Abstract. Martin's conjecture is an attempt to make precise the idea that the only natural functions on the Turing degrees are the constant functions, the identity, and transfinite iterates of the Turing jump. The conjecture is typically divided into two parts. Very roughly, the first part states that every natural function on the Turing degrees is either eventually constant or eventually increasing and the second part states that the natural functions which are increasing form a well-order under eventual domination, where the successor operation in this well-order is the Turing jump.

In joint work with Benny Siskind, we prove part 1 of Martin's conjecture for a class of functions that we call measure-preserving. This has a couple of consequences. First, it allows us to connect part 1 of Martin's conjecture to the structure of ultrafilters on the Turing degrees. Second, we also show that every order-preserving function on the Turing degrees is either eventually constant or measure preserving and therefore part 1 of Martin's conjecture holds for order-preserving functions. This complements a result of Slaman and Steel from the 1980s showing that part 2 of Martin's conjecture holds for order-preserving Borel functions.

Abstract. A Polish group \(G\) is tame if for any continuous action of \(G\), the corresponding orbit equivalence relation is Borel. Extending results of Solecki, Ding and Gao showed that if \(G\) is a tame non-archimedean abelian group, then in fact all actions of \(G\) are potentially \(\Pi^0_6\). That is, they are Borel reducible to a \(\Pi^0_6\) orbit equivalence relation. They noted that all previously known examples of such actions were in fact potentially \(\Pi^0_3\), and conjectured that their upper bound could be improved to \(\Pi^0_3\). We refute this by finding an action of a tame non-archimedean abelian group which is not potentially \(\Pi^0_5\). This is joint work with Shaun Allison.

Abstract. A well known and long-standing open problem in the theory of Borel equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. Previous progress on this problem has been confined to groups possessing coarse euclidean geometry and polynomial volume growth (ultimately leading to a positive answer for groups that are either virtually nilpotent or locally nilpotent). In this talk I will discuss the coarse geometric notion of asymptotic dimension and its recently discovered applications to this problem. Relying upon the framework of asymptotic dimension, it is possible to both significantly simplify the proofs of prior results and uncover the first examples of solvable groups of exponential volume growth all of whose Borel actions generate hyperfinite equivalence relations. This is joint work with Clinton Conley, Steve Jackson, Andrew Marks, and Robin Tucker-Drob.

Abstract. Descriptive combinatorics is the study of combinatorial problems (such as graph coloring) under additional topological or measure-theoretic regularity restrictions. It turns out that there is a close relationship between descriptive combinatorics and distributed computing, i.e., the area of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. In this talk I will outline this relationship and present a number of applications.

Abstract. NOTE THE SEMINAR MEETS ONLINE AT https://ucla.zoom.us/j/672060601. PLEASE MUTE YOUR MICROPHONE EXCEPT WHEN SPEAKING.

An important way to study the complexity of Borel equivalence relations is in terms of whether they can be written as increasing unions of simpler equivalence relations. For example, the equivalence relations that can be written as increasing unions of Borel equivalence relations with finite classes are called the hyperfinite Borel equivalence relations. They are an important and well-studied class.

We consider an old question of Slaman and Steel: whether Turing equivalence is an increasing union of Borel equivalence relations none of which contain a uniformly computable infinite sequence. We show this question is deeply connected to problems surrounding Martin's conjecture, and also in countable Borel equivalence relations.

Abstract. Understanding structures in terms of what combinatorial properties they can express is an important part of Model Theory. One such combinatorial property is called independence, and structures not expressing it are called NIP. Known examples of NIP fields are all either finite, separably closed, real closed or henselian. Bearing in mind the example of Qp, and with the help of a recent result of S. Anscombe and F. Jahnke, we will explore some conjectures around NIP fields, and give an explicit formula defining henselian valuation rings under some of these conjectures.

Abstract. In joint work with Lou van den Dries and Joris van der Hoeven, we constructed the field of logarithmic hyperseries. This is a proper class-sized ordered differential field which is also equipped with a composition and hyperlogarithm functions (which act like transfinite iterates of a logarithm function). In subsequent joint work with Vincent Bagayoko and Joris van der Hoeven, we introduced the more general notion of a hyperserial field (of type \({On}\)). These proper class-sized fields admit an external composition over the field of logarithmic hyperseries. We showed that each hyperserial field has a minimal hyperserial field extension in which each hyperlogarithm is bijective on the class of positive infinite elements. In this talk, I will discuss both of these works and their connection with transseries, logarithmic transseries, and the surreal numbers.

Abstract. The burden is a notion of dimension associated to any NTP2 theory, i.e. to a large class of relatively tame first order theories. I will present a reduction principle for abelian groups: the burden of a pure short exact sequence of abelian groups A->B->C, seen as a three-sorted structure, can be computed in terms of the burden of \(A\) and that of \(B\). This result generalizes a work of Chernikov and Simon and uses a new quantifier elimination result of Aschenbrenner, Chernikov, Gehret and Ziegler.

Abstract. Pre-\(H\)-fields were introduced by Aschenbrenner and van den Dries to study Hardy fields and transseries from an algebraic and model-theoretic perspective. Aschenbrenner, van den Dries, and van der Hoeven isolate the model companion of the theory of pre-\(H\)-fields and show that the ordered valued differential field of logarithmic-exponential transseries is a natural model. In this talk, I will define a theory of pre-\(H\)-fields whose models are transexponential, unlike transseries, and describe its model companion; moreover, this model companion has quantifier elimination and NIP.

Abstract. An orientation of a graph is called a \(k\)-orientation if the out degree of every vertex is bounded by \(k\), and the orientation number of a graph is the least \(k\) so that the graph admits a \(k\)-orientaton. In finite combinatorics, the orientation number is one of several important and closely related measures of sparsity. In this talk I will adapt the notion to the descriptive set theoretic setting and give a measurable generalization of Edmonds' formula for the orientation number.

Abstract. We show that the category of standard Borel spaces is the free or "universal" category equipped with some familiar set operations of countable arity (e.g., products) obeying some simple compatibility conditions (e.g., products distribute over disjoint unions). In this talk, we will discuss the precise formulation of this result, its connection with the amalgamation property for \(\kappa\)-complete Boolean algebras, and its proof using methods from categorical logic.

Abstract. Pierre Simon introduced distality to better understand unstable NIP theories by studying their stable and "purely unstable," or distal, parts separately.

We introduce distality rank as a property of first-order theories and give examples for each rank \(m\) such that \(1\leq m \leq \omega\). For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called \(m\)-determinacy and show that theories of distality rank \(m\) require certain products to be \(m\)-determined. Furthermore, for NIP theories, this behavior characterizes distality rank \(m\).

The preprint containing these results can be found at https://arxiv.org/abs/1908.11400. We will also discuss some new results not contained in the paper.

Abstract. I will introduce two closely-related properties, which we've been calling "semi-equational" and "strongly semi-equational", which can be thought of as attempts to complete the analogy "stable : equational :: NIP : ?", or as one-sided versions of distality. The relationship between these and other model-theoretic properties will be discussed, and I will give some examples of (strongly) semi-equational theories, and describe some techniques for showing that theories are not (strongly) semi-equational. The results in this talk are joint work with Artem Chernikov.

Abstract. In this talk I will first define and describe the mathematical object \(\mathbb{T}_{\log}\): the ordered valued differential field of logarithmic transseries. I will then discuss a strategy I have developed for proving \(\mathbb{T}_{\log}\) is model complete in a certain language that I will introduce. I reduce the problem of model completeness down to two precise conjectures concerning the nature of logarithmic derivatives, solutions of linear differential equations, and differential-transcendence. Note: this is an updated version of a talk I gave in the Cabal Seminar in Spring 2018, although I will mention a few new things.

Abstract. Given a countable group Gamma, the outer automorphism group Out(Gamma) is either countable or of cardinality continuum. A finer and more suitable notion is to consider the Borel complexity of Out(Gamma) as a Borel equivalence relation. We show that in this context, Out(Gamma) is of rather low complexity, namely that it is a hyperfinite Borel equivalence relation. In general, we show that for any Polish group G and any countable normal subgroup Gamma, the quotient group G/Gamma is hyperfinite. This is joint work with Joshua Frisch.

Abstract. Assume the axiom of determinacy. Martin's partition relations on \(\omega_1\) imply that the \(\epsilon\)-partition measure on \([\omega_1]^{\epsilon}\) are countably complete ultrafilter for each \(\epsilon \leq \omega_1\).

An almost everywhere (according to the partition measures) selection principle for club subsets of \(\omega_1\) will be discussed.

From this uniformization, one can show that every function \(\Phi : [\omega_1]^{\omega_1} \rightarrow \omega_1\) is continuous almost everywhere with respect to the strong partition measures.

Since the \(\epsilon\)-partition measure is an ultrafilter, for any sentence of set theory, for almost all increasing \(\epsilon\)-sequence of countable ordinals, \(L[f]\) satisfies this sentence or its negation. The \(\epsilon\)-stable theory is the collection of statements that hold for almost all \(\epsilon\)-sequences. We will discuss some natural statements which belong to the various stable theory. For example, for all \(\epsilon \leq \omega_1\), there is a club \(C\) subset of \(\omega_1\) so that for all \(\epsilon\)-length increasing sequences \(f\) through \(C\), \(L[f]\) satisfies the generalize continuum hypothesis. The club selection principle is important for studying the \(\omega_1\)-stable theory.

This is joint work with Jackson and Trang.