
Zoom ID: 884 0948 8227 The seminar will focus on various expansions of ominimal structures, such as those with ominimal open core, dminimal structures, Hstructures, lovely pairs, RCVFs, CODFs, distal and general NIP. We target talks in pure model theory and applications. The intention is to run the seminar once every two weeks. The exact times may slightly vary, so please check below. Upcoming talks:Title: TBA Abstract: TBA Past talks:Title: Elimination of imaginaries in lexicographic products of ordered abelian groups Abstract: We will investigate the property of elimination of imaginaries for some special cases of ordered abelian groups. As main result, we will show that the lexicographically ordered groups Z^n and Z^n \times Q eliminate imaginaries once we add finitely many constants to the language of ordered abelian groups. ➯ RECORDING Title: Pseudofinite sets, pseudoominimality Abstract: Given a language L, the class of ominimal Lstructures is not elementary, e.g., an ultraproduct of ominimal structures need not be ominimal. This fact gives rise to the following notion, introduced by Hans Schoutens: Given a language L, an Lstructure is pseudoominimal if it satisfies the common theory of ominimal Lstructures. Of particular importance in pseudoominimal structures are pseudofinite sets. A definable set in an ordered structure is pseudofinite if it is closed, bounded, and discrete. Many results from ominimality translate to pseudoominimality by replacing finite with pseudofinite. We will review the key role that pseudofinite sets play in pseudoominimality, as well as other firstorder properties of ominimality such as definable completeness* and local ominimality**. Finally, we will see how pseudofinite sets can be used to prove distinctions between generalizations of ominimality and answer two questions by Schoutens, one of them is whether there is an axiomatization of pseudoominimality by firstorder conditions on onevariable formulae only. This also partially answers a conjecture by Antongiulio Fornasiero. No knowledge in ominimality or generalizations thereof will be assumed throughout the talk. * An ordered structure is definably complete if every bounded definable subset has a supremum. ** An ordered structure is locally ominimal if, for every definable subset D and every point x, there is an interval with an endpoint in x that is either contained in D or disjoint from D. ➯ RECORDING Title: On the PilaWilkie theorem Abstract: I’ll give an account of the PilaWilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. Very basic knowledge of ominimality will be assumed; this is joint work with Prof. Lou van den Dries. ➯ RECORDING Title: Some finiteness results concerning points on a curve with a power on a curve Abstract: Let X and Y be two geometrically irreducible closed algebraic curves in the algebraic torus of dimension 3. Suppose a generic point of X satisfies no nontrivial multiplicative relations. We denote by [n]X the set of all nth powers of points in X. It is a conjecture that at most finitely many x in X will have the property that there is a positive integer n such that x^n is in Y and [n]X is not contained in Y. I shall discuss proofs of several cases of this conjecture. Work of Bays, Kirby and Wilkie, which established an analogue of Schanuel's conjecture for the operation of raising to an exponentially transcendental power, plays an important role. ➯ RECORDING Title: PilaWilkie in Tame Expansions of a Padic Structure Abstract: We adapt a paper of Eleftheriou on an analogue for the PilaWilkie theorem in expansions of the real analytic structure to a padic context. We prove a structure theorem for definable sets in an expansion o the padic subanalytic structure by a dense independent set along the way. ➯ RECORDING Title: Ominimal structures with many dense substructures. Abstract: We study expansion of ominimal structures with two or more predicates dense "independent" substructures. ➯ RECORDING Title: The definable (p,q)theorem for dense pairs of distal geometric structures where acl=dcl. Abstract: The definable (p,q)conjecture is a modeltheoretic version of the combinatorics (p,q)theorem. In this talk, I will discuss how we proved that the definable (p,q)conjecture holds for dense pairs of distal geometric structures where the algebraic closure is equal to the definable closure. This is a joint work with Gareth Boxall. ➯ RECORDING Title: A strong version of Cobham’s theorem and other decidability results in expansions of Presburger arithmetic Abstract: Let k,l>1 be two multiplicatively independent integers. Cobham’s famous theorem states that a subset of the natural numbers is both krecognizable and lrecognizable if and only if it is Presburgerdefinable. We show the following strengthening. Let X be krecognizable, let Y be lrecognizable such that both X and Y are not Presburgerdefinable. Then the theory of (N,<,+,X,Y) is undecidable. This is joint work with Christian Schulz. ➯ RECORDING Title: Expanding the Additive Group of Integers by Beatty Sequences Abstract: The Beatty Sequence generated by an irrational r>1 is ([nr] : n>0), where [a] denotes the integer part of a real number a. After presenting some properties of such a sequence, we will investigate the expansion of (Z, +) by the unary subset B consisting of the terms of a Beatty sequence. We will mention a quantifier elimination result and an axiomatization for the theory of such an expansion. If time allows, we will also show that there are no intermediate structures between (Z, +) and (Z, +, B). (This is a joint work with Melissa Özsahakyan.) ➯ RECORDING Title: Tameness properties of theories of valued fields with analytic functions Abstract: An important motif in modeltheoretic algebra over the last thirty years has been the concept of tameness and the impact it has for understanding the definable sets of a structure. In this talk, I will describe some of the ways this motif occurs in the case of valued fields, especially ordered convexly valued fields, when equipped with additional function symbols which, on the standard model, are interpreted by functions defined by convergent power series. All of these notions will be defined in the course of the talk. ➯ RECORDING Title: Interpretable fields in various valued fields Abstract: Difficulties in analyzing interpretable objects arise when we lack (a simple) elimination of imaginaries. It turns out that in several dpminimal settings it is possible to circumvent this difficulty by focusing on onedimensional subsets and by reducing these to several relatively understandable sorts. We consider an interpretable field F in either a real closed valued field K or Tconvex expansions of K. In this case one can reduce the analysis to the four sorts K, k, the value group, and K/O (for O the valuation ring), then eliminate the last two sorts, and conclude that F is either definable in the field K or in k. As a result, F is definably isomorphic to K,K(i), k or k(i). Similar analysis can be carried out in certain Pminimal structures (in particular, in padically closed fields), and probably more. (part of a joint work with Y. Halevi and A. Hasson) ➯ RECORDING Title: Fractal Dimensions and Definability from Büchi Automata Abstract: Büchi automata are the natural extension of finite automata, also called finitestate machines, to a "machine" that accepts infinitelength inputs. We say a subset X of the reals is rregular if there is a Büchi automaton that accepts (one of) the baser expansions of every element in X, and rejects the baser expansions of each element in its complement. We can analogously define rregular subsets of higher arities of the reals, and these sets often exhibit fractallike behaviore.g., the Cantor set is 3regular. There are several knownand remarkableconnections in logic to Büchi automata, including the fact that the expansion of the real additive group by every rregular subset of [0,1] (for some fixed positive integer r) interprets the monadic secondorder theory of the natural numbers with successor. In this talk, I will focus on some of the geometric behavior of closed rregular sets in terms of fractal dimensions, and discuss how closed rregular subsets of [0,1] with and without integer Hausdorff dimension form a dichotomy in terms of first order definability in expansions of the real additive group by a predicate for a specific rregular set. ➯ RECORDING Title: A survey on exponentialalgebraic closure Abstract: Zilber conjectured that complex exponentiation is quasiminimal in 1997 (if not before) and produced different quasiminimal structures. He later formulated the exponentialalgebraic closedness conjecture (EAC), which would imply quasiminimality of complex exponentiation. I will summarise what has been proved so far around EAC, including extensions to abelian exponentials, modular functions, the special case of raising to powers, and an odd spinoff with ominimal open core. ➯ RECORDING Title: The domination monoid in ominimal theories Abstract: The product of invariant types and dominationequivalence are not always compatible but, when they are, they allow to define the "domination monoid" associated to a monster model U of a firstorder theory. In the superstable case, this object parameterises "finitely generated saturated extensions of U" and how they can be amalgamated independently. After recalling the basic definitions and facts, I will talk about some results from my thesis, concerning the study of this monoid in a different context, that of ominimal theories. This includes a reduction of the problem to showing generation by classes of 1types, and a proof that this holds in RCF. I will then discuss the open problem of showing generation by 1types in general, and some possible lines of attack. ➯ RECORDING Title: Blurrings of the jfunction Abstract: I will define blurred variants of the jfunction and its derivatives, where blurring is given by the action of a subgroup of GL_{2}(ℂ). For a dense subgroup (in the complex topology) an Existential Closedness theorem holds which states that all systems of equations in terms of the corresponding blurred j with derivatives have complex solutions, except where there is a functional transcendence reason why they should not. The proof is based on the AxSchanuel theorem and Remmert's open mapping theorem from complex geometry. For the jfunction without derivatives we prove a stronger theorem, namely, Existential Closedness for j blurred by the action of a subgroup which is dense in GL_{2}^{+}(ℝ), but not necessarily in GL_{2}(ℂ). In this case apart from the AxSchanuel theorem and some basic complex geometry, ominimality is also used in the proof (I will present the proof in this case). If time permits, I will also discuss some model theoretic properties of the blurred jfunction such as stability and quasiminimality. This is a joint work with Jonathan Kirby. ➯ RECORDING Title: Generic derivations on ominimal structures Abstract: Let T be a model complete ominimal theory which extends the theory of real closed ordered fields (RCF). We introduce Tderivations: derivations on models of T which cooperate with Tdefinable functions. The theory of models of T expanded by a Tderivation has a model completion. If T = RCF, then this model completion is the theory of closed ordered differential fields (CODF) as introduced by Singer. We can recover many of the known facts about CODF (open core, distality) in our setting. This is joint work with Antongiulio Fornasiero. ➯ RECORDING Title: Compressible types in NIP theories Abstract: I will present some work in progress joint with Martin Bays and Pierre Simon. I will discuss compressible types and relate them to uniform definability of types over finite sets (UDTFS), and to uniformity of honest definitions. All notions will be defined during the talk. ➯ RECORDING Title: Geometric random graphs Abstract: Geometric random graphs are graphs on a countable dense set of some underlying metric space such that locally in any ball of radius one, it is a random graph. The geometric random graphs on R^n and on circles have been studied by probabilists and graph theorists. In this talk we will present some model theoretic views. In particular, we will show that under some mild assumptions, the geometric random graphs based on a fixed metric space will have the same theory. We will also talk about some geometric properties of the underlying metric space that can be recovered from the graphs. This is a work in progress joint with Omer BenNeria and Itay Kaplan. ➯ RECORDING Title: Connectedness in structures on the real numbers Abstract: We consider structures on the set of real numbers having the property that connected components of definable sets are definable. Our main analyticgeometric result is that any such expansion of the real additive group by boolean combinations of closed sets (of any arities) is either ominimal (with respect the usual order) or undecidable, and if the set of integers is definable, then so is integer multiplication. It is known that all ominimal structures on the real line have the property, as do all expansions of the real field that define the integers (easy modulo some basic descriptive set theory). We show that fusions of the real ordered additive group with expansions of the ring of integers are also examples. All results hold with "connected component" replaced by "path component" or "quasicomponent". (Joint with A. Dolich, A. Savatovsky and A. Thamrongthanyalak. Preprint available on MODNET and arXiv.) ➯ RECORDING Title: Generic expansions by a reduct Abstract: Consider the expansion T_{S} of a theory T by a predicate for a submodel of a reduct T_{0} of T. This generalizes the generic predicate construction and some theories of lovely pairs. We present a setup in which this expansion admits a model companion TS. We show that the nice features of the theory T transfer to TS. In particular, by studying independence relations, we find conditions for which this expansion preserves the NSOP_{1}, simplicity or stability of the starting theory T. We will also give concrete examples of new modelcompanion obtained by this process, among them new NSOP_{1} theories such as the expansion of an algebraically closed field of positive characteristic by an additive subgroup (ACFG) and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup. This construction also gives some very wild expansions of fields, such as the expansion of an algebraically closed valued field of positive characteristic by a generic additive subgroup, which has TP_{1} and TP_{2}. ➯ RECORDING Title: Distality in valued fields and related structures Abstract: In this talk we discuss distality, a modeltheoretic notion of tameness generalizing ominimality, in valued fields and related structures. In particular, we characterize distality in certain ordered abelian groups, provide an AxKochenErshov style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g. the valued differential field of logarithmicexponential transseries, are distal. This relies in particular on a general quantifier elimination result for pure short exact sequences of abelian groups. Joint work with Matthias Aschenbrenner, Allen Gehret and Martin Ziegler. ➯ RECORDING Title: A review of expansions by predicates and some preservation theorems Abstract: We say that a theory is geometric if the algebraic closure satisfies the exchange property and eliminates the quantifier exists infinitely. Examples include dense ominimal theories, strongly minimal theories and SUrank one theories. In this talk we will introduce geometric theories, review some of its expansions by predicates and the structural properties (stability, simplicity, NIP, NTP2) that these expansions preserve. ➯ RECORDING 