | ||||

Friday Feb 09 2018 | ||||

14:00-15:30 (MS 6221) | Aristotelis Panagiotopoulos (Caltech) | A combinatorial model for the Menger curve | ||

Abstract. We will represent the universal Menger curve as a canonical quotient of a projective Fraisse limit
and we will illustrate how various homogeneity and universality properties of this space reduce to standard Fraisse theory and basic combinatorics. Finally we will discuss how our approach can be extended to higher dimensions, where it suggests the existence of homology versions of the $n$-dimensional Menger space, as well as a homology version of the Hilbert cube.
This is a joint work with Slawomir Solecki. | ||||

Friday Dec 08 2017 | ||||

14:00-15:30 (MS 6221) | Henry Towsner (U Penn) | Relatively Random First-Order Structures | ||

Abstract. The Aldous-Hoover Theorem gives a characterization of those random processes which generate "exchangeable" first-order structures. Exchangeable structures are precisely those where the "labels" of the points do not matter — they are random structures whose distribution remains the same when we permute the points. They therefore correspond to the structures we get when we sample countable substructures randomly from an ultraproduct; indeed, the original proof of the full Aldous-Hoover Theorem used ultraproducts, and the topic remains intimately tied to the way probability measures behave in ultraproducts.
For some purposes, full exchangeability is too strong: we should consider only those permutations respecting some existing structures. A full Aldous-Hoover theorem is not always possible in this setting, and how much we recover turns out to depend on the amalgamation properties of M. Our goal in this talk is to explain the connection between exchangeability and model theoretic notions, without assuming much prior expertise in either. | ||||

Friday Dec 01 2017 | ||||

14:00-15:30 (MS 6221) | Omer Ben-Neria (UCLA) | Singular stationarity | ||

Abstract. I will talk about Mutually Stationary sequences of sets.
The notion of Mutual Stationarity was introduced by Foreman and Magidor in the 90s, and has been developed as a notion of stationarity for subsets of singular cardinals and as means to address some classical problems in set theory.
We will discuss the basic concepts and describe several known and recent results. | ||||

Friday Nov 03 2017 | ||||

14:00-15:30 (MS 6221) | Matthew Foreman (UCI) | Classifying diffeomorphisms of the torus (part 1) | ||

Abstract. In 1932 von Neumann proposed classifying the statistical behavior of diffeomorphisms of compact manifolds. Notable progress was made by Halmos and von Neumann using spectral invariants. Later, Bernoulli shifts were classified by Ornstein using Kolmogorov's notion of entropy.
In these talks we show that the general von Neumann problem is intractable in a rigorous sense: the collection of pairs of diffeomorphisms of the torus that are isomorphic is complete analytic — in particular not Borel. It follows that there is no inherently countable procedure for determining whether a given pair (S,T) is isomorphic. A problem closely related to the isomorphism problem is the Realization Problem: which abstract measure preserving transformations can be realized as diffeomorphisms of a compact manifold. The only known restriction is that the transformations must have finite entropy. As a byproduct of the anti-classification result stated above, we show that a large class of previously unknown examples can be realized smoothly. For example there are measure distal diffeomorphisms of the 2-torus of arbitrary countable ordinal height, and diffeomorphisms of the torus whose simplex of invariant measures is affinely homeomorphic to an arbitrary compact Choquet simplex. This latter results are proved by establishing a Global Structure Theorem that says that there is a categorical isomorphism between the collection of ergodic transformations with odometer factors and diffeomorphisms realized by the Anosov-Katok method off conjugacy. | ||||

Friday Oct 20 2017 | ||||

14:00-15:30 (MS 6221) | Jeffrey Bergfalk (Cornell U.) | The first omega alephs | ||

Abstract. In the early seventies, several decisive relations were noticed between the homological dimension and the cardinality of a small category - particularly for the cardinalities $\aleph_n$ $(n\in\mathbb{N})$. Those relations, in the case of $n=1$, are best understood in terms of Todorcevic's method of minimal walks on the countable ordinals; the higher-$n$-cases point, similarly, to generalizations of that method, and to $n$-dimensional incompactness principles correlated, for each $n$, to $\aleph_n$. All these are ZFC phenomena. How these principles behave on other cardinals is a largely open question. We discuss these matters, and some of their implications both in set theory and in algebraic topology. | ||||

Friday Oct 13 2017 | ||||

14:00-15:30 (MS 6221) | Kota Takeuchi (University of Tsukuba) | Ramsey property and 2-Order Property | ||

Abstract. The notion of n-dependence was introduced by Shelah motivated to treat vector spaces with a bilinear form along NIP theories. One of the important tools analyzing the property is the Ramsey property of the class of ordered finite hyper graphs. Actually many of known results, preservation theorem under boolean combinations, finding a witness in a single variable and characterizing the property by generalized indiscernible can be implied from the Ramsey property. In this point of view we introduce the notion of 2-Order Property, which lies between Independent Property and 2-Independent Property, related a Ramsey class consisting of ladder-like graphs, and demonstrate that how the Ramsey property works well to prove similar results. The typical examples of 2-dependence are not 2-OP. So far, It is open if 2-OP is strictly weaker than 2-IP. | ||||

Wednesday Oct 04 2017 | ||||

16:00-17:30 (MS 6221) | Tobias Kaiser (Universitaet Passau) | Asymptotics of parameterized exponential integrals given by Brownian motion on globally subanalytic sets | ||

Abstract. (Joint work with Julia Ruppert) Understanding integration in the o-minimal setting is an important and difficult task. By the work of Comte, Lion and Rolin, succeeded by the work of Cluckers and Miller, parameterized integrals of globally subanalytic functions are very well analyzed. But very little is known when the exponential function comes into the game. We consider certain parameterized exponential integrals which come from considering the Brownian motion on globally subanalytic sets. We are able to show nice asymptotic expansions of these integrals.
(Note unusual day and time: Wednesday, 4pm.) | ||||

Friday Sep 29 2017 | ||||

14:00-15:30 (MS 6221) | Dana Bartosova (University of Sao Paulo) | Free actions via graphs | ||

Abstract. A topological group admits a free action if there is a
compact space on which it acts without fixed points. We translate this property into colorability of graphs, which leads us to questions of combinatorial nature. This is a joint work in progress with Vladimir Pestov. |