Event Calendar

Two tracial von Neumann algebras are elementarily equivalent if they cannot be distinguished by first-order sentences or, more algebraically, if they have isomorphic ultrapowers. The same definition can be made for (countable, discrete) groups, and it is natural to wonder whether or not there is a connection between two groups being elementarily equivalent and their corresponding group von Neumann algebras being elementarily equivalent. In the first part of the talk, I will give examples to show that, in general, there is no connection in either direction. In the second part of the talk, I will introduce a strengthening of elementary equivalence, called back-and-forth equivalence (in the sense of computability theory) and show that back-and-forth equivalent groups have back-and-forth equivalent group von Neumann algebras. I will also discuss how the same is true for the group measure space von Neumann algebra associated to the Bernoulli action of a group on an arbitrary tracial von Neumann algebra. The latter half of the talk represents joint work with Matthew Harrison-Trainor.

This talk is based on a joint work with Jennifer Pi and Mira Tartarotti. One of the major goals of operator theory is to classify (a large class of) closed *-algebras of bounded operators on Hilbert spaces, i.e., to classify C*-algebras. In 1976 Elliott proved that a class of AF algebras can be classified by their ordered K-theory. This theorem was generalised to a bigger class of C*-algebras, called classifiable. We prove that there is a transfer of strategies between Ehrenfeucht-Fraïssé games (of various lengths) on classifiable C*-algebras and their invariants consisting of K-theory and traces answering a question from a work of Ben De Bondt, Andrea Vaccaro, Boban Velickovic and Alessandro Vignati.

Fix a class of groups (e.g., simple, perfect, nilpotent). Then one can easily construct automorphisms of products of groups from this class by permuting indices (with isomorphic projections) and considering automorphism of each coordinate individually. However, the natural question then arises: when does every automorphism of every product of groups from the class essentially decompose into the form described above? In general, we are interested in when classes of groups (or other structures) have such property with respect to all (reduced) products. Using model theoretic methods, one can show that certain natural families of groups have such property and that a total characterization is quite complicated. This is joint work with Ilijas Farah and Pierre Touchard.

The Sauer–Shelah lemma is a fundamental tool in the combinatorics of set systems. It says that if an $n$-element set system does not contain $d$ elements on which all $2^d$ possible intersections occur, then it has at most $n^d$ sets. We prove that if this set system comes from a graph in a monadically dependent class, then the bound can be improved from polynomial to almost linear, i.e. $n^{1+o(1)}$.

This research program aims to finitize Shelah's work on theories with NIP. The hope is that we can use these tools to prove a dichotomy conjecture about the computational complexity of first-order model-checking.

This is joint work with Jan Dreier, Nikolas Mahlmann‚ Michal Pilipczuk, and Szymon Torunczyk.

Historically, proofs of BPI in models without choice have used a contradiction framework introduced by Halpern. We introduce a framework for the direct proof of BPI in permutation models (of ZFA) and in symmetric extensions (of ZF). In permutation models, we prove that BPI holds if and only if there is a precise manner in which it can be seen to be inherited from an outer model of choice. We then generalize this method to prove BPI in the Cohen model, and discuss a new connection between this result and the Halpern-Lauchli theorem.

We examine relationships between covering properties, large cardinals, and the derived model. One natural question is whether PFA + kappa a limit of Woodin cardinals implies the derived model D(V,kappa) satisfies AD_R. A weaker conjecture would be that Theta of this derived model is below kappa^+. We make partial progress on each conjecture. This is joint work with Nam Trang, extending results of Trevor Wilson.

Using filtrations between topologies, we present a general form of Scott analysis for definable equivalence relations. We develop a theory using this notion. We show that the speaker's canonical approximations of Polishable subgroups, Hjorth's Scott analysis of Polish group actions, and the Scott analysis in continuous model theory described by Doucha, Ben Yaacov, Nies, and Tsankov are special cases, appropriately understood, of our approach. Parts of this work are joint with M. Lichman and J. Swaczyna.

Notions of reducibility between various kinds of structures play an important role in mathematical logic, especially in set theory and computability theory. Examples include Borel and Turing reducibility. However, the only model-theoretic notion of reducibility that I am aware of is interpretability, which is rather rigid. We introduce a novel notion of model-theoretic reducibility and discuss a recently developed connection with constraint satisfaction problems. Along the way we will attempt to convince the audience that the trivial theory of an infinite set with equality is surprisingly interesting.

Hardy fields are differential fields of (germs at infinity of) real-valued functions. Interest in them comes from several areas of mathematics, including asymptotic analysis, dynamical systems, and model theory of the real numbers (more precisely, o-minimality). The first-order theory of existentially closed Hardy fields is completely axiomatizable and model complete in the language of ordered valued differential fields, as M. Aschenbrenner, L. van den Dries, and J. van der Hoeven have shown in a long and impressive series of works; in particular, all maximal Hardy fields are elementarily equivalent. Moreover, each maximal Hardy field can be equipped with an elementary differential subfield that is Dedekind complete in the maximal Hardy field. Along the lines of tame pairs of real closed fields (or tame pairs of o-minimal fields, more generally), the theory of such pairs is axiomatized by the notion of a transserial tame pair, the subject of this talk. After introducing these objects, I will explain how they are topologically tame, for example satisfying a definable Baire Category Theorem. Although self-contained, this is a sequel to my talk "Tame pairs of transseries fields" at the UCLA Logic Colloquium of October 2, 2024.

Many concepts in set theory are considered through the lens of infinite games. One particular genre goes at least back to work of Ulam and others in the 1960's, and is sometimes called the cut and choose game. The focus of this talk will be on some forcing arguments that shed some light on problems related to the version of this game for complete boolean algebras. The context is a 1995 theorem of Zapletal that, assuming the consistency of a supercompact cardinal, it is consistent that for all $\le \omega_1$-distributive boolean algebras, the choosing player has a winning strategy. In particular, we answer a question of his by extending the result to include $(\lambda,2)$-distributive boolean algebras for $\lambda$ a successor of a regular cardinal.

Measures on uncountable cardinals play a central role in both large cardinal theory and determinacy theory. The Ultrapower Axiom (UA) is a structure principle for such measures abstracted from inner model theory. Though independent of ZFC, UA holds in all known canonical inner models of large cardinals. This talk concerns the recent result that UA also holds in natural models of ZF plus the Axiom of Determinacy, far from the ZFC context in which it was originally formulated. We plan to present aspects of the proof, which relies on a generic ultrapower construction due to Woodin and the Kechris coding theorem for measures. We also hope to discuss some corollaries and speculate about the role of UA in the theory of very large cardinals.

Dana Scott's ordinal analysis of countable structures (models) in a countable languages is a well-known and celebrated tool for analyzing the isomorphism classes of countable structures. It is well-known that these ideas can be put into the context of descriptive set theory by introducing the Polish space of L-structures with a countable universe. In that setting, Scott's analysis gives us Borel descriptions of the isomorphism classes, and more generally, descriptions of Borel sets that are invariant under the "logic action" of the infinite symmetric group $S_\infty$ (see e.g. 16.C in Kechris' Classical Descriptive Set Theory).

All of the above can be viewed as facts about the logic action of $S_\infty$. Taking the group action view, Greg Hjorth generalized the Scott analysis to Polish group actions. In this talk I will discuss a different approach to Hjorth's Scott analysis of Polish group actions.

It is known that the so called pseudoarc can be represented as a quotient of a zero dimensional compact "prespace" under an appropriate equivalence relation due to Irwin–Solecki (which is an inverse limit of linear graphs), and the automorphisms of this prespace densely embeds into the homeomorphism group of the pseudoarc. Although this embedding is only continuous, not a homeomorphic embedding, we can actually characterize the topology inherited from the homeomorphism group intrinsically, only in terms of the prespace. Using this characterization we prove that not all homeomorphisms are conjugate to an automorphism. Moreover, we generalize theorems of Kechris–Rosendal to characterize when the homeomorphism group of such a continuum (i.e. one that can be represented via a prespace) admits a dense or comeager conjugacy class, and we improve a theorem of Bice–Malicki showing that the diagonal action of the homeomorphisms of the pseudoarc on its countable product admits a dense conjugacy class. Joint with Slawomir Solecki.

A tremendous amount of mathematics concerns groups acting on sets and the primitive permutation groups are the building blocks of all group actions. A primitive permutation group (X,G) is a group G together with an action of G on X such that there are no nontrivial equivalence relations on X preserved by G. An important line of work in model theory attempts to classify primitive permutation groups that arise in certain tame settings. For example, a rough classification of primitive permutation groups of finite Morley rank has been carried out by Macpherson and Pillay and this classification was then used by Borovik and Cherlin to prove that if (X,G) is a primitive permutation group of finite Morley rank, the rank of G can be bounded in terms of the rank of X. What made this work possible was a combination of progress on the Cherlin-Zilber conjecture, which predicts that simple groups of finite Morley rank are algebraic, and the remarkably nice properties of Morley rank. We study an analogous situation for pseudo-finite primitive permutation groups of finite SU-rank, making the problem at once harder and easier: the (known) classification of finite simple groups replaces the (wide open) Cherlin-Zilber conjecture, at the cost of working with a more unwieldy notion of rank. This is joint work with Ulla Karhumäki.

The Zilber Trichotomy is a highly influential program in modern model theory: roughly, it refers to the general expectation that sufficiently model-theoretically tame structures, if non-trivial, should always arise from modules and fields. Put another way: any sufficiently tame model-theoretic setting should admit a linear/non-linear dividing line, so that non-trivial structures in that setting only arise from (a) modules, if on the linear side, and (b) fields, if on the non-linear side. The trichotomy was classically stated in the 1970s-1980s for combinatorially tame (e.g. stable) structures, but turned out to be false in essentially every possible formulation; the only truly abstract stable setting where it worked was extremely restrictive and required the structure be controlled by a well-behaved system of Noetherian topological spaces. Later (1990s), a true (and simpler) trichotomy theorem was found for ordered (o-minimal) structures: these are unstable structures, but they similarly carry a natural topology, and this topology leads to `stability-like’ phenomena in the class of definable sets. In this talk, I propose that the Zilber trichotomy is really a topological phenomenon rather than a stability-theoretic one, and thus the abstract study of the Zilber trichotomy should proceed in a general setting of topological structures. As a first step, I will discuss a topological linear/non-linear dividing line, paired with a theorem giving the strongest possible classification for structures on the linear side. This is joint work with Assaf Hasson.

A random rooted graph is said to be sofic if it is the Benjamini-Schramm limit of a sequence of finite graphs. Perhaps surprisingly, our understanding of which graphs are sofic is still quite limited. For starters, sofic graphs are known to possess a certain property known as unimodularity. (Unimodular random rooted graphs can also be encoded by graphings of pmp Borel equivalence relations.) However, in a recent breakthrough, Bowen, Chapman, Lubotzky and Vidick have shown that not all unimodular graphs are sofic. In this talk, I will give an overview of what is known in the other direction: Which additional conditions on the graph are known to imply soficity? Two important properties which I will talk about here are hyperfiniteness and treeability. Then, I will discuss a novel result along these lines: For any finite graph H, every one-ended, unimodular graph which does not have H as a minor must be sofic. The proof of this result proceeds by showing that all unimodular graphs of this kind are "almost" treeable.

Countable first-order structures are classified up to isomorphism according to their Scott sentences. There does not exist analogous complete classification for uncountable models. The study of wide Aronszajn trees - trees of size $\aleph_1$ with no uncountable branches - arose in an attempt to extend the Scott analysis to models of size $\aleph_1$. I will discuss the combinatorics of the class of wide Aronszajn trees as well as its implications in the structure of uncountable models. This is joint work with Omer Ben-Neria, Menachem Magidor and Jouko Väänänen.

I will discuss a joint work with Anand Pillay, in which we give an "arithmetic" version of Tao's "algebraic regularity lemma" for graphs definable in finite fields; in particular we show that, if $F$ is a finite field, and $G$ is a definable group in $F$ and $D\subseteq G$ a definable subset, both of bounded complexity, then $G$ has a definable normal subgroup $H$ of bounded complexity and index such that, for any cosets $V,W$ of $H$, the bipartite graph $(V,W,xy^{-1}\in D)$ is quasirandom. I will largely discuss the result's statement and the broader context for it, including Tao's original theorem and other arithmetic regularity results in the literature.

A semi-retraction is a pair of maps g from structure A to structure B, and f from B back to A again, whose composition fg is an embedding. In this talk, I will describe how semi-retractions transfer partition properties from B to A and give examples showing the necessity of certain assumptions.

Namba forcing was originally devised to show that $\aleph_2$ can be forced to be an ordinal of countable cofinality without collapsing $\aleph_1$, but has since found many other uses. We will discuss some applications of Namba forcing to the theory of singular cardinals like $\aleph_\omega$ and their successors, namely pertaining to the distinction between so-called good points---which indicate a well-behaved enumeration of elements of a product e.g. $\prod_{2 \le n < \omega}\aleph_n$ in length $\aleph_{\omega+1}$---and approachable points---which indicate a well-behaved expression of elements of $[\aleph_{\omega+1}]^{<\aleph_\omega}$ in length $\aleph_{\omega+1}$. In joint work with Heike Mildenberger last year, we began to delve into the tension between these concepts. In joint work with Hannes Jakob this year, we forced a distinction between good and approachable points in $\aleph_{\omega+1}$, and we even obtained non-approachable points of arbitrarily high cofinality, answering longstanding open questions in PCF theory.

In joint work with Gianluca Basso, we explore the class of Polish groups whose universal minimal flows admit a comeager orbit. By work of Ben Yaacov, Melleray, and Tsankov, this class contains all Polish groups with metrizable universal minimal flow, and by an example of Kwiatkowska, this inclusion is strict. We isolate the correct generalization of this class of Polish groups to the class of all topological groups. We call these the topological groups with "tractable minimal dynamics (TMD)." One way of phrasing what makes this class "tractable" is an "abstract Kechris-Pestov-Todorcevic correspondence" which characterizes membership in TMD using a Ramsey-theoretic property of the group. In particular, this implies that TMD is absolute between models of set theory. We also state some conjectures to the effect that any topological group not in TMD has "wild" minimal dynamics.

An $L$-theory $T$ is "finitely undecidable" if and only if every nonempty finitely axiomatised subtheory of $T$ is undecidable. In 1982, Ziegler proved specific field theories have this property; during my PhD, I proved large classes of field theories have this property too. (And consequently, if you assume certain model-theoretic conjectures, all NIP, simple, or superrosy fields have a finitely undecidable $L_r$-theory.) In this talk, I will indicate the highlights of proving finite undecidability in number-theoretic and algebro-geometric fields, and discuss quantitative statements one can make following from these results (such as: "almost all subfields of $\tilde{\mathbb{Q}}$ or $\tilde{\mathbb{F}_p}$ have a finitely undecidable $L_r$-theory").

The Ultrapower Axiom (UA) states that any pair of ultrapowers, taken via σ-complete ultrafilters, can be compared by taking further internal ultrapowers. In recent years, UA has been extensively studied by Goldberg, leading to a series of striking results concerning the structure of the set-theoretic universe. One such result is that, under UA, the class of all σ-complete ultrafilters is well-ordered by a natural order originally introduced by Ketonen. Goldberg observed that UA is, in fact, equivalent to the linearity of Ketonen’s order. He also proved that Ketonen relations between pairs of ultrafilters imply that certain games involving these ultrafilters are determined. The determinacy of these games is used in descriptive set theory to define a partial order on subsets of Cantor space (2^κ for some cardinal κ), known as the Lipschitz order. In this talk, we will present the Ketonen and Lipschitz orders, explore the relationships between them and other natural orders such as the Mitchell order and the Rudin–Keisler order, and, time permitting, outline a recent result demonstrating that the Ketonen and Lipschitz orders can be separated— answering a question posed by Goldberg.

In 1926, Lindenbaum proved the striking fact that the class (LO, +) of linear orders equipped with the operation of ordered sum has a Euclidean algorithm. The only published proof of this result did not appear until some 30 years later, after Lindenbaum's death, in a book by Tarski. Tarski's proof is surprisingly difficult, and does not clearly relate the Euclidean algorithm in (LO, +) to its familiar analogues in (N, +) or (R, +).

In the 1960s and 1970s, Holland, McCleary, and others began the systematic study of automorphism groups of linear orders, building on earlier work of Hoelder and Conrad concerning Archimedean ordered groups. These authors were seemingly unaware of Lindenbaum's earlier results or of any relation between their work and the arithmetic of (LO, +).

In recent joint work with Eric Paul, we bring these threads together. We give a new proof of Lindenbaum's theorem using the theory of automorphism groups of linear orders that gives much more structural information than Tarski's, and directly relates arithmetic in (LO, +) to arithmetic in the ordered group (R, +). In the other direction, we show how combinatorial facts about sums of linear orders can be used to give new proofs of some of Holland's and McCleary's algebraic theorems. In this talk I will discuss the history of these results and give an overview of our approach.

This is on joint work with John Baldwin and James Freitag. One of the central projects of model theory, initiated by Shelah in his book "Classification Theory," is to classify unstable first-order theories. As part of this program, Koponen proposes to classify simple homogeneous structures, such as the random graph. More precisely, she conjectures (2016) that all simple theories with quantifier elimination in a finite relational language are supersimple of finite rank, and asks (2014) whether they are one-based. In this talk, we discuss our resolution of the Koponen conjecture, where we show that the answer to this question is yes. In the process, we further demonstrate what Kennedy (2020) calls "the fragility of the syntax-semantics distinction."

A difference field is a field equipped with a distinguished automorphism and a difference variety is the natural analogue of an algebraic variety in this setting. Complex numbers with complex conjugation or finite fields with the Frobenius automorphism are natural examples of difference fields. For finite fields and varieties over them, the celebrated Lang-Weil estimate gives a universal estimate of number of rational points of varieties over finite fields in terms of several notions of the complexities of the given variety. In this talk, we will discuss an analogue to the Lang-Weil estimate for difference varieties in finite difference fields. The proof uses the model theory of pseudofinite difference fields and $\omega$-increasing valued difference fields. Particularly, some interesting interactions between the non-standard Frobenius and the non-archimedean topology occur. This is joint work with Martin Hils, Ehud Hrushovski and Tingxiang Zou.

O-minimal structures satisfy many "tame topology" properties, such as a cell decomposition theorem for definable sets, generic continuity for definable functions, and a sensible notion of dimension for definable sets. It turns out that these tame topology properties hold in a number of other settings, such as weak o-minimality, C-minimality, P-minimality, and (more recently) Hensel minimality.

One very weak condition, which generalizes all these, is "topological minimality" (t-minimality) in the sense of Mathews. A theory is t-minimal if there is a definable topology on models M such that no points are isolated, and any definable subset D ⊆ M has finite boundary. A rudimentary amount of "tame topology" continues to work for t-minimal theories.

This talk will discuss an ongoing investigation into definable fields in t-minimal theories. Although t-minimality is a very weak assumption, one can somehow prove non-trivial things about definable groups and fields. For example, if K is a definable field in a t-minimal theory, then K must be perfect. Under some weak assumptions (which I hope to remove), K must also satisfy a technical condition called "largeness", which rules out possibilities like number fields and function fields. Beyond these, I expect there to be more substantial constraints on K, as I will discuss.

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility in much the same way that the tree property characterizes weak compactness. That is, an inaccessible cardinal κ is strongly compact if and only if the strong tree property holds at κ, and supercompact if and only if ITP holds at κ. It is a longstanding project in set theory to obtain large-cardinal properties, like the strong tree property or ITP, at all possible cardinals simultaneously. I will discuss several results related to this program.

TBD

A fourfold classification of Subject quantifiers and their basic logical properties Intersective, Co-Intersective, Proportional, Definite All satisfy one general, strong, constraint: Conservativity Natural languages are provably not sortally reducible Logical subclasses given by PI (Permutation Invariance) & BCL (Boolean closure) e.g. Over finite domains Cardinal quantifiers = Intersective + PI BCL(Intersective Co-intersective) = CONS (the set of conservative quantifiers) Two novel entailment paradigms: e.g. Midpoint QPs hold of predicates and their negations (QxPx and QxPx) are never logically equivalent for Q = or )

Object Quantifiers (The arguments of P2s, two place predicates, behave asymmetrically) Subject QPs extend, and new types arise: nominal and predicate anaphors including one new self-dual quantifier One new entailment paradigm (Facing Negations Theorem) using P2s: For what QPs F,G,F’,G’ do F(G(R)) = F’(G’(R)), all binary relations R?

Open Questions: Can we find an elegant characterization of expressible functions from P2s to P1s? From P2s directly to {0,1}? (e.g. Different people like different things). Is the richness of Object quantifiers related to the Boolos & Jeffrey theorem on increased expressive power of FOL when P2s are added to monadic FOL?

We consider "local" combinatorial problems on graphs. I.e, problems in which we seek a global labelling of vertices, edges, etc. satisfying some set of local constraints. Typical examples include proper coloring and perfect matching. We are moreover interested in finding solutions which are in some sense "constructive" or "definable". We will focus on two specializations of this: finding Baire measurable solutions for Borel graphs on Polish spaces, and finding computable solutions for computable graphs on the natural numbers. Recent investigations have uncovered a large number of similarities between these two settings, but there are interesting questions about how deep the relationship really is. We will attempt to survey some recent positive and negative results in this direction.

Includes joint work with Berlow, Bernshteyn, Bowen, Conley, Lyons, and Qian.

Transseries emerged in connection with Écalle's work on Dulac's problem and Dahn and Göring's work on nonstandard models of real exponentiation, and some can be viewed as asymptotic expansions of solutions to differential equations. More recently, Aschenbrenner, Van den Dries, and Van der Hoeven completely axiomatized the elementary theory of the differential field of (logarithmic-exponential) transseries and showed that it is model complete. This talk concerns pairs of models of this theory such that one is a tame substructure of the other in a certain sense. I will describe the model theory of such transserial tame pairs, including a model completeness result for them, which can be viewed as a strengthening of the model completeness of large elementary extensions of the differential field of transseries, such as hyperseries, surreal numbers, or maximal Hardy fields.

It has been known for some time that any group definable in an o-minimal expansion of the real field can be endowed definably with the structure of a Lie group, and that any definable homomorphisms between definable groups is a Lie homomorphism (under the above mentioned Lie structure).

In this talk we explore the converse: We will characterize when a Lie group has a Lie isomorphic group which is definable in an o-minimal expansion of the real field, when Lie isomorphisms between such definable groups is definable, and whether one can achieve a definable Lie analytic structure in any such definable group.

My attempt is to give the talk with almost no assumptions on background, other than first order logic. In particular I will recap most of the relevant algebraic and geometric definitions.

O-minimal structures have been extensively studied as a framework for 'tame topology', in particular in terms of the underlying euclidean (order) topology. In addition, studies of the topological nature of various definable objects, such as groups, manifold spaces, orders, function spaces and metric spaces, have also been key to the development of o-minimality. Our work is directed towards a more general understanding of the nature of topological spaces definable in o-minimal structures (where, in any model-theoretic structure, a 'definable topological space' is a definable set together with a (uniformly) definable family forming a basis for a topology on that set). So far, our focus has mainly been on one-dimensional definable topological spaces but, even in this setting, there are examples exhibiting a wide variety of topological properties, including various classical topological counterexamples.

We present a number of classification results given in terms of decomposition and embedding theorems and, in parallel, identify suitable definable analogues of classical properties such as separability, compactness and metrizability. This leads to a variety of applications, including definable versions of conjectures from classical topology due to Gruenhage and Fremlin (on the nature of regular Hausdorff and perfectly normal compact Hausdorff spaces), as well as universality results for certain classes of spaces. This is part of a long-term joint project with Pablo Andújar Guerrero and Erik Walsberg, and intersects with work carried out independently by Peterzil and Rosel.

In this talk I will discuss realizations of countable Borel equivalence relations by continuous actions of

countable groups, focusing in particular on the problem of realization by continuous actions on compact

spaces and more specifically subshifts. This also leads to considering a natural universal space for actions

and equivalence relations via subshifts and the study of the descriptive and topological properties in this

universal space of various classes of countable Borel equivalence relations, especially the hyperfinite ones.

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

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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?

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.

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

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.