The UCLA Logic Colloquium meets on alternate Wednesdays,
at 4 p.m., in MS 6221.

The Logic Colloquium is organized by
Mariana Vicaria and
Jan Grebik.

Here are links to the
UCLA Logic Center, the
Caltech-UCLA Logic Seminar, and the
Philosophy Colloquium.

Talks are listed here in ** reverse chronological order. **

Wednesday May 08 2024 | ||||

16:00-16:50 (MS 6221 (now online)) | Amanda Wilkens (University of Texas at Austin) | Poisson--Voronoi tessellations and fixed price in higher rank | ||

Abstract. We define and motivate the Poisson point process, which is, informally, a "maximally random" scattering of points in space. We introduce the ideal Poisson--Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group. We give some intuition for the proof, which relies on Gaboriau's theory of cost for measure-preserving actions. No prior knowledge on Poisson point processes or symmetric spaces will be assumed.
**Online only.** Zoom link: https://ucla.zoom.us/j/98099215455 Meeting ID: 980 9921 5455 | ||||

Wednesday Apr 24 2024 | ||||

16:00-16:50 (MS 6221) | David Gonzalez (UC Berkeley) | Semi-periodic Functions and the Scott Analysis of Linear Orderings | ||

Abstract. The concept of Scott complexity was introduced by Alvir, Greenberg, Harrison-Trainor and Turetsky and gives a way of assigning countable structures to elements of the Borel hierarchy that correspond to their descriptive complexity. This concept refines the previous notions of Scott rank. In computable structure theory, Scott analysis refers to a wide variety of pursuits related to the concepts of Scott rank and Scott complexity. For example, it is typical to study the sorts of Scott ranks and Scott complexities that can appear in a given class of structures or the sorts of structures from a class that can have a given Scott rank or Scott complexity.
I will describe recent work that solves a number of open questions regarding the Scott analysis of linear orderings (and of structures in general). Central to this work is a new construction of a linear ordering given a so-called semi-periodic function. We will discuss this construction and how to use the combinatorial structure of semi-periodic functions to extract Scott analytic facts about their corresponding linear orderings. | ||||

Wednesday Apr 10 2024 | ||||

16:00-16:50 (MS 6221) | Benjamin Castle (University of Maryland-Urbana) | Zilber's Restricted Trichotomy via Valued Fields | ||

Abstract. This talk will survey the recent solution of Zilber's `Restricted Trichotomy Conjecture' -- which asserts that every sufficiently `non-degenerate' reduct of an algebraically closed field interprets a copy of the same field. Special cases of this conjecture emerged during the 1980s and 1990s in work of Martin, Rabinovich, Rabinovich-Zilber, and Marker-Pillay; while Zilber showed in 2014 that a positive answer has applications in algebraic geometry. In the end, the problem was solved in two papers by considering (more generally) reducts of algebraically closed valued fields. I will attempt to explain the history of the problem, the general proof strategy, and the reason valued fields ultimately play a key role. | ||||

Wednesday Mar 27 2024 | ||||

16:00-16:50 (MS 6221) | Pablo Cubides (Universidad de los Andes) | Geometric spaces from the model-theorist point of view | ||

Abstract. In this talk I will informally show how different geometric spaces can be seen model-theoretically as spaces of definable types. I will concentrate on three kinds of geometric spaces having an algebraic nature: affine algebraic varieties over an algebraically closed field, affine real algebraic varieties over a real closed field, and, if time allows, the Berkovich analytification of an affine algebraic variety over a non-archimedean algebraically closed valued field. A short word about the (strict) pro-definability of such spaces will be discussed. No background in algebraic geometry will be assumed. | ||||

Wednesday Mar 13 2024 | ||||

16:00-16:50 (MS 6221) | Nam Trang (University of North Texas) | Ideals and Strong Axioms of Determinacy | ||

Abstract. We present the main ideas behind the proof of the equiconsistency of the theories:
(1) ZF + AD_R + \Theta is regular. (2) ZFC + CH + there is an \omega_1-dense ideal on \omega_1. (3) ZFC + the nonstationary ideal on P_{\omega_1}(R) is strong and pseudo-homogeneous. This resolves a long-standing open problem asked by W.H. Woodin in the 1990's. In the first talk, we discuss some history related to this problem and the general program in descriptive inner model theory that aims to calibrate the consistency strength of theories like the above. The results from this talk are from the paper Ideals and Strong Axioms of Determinacy, by Adolf, Sargsyan, Trang, Wilson and Zeman: https://arxiv.org/abs/2111.06220 | ||||

Wednesday Feb 28 2024 | ||||

16:00-16:50 (MS 6221) | Gabriel Goldberg (UC Berkeley) | Ordinal definability and large cardinals. | ||

Abstract. Woodin's HOD conjecture is one of the most important open problems in large cardinal set theory. A particularly simple formulation of the conjecture states that under large cardinal assumptions (in particular a supercompact cardinal), one can define a well order of the class Ordω of countable sequences of ordinals. This talk will discuss the motivation behind Woodin's conjecture, its connection to some of the speaker's recent ZFC theorems on ordinal definability, and the current status of a cluster of related problems in inner model theory and large cardinals. | ||||

Wednesday Feb 14 2024 | ||||

16:00-16:50 (MS 6221) | Diego Bejarano (UC Berkeley) | Separable structure theory | ||

Abstract. In [1], Ben Yaacov et. al. extended the basic ideas of Scott analysis to metric structures in infinitary continuous logic. These include back-and-forth relations, Scott sentences, and the Lopez-Escobar theorem to name a few.
In this talk, I will talk about joint work with Dino Rossegger connecting the ideas of Scott analysis to the definability of automorphism orbits and “isolation” of types within separable metric structures.
Our results are a continuous analogue of the robuster Scott rank developed by Montalbán [2] for countable structures in discrete infinitary logic. However, there are some differences arising from the subtleties behind the notion of definability in continuous logic.
[1] Itaï Ben Yaacov, Michal Doucha, Andre Nies, and Todor Tsankov. “Metric Scott analysis”. In: Advances in Mathematics 318 (2017), pp. 46–87. [2] Antonio Montalbán. “A robuster Scott rank”. In: Proceedings of the American Mathematical Society 143.12 (Apr. 2015), pp. 5427–5436. | ||||

Wednesday Jan 31 2024 | ||||

16:00-16:50 (MS 6221) | Isaac Goldbring (UC Irvine) | Preliminary remarks on the first-order free group factor problem | ||

Abstract. A standard construction in von Neumann algebra theory is to construct the group von Neumann algebra L(G) associated to any discrete group G. This process can “forget” much of the algebraic information about the group. For example, a celebrated result of Connes implies that any two discrete amenable groups all of whose nontrivial conjugacy classes are infinite yield the same von Neumann algebra. A famous open question in the subject is whether or not L(F_m) and L(F_n) are isomorphic for distinct m and n, where F_m denotes the nonabelian free group on m generators. In this talk, we will discuss some preliminary observations about the model-theoretic version of this question, which asks whether or not L(F_m) and L(F_n) are elementarily equivalent for distinct m and n (which can be viewed as a noncommutative version of the famous Tarski problem, which asks whether or not F_m and F_n are elementarily equivalent and for which the problem is now known to have a positive solution). The work presented in this talk is joint with Jennifer Pi. We will assume no prior knowledge of von Neumann algebra theory. | ||||

Wednesday Jan 24 2024 | ||||

16:00-16:50 (MS 6221) | Asger Tornquist (University of Copenhagen) | Some set theoretic aspects of a model of the mind proposed in psychology | ||

Abstract. This talk is about a somewhat unusual topic for me to work with: A psychology professor in Denmark, Jens Mammen, has developed a "model of the human mind", which is formulated in terms of some simple mathematical objects: The model consists of a "universe", which is a set U (whose elements are "objects" or "individuals" in this theory; the objects in the universe are meant to represent the things/people/pets in the world that the mind can potentially sense or interact with: your car, your father, your cat, etc.), and additionally the model has two collections of subsets of the universe U, called C and S. The subsets of U which are elements of S are called the "sense" categories, and they represent broad categories that the mind can form (e.g., the category of all cats). Subsets of U which are in C are called "choice categories", and are supposed to represent those things in the universe the mind can single out (for instance, _your_ cat (or cats)) among all the things in the broad categories.
Mammen formulated a number of axioms that C and S must satisfy to reasonably represent the human mind, and then asked a number of questions about what sort of models were possible to have. Remarkably, several of these problems require some actual mathematics, and some problems even require resonably serious set theory. To appreciate why, note that when the universe U is countable, then C and S will be subsets of P(U), i.e. of Cantor space, so it makes sense to ask how definable in the descriptive set theoretic sense C and S can be and still satisfy Mammen's axioms (as well as additional desirable properties). Purely combinatorial problems (including using ultrafilters, AC, and cardinal invariants!) also appear when trying to obtain various kinds of models of Mammen's axiom system. In this talk, I'll give a brief overview of Mammen's theory of the mind, and then move on to discuss the set-theoretic problems that Mammen's theory poses, with an emphasis on the descriptive set theory side of things. Towards the end, I'll mention some open problems (of a mathematical nature) that are still around. | ||||

Wednesday Jan 17 2024 | ||||

16:00-16:50 (MS 6221) | Samaria Montenegro (Universidad de Costa Rica) | Fields and NTP2 | ||

Abstract. One of the objects of study of model theory are the complete first order theories and their classification. Shelah classified complete first order theories by their ability to encode certain combinatorial configurations. For example, the theories that are not able to encode linear orders are the stable theories. Shelah and others produced important results and techniques for analyzing types and models within this classification. In algebraic structures such as groups or fields, these model-theoretic properties are related to algebraic properties of the structure.
We are going to focus on the class of NTP2 theories (theories without the tree property of the second kind), Shelah defined this class in the 1980s and contains strictly the class of simple and NIP theories. We will focus on fields that are NTP2, we will explain the case of bounded PAC, PRC, and PpC fields. Then we propose a unified framework for studying these fields - the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a "local-global" principle for the existence of points on varieties based on models of T. This approach also enables a good description of some fields equipped with multiple V-topologies, particularly pseudo-algebraically closed fields with a finite number of V-topologies. We are going to show how we can use this approach to produce many new examples of NTP2 fields. Part of this talk is a joint work with Silvain Rideau-Kikuchi. | ||||

Wednesday Nov 29 2023 | ||||

16:00-16:50 (MS 6221) | Kyle Gannon (Peking University) | Generically stable idempotent measures in abelian NIP groups | ||

Abstract. Given a locally compact topological group, there is a correspondence between idempotent probability measures and compact subgroups. An analogue of this correspondence continues into the model theoretic setting. In particular, if G is a stable group, then there is a one-to-one correspondence between idempotent Keisler measures and type-definable subgroups. The proof of this theorem relies heavily on the theory of local ranks in stability theory. Recently, we have been able to extend a version of this correspondence to the abelian NIP setting. In this context, we prove that generically stable idempotent Keisler measures correspond to fsg subgroups. These results rely on recent work connecting generically stable measures to generically stable types over the randomization. This is joint work with Artem Chernikov and Krzysztof Krupinski. | ||||

Wednesday Nov 08 2023 | ||||

16:00-16:50 (MS 6221) | Joshua Frisch (UC San Diego) | Equivalence Relations Classifiable by Polish Abelian Groups | ||

Abstract. The theory of Borel reducibility gives a way to formalize when one equivalence relation is less complicated than another. Since the founding of the theory, a common leitmotif has been to analyze, in particular, how complicated those equivalence relations coming from group actions are. In this talk, I will discuss some recent work joint with Forte Shinko about the following question: when does a countable Borel equivalence relation reduce to one generated by a Polish abelian group? | ||||

Wednesday Oct 25 2023 | ||||

16:00-16:50 (MS 6221) | Deirdre Haskell (McMaster University) | Residue field domination in some theories of valued fields | ||

Abstract. A paraphrase of the Ax-Kochen-Ersov theorem for some theories of valued fields is that the elementary theory is determined by the theory of the value group and the residue field. At the level of types, the intuition is that a type should be controlled by its trace in each of the residue field and value group.
In this talk, I will first talk about the algebraic structure of valued fields, and draw some pictures to provide some intuition for those who do not work with them all the time. Then I will explore some ways in which the intuition that stems from the AKE theorem can be made precise, and also some limitations to that preliminary intuition. I will try to give lots of examples to keep the discussion concrete. | ||||

Wednesday Oct 11 2023 | ||||

16:00-16:50 (MS 6221) | Jan Grebik (UCLA) | Complexity of Borel colorings | ||

Abstract. The fact that a finite graph admits a proper k-coloring of vertices, i.e., its chromatic number is at most k, does not necessarily mean that there is an efficient way to produce such a k-coloring -- this phenomenon has been intensively studied in various areas of mathematics and theoretical computer science. In recent years, Bernshteyn discovered formal connections between the existence of definable coloring of Borel graphs and the existence of an efficient algorithm in the so-called LOCAL model of distributed computing that produces such a coloring. Unlike in the finite setting, this result suggests that in the setting of descriptive graph combinatorics (an area of descriptive set theory that sits at the intersection of set theory, dynamics, and combinatorics, and studies definable analogues of graph properties on definable graphs) Borel chromatic number at most k is equivalent with the existence of efficient algorithm that produces a k-coloring. In this talk I will discuss a related question; how complex is the class of Borel graphs of Borel chromatic number at most k? In particular, is it possible to decide whether a given Borel graph of degree bounded by k satisfies the Borel analogue of Brook's theorem?
This is a joint work with Brandt, Chang, Grunau, Rozhon and Vidnyánszky. Zoom recording available here: https://ucla.zoom.us/rec/share/OAygOAntAAtDXFmtN03kOW7mdUnP9bt2v7T2F2acwZJrcWpDDcXwS_XlffILPRd1.wMAtWTRtMJYE-LjL | ||||

Wednesday Jun 07 2023 | ||||

16:00-16:50 (MS 6221) | Jinhe (Vincent) Ye (Oxford) | Curve-excluding fields | ||

Abstract. Given C a curve over $\mathbb{Q}$ with genus at least 2 and $C(\mathbb{Q})$ is empty, the class of fields K of characteristic 0 such that $C(K)=\emptyset$ has a model companion, which we call CXF. Models of CXF have interesting combinations of properties. For example, they provide an example of a model-complete field with unbounded Galois group, answering a question of Macintyre negatively. One can also construct a model of it with a decidable first-order theory that is not "large'' in the sense of Pop. Algebraically, it provides a field that is algebraically bounded but not "very slim" in the sense of Junker and Koenigsmann. Model theoretically, we find a pure field that is strictly NSOP_4. | ||||

Wednesday May 24 2023 | ||||

16:00-16:50 (MS 6221) | Elliot Glazer (Harvard) | Choiceless analysis of coin-flipping measures | ||

Abstract. In the presence of countable choice, one can construct for an arbitrary $X$ the completed product measure on $2^X$ (e.g., for $X=\omega,$ this is the Lebesgue measure). Quotienting out the null ideal, we then get a well-behaved measure algebra, on which we have the $L^p$ spaces. We show that under reasonable definitions, the basic theory of these measure algebras and the analysis of the corresponding $L^p$ spaces can be derived just in ZF. We will be particularly interested in the case of $X = \omega,$ which will allow us to make sense of the choiceless theory ZF + "all sets of reals are Lebesgue measurable" and verify it to be equiconsistent with ZF, despite the famous equiconsistency of ZF + DC + "all sets of reals are Lebesgue measurable" with an inaccessible. | ||||

Wednesday May 10 2023 | ||||

16:00-16:50 (MS 6221) | Jing Yu (Georgia Tech) | Large scale geometry of graphs of polynomial growth | ||

Abstract. In 1995, Levin and Linial, London, and Rabinovich conjectured that every connected graph $G$ of polynomial growth admits an injective homomorphism to the $n$-dimensional grid graph for some $n$. Moreover, they conjectured that if every ball of radius $r$ in $G$ contains at most $O(r^\rho)$ vertices, then one can take $n = O(\rho)$. Krauthgamer and Lee confirmed the first part of this conjecture and refuted the second in 2007. By constructing some finite expander graphs, they showed best possible upper bound on $n$ is $O(\rho \log \rho)$. Prompted by these results, Papasoglu asked whether a graph $G$ of polynomial growth admits a coarse embedding into a grid graph. We give an affirmative answer to this question. Moreover, it turns out that the dimension of the grid graph only needs to be linear in the asymptotic growth rate of $G$, which confirms the original Levin–Linial–London–Rabinovich conjecture "on the large scale." Besides, we find an alternative proof of the result of Papasoglu that graphs of polynomial growth rate $\rho < \infty$ have asymptotic dimension at most $\rho$. Furthermore, our proof works in the Borel setting and shows that Borel graphs of polynomial growth rate $\rho < \infty$ have Borel asymptotic dimension at most $\rho$. This is joint work with Anton Bernshteyn. | ||||

Wednesday Apr 26 2023 | ||||

16:00-16:50 (MS 6221) | Sean Walsh (UCLA, Philosophy) | Algorithmic randomness and Lévy's Upward Theorem | ||

Abstract. Much recent work in algorithmic randomness has concerned characterizations of randomness notions in terms of the almost-everywhere behavior of suitably effectivized versions of functions from analysis or probability. In this work, we examine the relationship between algorithmic randomness and Lévy's Upward Martingale Convergence Theorem, in the setting of arbitrary computable Polish spaces. We show that Schnorr randoms are precisely the points at which the conditional expectations of L^1-computable functions converge to their true value. This result has natural applications to formal epistemology and the philosophical interpretation of probability: for, the natural Bayesian interpretation of this result is that belief, in the form of an agent's best estimates of the true value of a random variable, aligns with truth in the limit, under appropriate effectiveness and randomness assumptions. We also consider other randomness notions such as Martin-Löf Randomness and density randomness. This is joint work with Simon M. Huttegger (UC Irvine) and Francesca Zaffora Blando (CMU). | ||||

Wednesday Apr 12 2023 | ||||

16:00-16:50 (MS 6221) | Pieter Spaas (Copenhagen) | Stable decompositions for countable equivalence relations | ||

Abstract. We will start with some motivation and background for the talk, and then discuss stable decompositions of a countable ergodic p.m.p. equivalence relation. We will explain the definition and show that the stabilization of any equivalence relation without central sequences in its full group (i.e. it is not ''Schmidt'') has a unique stable decomposition. This provides the first non-strongly ergodic such examples. | ||||

Friday Mar 24 2023 | ||||

16:00-16:50 (MS 6221) | Alexis Chevalier | An algebraic hypergraph regularity lemma | ||

Abstract. In "Expanding polynomials over finite fields…" (2012), Tao proves the algebraic regularity lemma. This is a strong form of the Szemeredi regularity lemma for definable graphs in the language of rings in finite fields. The algebraic regularity lemma improves the Szemeredi regularity lemma by providing definable regular partitions of definable bipartite graphs which have no irregular pairs and such that the error bounds on regularity vanish as the size of the finite field grows.
Tao asks if the algebraic regularity lemma can be extended to definable hypergraphs, in the same way that the Szemeredi regularity lemma extends to hypergraphs in the style of Rodel and Skokan (2004) or Gowers (2006). We answer this question positively by giving a new analysis of the algebraic regularity lemma. We use the model theory of pseudofinite fields to relate the combinatorial notion of regularity (for graphs and for hypergraphs) to Galois-theoretic information associated to definable sets. With this new analysis in hand, the algebraic hypergraph regularity lemma follows by classical results of Gowers, albeit with some interesting technical details. | ||||

Friday Mar 10 2023 | ||||

16:00-16:50 (MS 6221) | Tom Benhamou | The Galvin property and its applications | ||

Abstract. We present a property of filters discovered by F. Galvin which he proved to hold for normal filters over strongly regular cardinals, and which gained renewed interest due to recent developments in set theory. In the first part of the talk, we will provide applications of this property. The second goal will be to discuss a strengthening of Galvin's theorem, and the situation in some canonical inner models. We will also present relevant constructions of filters and ultrafilters without the Galvin property, answering several questions. If time permits, we shall present extensions of the work of U. Abraham and S. Shelah, who produced a model where the club filter fails to satisfy the Galvin property in a strong sense at $\kappa^+$, where $\kappa$ is a regular cardinal and $2^{\kappa}>\kappa^+$. We will produce a model where the club filter fails to satisfy the Galvin property at $\kappa^+$, where $\kappa$ is singular and $2^{\kappa}>\kappa^+$. We will obtain this model from the optimal large cardinal assumptions and explore the possibility of obtaining the stronger form of failure as in the Abraham and Shelah model. This is partially a joint work with M. Gitik, S. Garti, and A. Poveda. | ||||

Friday Feb 24 2023 | ||||

16:00-16:50 (MS 6221) | Srivatsav Kunnawalkam Elayavalli | Generic algebraic properties in spaces of enumerated groups | ||

Abstract. We will introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate the natural question: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer the question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. This is joint work with Goldbring and Lodha. | ||||

Friday Jan 27 2023 | ||||

16:00-16:50 (MS 6221) | Asger Tornquist | Maximal orthogonal families of probability measures: An overview | ||

Abstract. Let $X$ be a Polish space. Two Borel measures $\mu$ and $\nu$ on $X$ are called orthogonal if there is a Borel set $B\subseteq X$ which is null or $\mu$ and co-null for $\nu$. In the early 1980s, Daniel Mauldin asked if a maximal orthogonal family (“mof”) of probability measures on an uncountable Polish space $X$ can be analytic, and this was quickly answered in the negative by Rataj and Preiss (1985), who used Baire category methods to prove this. Over the years, other proofs of this result have been given, most notably by Kechris and Sofronidis, who gave an elegant proof using turbulence; and recently, gave a simplified version of Rataj and Preiss proof that relies on the Kuratowski-Ulam theorem rather than using a Banach-Mazur game. Some other known proofs use other notions of regularity than Baire category, such as completely Ramsey and Lebesgue measurability. On the other hand, David Schrittesser and I proved that Baire category can’t be replaced by Sacks and Miller measurability to prove that there are no mofs in the lower rungs of the projective hierarchy (Pi-1-1 and Sigma-1-2). In this talk, I will give an overview of the subject. | ||||

Friday Jan 13 2023 | ||||

16:00-16:50 (MS 6221) | Andrew Marks | A dichotomy characterizing piecewise Baire class $\alpha$ functions | ||

Abstract. In the 1920s, Lusin asked whether every Borel function on
$2^\omega$ is a union of countably many partial continuous functions (i.e. whether every Borel function is piecewise continuous). This question has a negative answer; an example of a non-piecewise continuous Borel function is the Turing jump. This is the only counterexample in one sense. Solecki and Zapletal have shown that every Borel function $f$ is either piecewise continuous, or the Turing jump continuously reduces to $f$.
We generalize the Solecki-Zapletal dichotomy throughout the Borel hierarchy. Recall that a Borel function is Baire class $\alpha$ if and only if it is $\mathbf{\Sigma}^0_{\alpha+1}$ measurable. We show that every Borel function $f$ is either piecewise Baire class $\alpha$, or the complete Baire class $\alpha+1$ function (an appropriate iterate of the Turing jump) continuously reduces to $f$. Our proof uses an adaptation of Montalban's game metatheorem for priority arguments to boldface descriptive set theory. This is joint work with Antonio Montalban. | ||||

Friday Dec 02 2022 | ||||

16:00-16:50 (MS 6221) | Don Stull | Pinned distance sets using effective dimension | ||

Abstract. Recent work has shown that effective techniques can be used to understand problems in (classical) geometric measure theory. An important open problem in geometric measure theory is to prove strong lower bounds on the Hausdorff dimension of pinned distance sets. Given a set E in the plane, and a point x, the pinned distance set of E with respect to x is the set of all distances between x and the points in E. In this talk, I will discuss how we can use effective methods to improve the bounds on the dimension of pinned distance sets. | ||||

Friday Nov 18 2022 | ||||

16:00-16:50 (MS 6221) | Scott Mutchnik | NSOP_2 Theories | ||

Abstract. Model theory has been described as a "geography of tame mathematics," creating a map of the universe of first-order theories according to various dividing lines, such as tree properties or order properties. While some regions of this map, such as the stable theories or simple theories, are well-understood to varying degrees, as we progress outward it even becomes open whether some regions are empty or not. Extending the NSOP_n hierarchy of Shelah [1995] defining an ascending chain of strong order properties for n > 2, Dzamonja and Shelah [2004] introduce two further tree properties, NSOP_1 and NSOP_2, and ask whether the implications between NSOP_1 and NSOP_2 and between NSOP_2 and NSOP_3 are strict. We have answered the first of these questions, showing that the class NSOP_1 coincides with NSOP_2. We discuss this result and some aspects of its proof, which incorporates ideas from various other regions of the model-theoretic map such as the NSOP_1, NSOP_3 and NTP_2 theories. | ||||

Friday Nov 04 2022 | ||||

16:00-16:50 (MS 6221) | Garrett Ervin | Decomposing the real line into everywhere isomorphic suborders | ||

Abstract. We show that it is impossible to decompose the real line (R, <) into two suborders that are everywhere isomorphic. That is, if R = A U B is a partition of R, then there is an open interval I such that A's restriction to I is not order-isomorphic to B's restriction to I. The proof depends on the completeness of R, and it turns out that in contrast there does exist a partition of the irrationals R - Q = A U B such that A and B are isomorphic on every open interval. I do not know whether it is possible to decompose R into three suborders that are everywhere isomorphic. | ||||

Wednesday Oct 26 2022 | ||||

16:00-16:50 (MS 6221) | Mariana Vicaría | Elimination of imaginaries in ordered abelian groups | ||

Abstract. I will present the current picture of the model theoretic study of ordered abelian groups. Their classification from a combinatorial point of view, results on quantifier elimination and model completeness. I aim to explain two main results on elimination of imaginaries in ordered abelian groups with finite spines, a class including the strongly dependent, dp-minimal and definably complete OAG.
No prior knowledge of advanced model theory will be assumed and everyone is very welcome to join. | ||||

Friday Oct 07 2022 | ||||

16:00-16:50 (MS 6221) | Meng-Che "Turbo" Ho | Torsion-free abelian groups of finite rank and fields of finite transcendence degree | ||

Abstract. In descriptive set theory, Borel reducibility is used to study the complexities of classes of countable structures. A classical example is the isomorphism problem on the class $TFAb_r$ of torsion-free abelian groups of rank r. Baer gave a simple invariant for $TFAb_1$, i.e., when two torsion-free abelian groups of rank 1 are isomorphic. On the other hand, Hjorth showed that $TFAb_1 <_B TFAb_2$ and Thomas generalized this to show that $TFAb_r <_B TFAb_{r+1}$. Recently, Paolini and Shelah, and independently Laskowski and Ulrich, showed that the class of torsion-free abelian group with domain $\omega$ is Borel complete.
The class $FD_r$ of fields over $\mathbb{Q}$ of finite transcendence degree r shares many features with $TFAb_r$. For instance, there is an r-tuple over which every element in the field is algebraic (definable in the case of groups). We compare the class of torsion-free abelian groups and the class of fields using the notion of Turing computable embedding defined by Knight, Miller, and Vanden Boom, and computable functors defined by Miller, Poonen, Schoutens, and Shlapentokh. In particular, we show that there are functorial Turing computable embeddings from $TFAb_r$ to $FD_r$ and from $FD_r$ to $FD_{r+1}$. Unlike in the results by Hjorth and Thomas, we do not know if these embeddings are strict. However, we show that under the computable countable reduction, these classes are all bi-reducible. This is joint work with Julia Knight and Russell Miller. | ||||

Friday May 22 2020 | ||||

15:00-15:50 (https://ucla.zoom.us/j/672060601) | Henry Towsner (University of Pennsylvania) | Removal and Amalgamation | ||

Abstract. (https://ucla.zoom.us/j/672060601)
The key step in the proof of the triangle removal lemma can be viewed as saying that we can identify a small number of edges in a graph as being the "exceptional" edges, and the remaining edges are sufficiently "representative of the neighborhood around them" that, if there are any triangles left, there must have been many triangles. This can be viewed as a amalgamation problem in the sense of model-theory: given types p(x,y), q(x,z), and r(y,z), each of which indicates that there is an edge between the vertices, when are the types p,q,r "large" in a way which guarantees that there are many (x,y,z) extending each of these types? The exceptional types can be characterized as the non-Lebesgue points - that is, the points which fail to satisfy the Lebesgue density theorem in the right measure space. We give a way to generalize this to types of higher arity and use this to prove a new generalization, an "ordered hypergraph removal lemma", extending the recent ordered graph removal lemma of Alon, Ben-Eliezer, and Fischer. | ||||

Friday Apr 10 2020 | ||||

16:00-16:50 (https://zoom.us/j/8244003061) | Matthew Foreman (UC Irvine) | Attacking Classical Problems in Dynamical Systems with Descriptive Set Theory | ||

Abstract. In his classical 1932 paper, von Neumann asked 3 questions: Can you classify the statistical behavior of differentiable systems? Are there systems where time-forward is not isomorphic to time-backward? Is every abstract statistical system isomorphic to a differentiable system? These questions can be addressed with some surprising consequences by embedding them in Polish Spaces. Indeed the tools answer other questions from the 60's and 70's such as the existence of diffeomorphisms with arbitrary Choquet simplexes of invariant measures. Moreover there are surprising analogues to Hilbert's 10th problem.
In a different category, building on work of Poincar\`{e}, Smale proposed classifying the \emph{qualitative} behavior of differentiable systems on compact manifolds. His 1967 paper explicitly argued that the equivalence relation of ``conjugacy up to homeomorphism" captures this notion and he proposes classifying it. Call this notion \emph{topological equivalence}. Very recent joint results with A. Gorodetski show: - The equivalence relation $E_0$ is Borel reducible to topological equivalence of diffeomorphisms of any smooth 2-manifold. - The equivalent relation of \emph{Graph Isomorphism} is Borel reducible to topological equivalence of diffeomorphisms of any smooth manifold of dimension 5 or above. As corollaries, none of the classical numerical invariants such as entropy, rates of growth of periodic points and so forth, can classify diffeomorphisms of 2-manifolds, and there is no Borel classification at all of diffeomorphisms of 5-manifolds. In the same 1967 paper Smale asks (in different language) whether there is a generic class that can be classified. This is still an open problem. Link to the talk: https://zoom.us/j/8244003061 | ||||

Friday Mar 13 2020 | ||||

16:00-16:50 (MS 6221) | Anand Pillay (University of Notre Dame) | CANCELLED | ||

Friday Feb 21 2020 | ||||

16:00-16:50 (MS 6221) | Jose G. Mijares (CalState LA) | Metrically Baire sets and the Ramsey Property | ||

Abstract. There exist topological Ramsey spaces admitting metric projections where every Baire set has the Ramsey property. Jointly with N. Dobrinen, we gave a characterization of such spaces, answering a question of S. Todocervic. In this talk, we will discuss the characterization and present some examples (see https://drive.google.com/open?id=1l35j19JP9B6-fskoSc4I91VLnV-XABxw for the references). | ||||

Friday Feb 14 2020 | ||||

16:00-16:50 (MS 6221) | Katrin Tent (University of Muenster) | Automorphism groups of order and tournament expansions | ||

Abstract. Introducing the notion of a stationary independence relation, Tent and Ziegler developed a framework for proving abstract simplicity of automorphism groups of a broad range of mathematical structures.
In current work with Calderoni and Kwiatkowska we are extending this to cover in particular order and tournament expansions of such structures and show that their automorphism groups are again simple groups. | ||||

Friday Feb 07 2020 | ||||

16:00-16:50 (MS 6221) | Natasha Dobrinen (University of Denver) | Strong coding trees and Ramsey theory on ultrahomogeneous structures | ||

Abstract. Strong coding trees were invented in order to solve the problem of whether or not the triangle-free Henson graph has analogues of Ramsey's theorem for colorings of finite triangle-free graphs. Since then, the method has been extended to handle all k-clique-free Henson graphs. Further, it has been useful for extending Ellentuck's infinite dimensional Ramsey theory to the rationals, and for extending the Galvin-Prikry theorem to the Rado graph. We will present some of the main ideas in these results, the history of the area, and some future directions. | ||||

Friday Jan 24 2020 | ||||

16:00-16:50 (MS 6221) | Aristotelis Panagiotopoulos (Caltech) | Definable (co)homology and classification of solenoids | ||

Abstract. We will develop a framework for enriching various classical invariants from algebraic topology with descriptive set-theoretic information. Applying these ideas to
Steenrod homology theory we get a new invariant for compact metrizable spaces up to homotopy equivalence which we call ``definable homology.'' Similarly we get a dual notion of ``definable cohomology'' for locally compact metrizable spaces by enriching the classical \v{C}ech cohomology theory. Our invariants are strictly finer than the original ones.
In particular, we will show that $n$-dimensional (co)solenoids are completely classified up to homeomorphism by their definable (co)homology. In the process, we will generalize Veli\v{c}kovi\'c's rigidity theorem for definable automorphisms of $\mathcal{P}(\omega)/\mathrm{fin}$, to the arbitrary quotient $G/N$, where $G$ is a locally pro-finite abelian group and $N$ is a countable dense subgroup of $G$, and we will develop some basic definable homological algebra.
This is a joint work with J. Bergfalk and M. Lupini. | ||||

Friday Jan 10 2020 | ||||

16:00-16:50 (MS 6221) | Rizos Sklinos (Stevens Institute of Technology) | Fields definable in the theory of nonabelian free groups | ||

Abstract. In this talk I will show that only finite fields are definable in the theory of nonabelian free groups. This is joint work with Ayala Byron. | ||||

Friday Nov 15 2019 | ||||

16:00-16:50 (MS 6221) | Isaac Goldbring (UC Irvine) | The almost sure theory of finite metric spaces | ||

Abstract. In this talk, we show that the class of finite metric spaces (of diameter at most 1) has an almost sure theory in the following sense: for each sentence $\sigma$ in the language of metric spaces, there is a real number $r$ such that, for any $\epsilon>0$, for large enough finite metric spaces $X$, with high probability, the value of $\sigma$ in $X$ differs from $r$ by at most $\epsilon$. This almost-sure theory is in fact the theory of a particular metric space, which we call the almost-sure metric space AS, and which can be defined as the Fraisse limit of the class of all finite metric spaces with nontrivial distances in the interval [1/2,1].
This is joint work with Bradd Hart. | ||||

Friday Nov 01 2019 | ||||

16:00-16:50 (MS 6221) | Pieter Spaas (UCLA) | Almost invariant sets and stable equivalence relations | ||

Abstract. We are interested in structural properties of countable ergodic, but non-strongly ergodic, equivalence relations. Firstly, we will discuss a question posed by Jones and Schmidt in the 80s. In particular, we will provide examples and non-examples of equivalence relations all of whose almost invariant sets come from a single hyperfinite quotient. Secondly, we will study stable equivalence relations, i.e. those that can be written as a direct product of some equivalence relation with the hyperfinite one. We will then show that the stabilization of any strongly ergodic equivalence relation admits a unique stable decomposition in a precise sense. | ||||

Friday Oct 18 2019 | ||||

16:00-16:50 (MS 6221) | Mate Szabo (Université Paris 1 - Panthéon-Sorbonne) | Pepis' and Kalmár's Arguments Against Church's Thesis | ||

Abstract. In his famous paper, "An Unsolvable Problem of Elementary Number Theory," Alonzo Church (1936) identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. In this talk I consider two early arguments against it. Pepis criticized Church's Thesis already in 1937 in his dissertation and in a letter written to Church. I will display recently found documents showing Pepis's stance. The main focus of the talk will be László Kalmár's famous "An Argument Against the Plausibility of Church's Thesis" from 1959. As this paper is quite short, my aim will be to present Kalmár's argument and to fill in missing details based on his general philosophical thoughts on mathematics and his writings published on these issues in Hungarian. | ||||

Friday Oct 04 2019 | ||||

16:00-16:50 (MS 6221) | Erik Walsberg (UC Irvine) | Expansions of $(\mathbb{R},<,+)$ | ||

Abstract. I will survey recent work on first order expansions of $(\mathbb{R},<,+)$. Time permitting, I will discuss connections to automata theory and $\mathrm{NIP}$. |

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