Past Logic Seminar

Positive logic is a generalisation of full first-order logic, where negation is not built in, but can be added as desired. In joint work with Jan Dobrowolski we succesfully generalised the recent development on Kim-independence in NSOP1 theories to the positive setting. One of the important theorems in this development is the independence theorem, whose statement is very similar to the well-known statement for simple theories, and allows us to amalgamate independent types. In this talk we will have a closer look at the proof of this theorem, and what needs to be changed to make the proof work in positive logic compared to full first-order logic.

The Hardy–Littlewood circle method is a well-known analytic technique that has successfully solved several difficult counting problems in number theory. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces of low degree. I will explain how one can implement a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties, leading thus to a more precise description of the geometry of the above moduli spaces. This is joint work with Tim Browning.

I will recall the Zilber-Pink conjecture for Shimura varieties and give my perspective on current progress towards a proof.

In this talk I will present joint work with Ehud Hrushovski on imaginaries in the ring of adeles and more generally in products and restricted products of structures (including the generalised products of Feferman-Vaught).

We prove a general theorem on weak elimination of imaginaries in products with respect to additional sorts which we deduce from an elimination of imaginaries for atomic and atomless Booleanizations of a theory. This combined with uniform elimination of imaginaries for p-adic numbers in a language with extra sorts as p-adic lattices proved first by Hrushovski-Martin-Rideau and more recently by Hils-Rideau-Kikuchi in a slightly different language, yields weak elimination of imaginaries for the ring of adeles in a language with extra sorts as adelic versions of the p-adic lattices. 

The proofs of the general results on products use Boolean valued model theory, stability theory, analysis of definable groups and liaison groups, and descriptive set theory of smooth Borel equivalence relations including Harrington-Kechris-Louveau and Glimm-Efros dichotomy. 

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves contains a Zariski-dense set of special points if and only if it is special. I'll explain this result, and discuss an effective version that Gal Binyamini, Harry Schmidt, Margaret Thomas and I proved.

I will present a generalisation of the Elekes-Szabó result, that any ternary algebraic relation in characteristic 0 having large intersections with (certain) finite grids must essentially be the graph of a group law, to a version where one obtains an algebraic group action. In the end the conclusion will be similar, but with weaker assumptions. This is recent work with Tingxiang Zou.

A key ingredient in the proof of the model completeness of the real exponential field was a valuation inequality for polynomially bounded o-minimal structures. I shall briefly describe the argument, and then move on to the complex exponential field and Zilber's quasiminimality conjecture for this structure. Here, one can reduce the problem to that of establishing an analytic continuation property for (complex) germs definable in a certain o-minimal expansion of the real field and in order to study this question I propose notions of "complex Hardy fields" and "complex valuations".   Here, the value group is not necessarily ordered but, nevertheless, one can still prove a valuation inequality.

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of Fp(t)^h is an elementary substructure of the perfect hull of Fp((t)). Joint work with Franziska Jahnke.

(joint with Chernikov) 1-dependent theories, better known as NIP theories, are the first class of the strict hierarchy of n-dependent theories. The random n-hypergraph is the canonical object which is n-dependent but not (n−1)-dependent. We proved the existence of strictly n-dependent groups for all natural numbers n. On the other hand, there are no known examples of strictly n-dependent fields and we conjecture that there aren’t any. 

We were interested which properties of groups and fields for NIP theories remain true in or can be generalized to the n-dependent context. A crucial fact about (type-)definable groups in NIP theories is the absoluteness of their connected components. Our first aim is to give examples of n-dependent groups and discuss a adapted version of absoluteness of the connected component. Secondly, we will review the known properties of NIP fields and see how they can be generalized.

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves is special if and only if it contains a Zariski-dense set of special points. I'll discuss joint work with Gal Binyamini, Harry Schmidt, and Margaret Thomas in which we use pfaffian methods to obtain an effective uniform version of Manin-Mumford for products of CM elliptic curves. Using this we then prove an effective version of Habegger's result.

The Zilber-Pink conjecture predicts that if V is a proper subvariety of an arithmetic variety S (e.g. abelian variety, Shimura variety, others) not contained in a proper special subvariety of V, then the “unlikely intersections” of V with the proper special subvarieties of S are not Zariski dense in V. In this talk I will present a strong counterpart to the Zilber-Pink conjecture, namely that under some natural conditions, likely intersections are in fact Euclidean dense in V.  This is joint work with Tom Scanlon.

I shall present a new flexible method showing that every countable model of PA admits a pointwise definable end-extension, one in which every point is definable without parameters. Also, any model of PA of size at most continuum admits an extension that is Leibnizian, meaning that any two distinct points are separated by some expressible property. Similar results hold in set theory, where one can also achieve V=L in the extension, or indeed any suitable theory holding in an inner model of the original model.

A conjecture due to Zilber predicts that the complex exponential field is quasiminimal: that is, that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the complex exponential function are countable or cocountable.
Zilber showed that this conjecture would follow from Schanuel's Conjecture and an existential closedness type property asserting that certain systems of exponential-polynomial equations can be solved in the complex numbers; later on, Bays and Kirby were able to remove the dependence on Schanuel's Conjecture, shifting all the focus to the existence of solutions. In this talk, I will discuss recent work about the quasiminimality of a reduct of the complex exponential field, that is, the complex numbers expanded by multivalued power functions. This is joint work with Jonathan Kirby.

The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of “vectors” or by bi-operads and another based on “matrices” or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

Affine logic is the fragment of continuous logic in which the connectives are limited to affine functions. I will discuss the basics of this logic, first studied by Bagheri, and present the results of a recent joint work with I. Ben Yaacov and T. Tsankov in which we initiate the study of extreme types and extremal models in affine logic.

In particular, I will discuss an extremal decomposition result for models of simplicial affine theories, which generalizes the ergodic decomposition theorem.

In their seminal 2012 paper, Elekes and Szabó found that a certain weak combinatorial non-expansion property of an algebraic relation suffices to trigger the group configuration theorem, showing that only (approximate subgroups of) algebraic groups can be responsible for it. I will discuss some more recent variations and elaborations on this result, focusing on the case of ternary relations on varieties of dimension >1.

Let $k,l>1$ be two multiplicatively independent integers. A subset $X$ of $\mathbb{N}^n$ is $k$-recognizable if the set of $k$-ary representations of $X$ is recognized by some finite automaton. Cobham's famous theorem states that a subset of the natural numbers is both $k$-recognizable and $l$-recognizable if and only if it is Presburger-definable (or equivalently: semilinear). We show the following strengthening. Let $X$ be $k$-recognizable, let $Y$ be $l$-recognizable such that both $X$ and $Y$ are not Presburger-definable. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable. Our work strengthens and depends on earlier work of Villemaire and Bès. The essence of Cobham's theorem is that recognizability depends strongly on the choice of the base $k$. Our results strengthens this: two non-Presburger definable sets that are recognizable in multiplicatively independent bases, are not only distinct, but together computationally intractable over Presburger arithmetic. This is joint work with Christian Schulz.

Toward a characterization of modularity using shatter functions, we show that an o-minimal expansion of the  real ordered additive group $(\mathbb{R}; 0, +,<)$ does not define restricted multiplication if and only if the shatter function of every definable family is asymptotic to a polynomial. Our result implies that vc-density can only take integer values in $(\mathbb{R}; 0, +,<)$ confirming a special case of a conjecture by Chernikov. (Joint with Abdul Basit.)

We will define a notion of a semi-retraction between two first-order structures introduced by Scow. We show how a semi-retraction encodes Ramsey problems of finitely-generated substructes of one structure into the other under the most general conditions. We will compare semi-retractions to a category-theoretic notion of pre-adjunction recently popularized by Masulovic. We will accompany the results with examples and questions. This is a joint work with Lynn Scow.

The equivalence of NSOP${}_1$ and NSOP${}_3$, two model-theoretic complexity properties, remains open, and both the classes NSOP${}_1$ and NSOP${}_3$ are more complex than even the simple unstable theories. And yet, it turns out that classical geometric stability theory, in particular the group configuration theorem of Hrushovski (1992), is capable of controlling classification theory on either side of the NSOP${}_1$-SOP${}_3$ dichotomy, via the expansion of stable theories by generic predicates and equivalence relations. This allows us to construct new examples of strictly NSOP${}_1$ theories. We introduce generic expansions corresponding, though universal axioms, to definable relations in the underlying theory, and discuss the existence of model companions for some of these expansions. In the case where the defining relation in the underlying theory $T$ is a ternary relation $R(x, y, z)$ coming from a surface in 3-space, we give a surprising application of the group configuration theorem to classifying the corresponding generic expansion $T^R$. Namely, when $T$ is weakly minimal and eliminates the quantifier $\exists^{\infty}$, $T^R$ is strictly NSOP${}_4$ and TP${}_2$ exactly when $R$ comes from the graph of a type-definable group operation; otherwise, depending on whether the expansion is by a generic predicate or a generic equivalence relation, it is simple or NSOP${}_1$.

In his pioneering and celebrated 1968 paper on the elementary theory of finite fields Ax asked if the theory of the class of all the finite rings $\mathbb{Z}/m\mathbb{Z}$, for all $m>1$, is decidable. In that paper, Ax proved that the existential theory of this class is decidable via his result that the theory of the class of all the rings $\mathbb{Z}/p^n\mathbb{Z}$ (with $p$ and $n$ varying) is decidable. This used Chebotarev’s Density Theorem and model theory of pseudo-finite fields.

I will talk about a recent solution jointly with Angus Macintyre of Ax’s Problem using model theory of the ring of adeles of the rational numbers.

We will construct several algebras of functions definable in R_{an,\exp} which are stable under parametric integration. 

Given one such algebra A, we will study the parametric Mellin and Fourier transforms of the functions in A. These are complex-valued oscillatory functions, thus we leave the realm of o-minimality. However, we will show that some of the geometric tameness of the functions in A passes on to their integral transforms.

I will explain how the model-theoretic algebraic closure in Zilber’s pseudo-exponential field can be described in terms of the self-sufficient closure. I will sketch a proof and show how the Mordell-Lang conjecture for algebraic tori comes into play. If time permits, I’ll also talk about the characterisation of strongly minimal sets and their geometries. This is joint work (still in progress) with Jonathan Kirby.