Lancashire Yorkshire Model Theory Seminar
A regular series of meetings of the model theorists in Leeds, Manchester and Preston, supported by the London Mathematical Society
10.30-11.00 | Arrival and coffee | |
11.00-12.00 | Victoria Gould (York) | ℵ_{0}-categoricity for semigroups Abstract may be found here. |
12.00-13.00 | Mike Prest (Manchester) | Nori motives and model theory Abstract: Homology and cohomology theories attach algebraic and numerical invariants to varieties and schemes. There are many such (co)homology theories and the idea (of Grothendieck) is that there should be a universal such theory - one through which all the others factor. This would be an abelian category of "motives" built from a suitable category of varieties. In the 90s Nori gave a construction of a category with some of the desired features. A recent paper of Barbieri-Viale, Caramello and Lafforgue gives a much more direct construction, using (topos-theoretic) model theory. Barbieri-Viale and I subsequently described an even more direct approach using (classical-style) model theory. In these approaches Nori motives are imaginary sorts in an appropriate language. I will talk about this and some current work with Luca Barbieri-Viale and Annette Huber. |
13.00-14.30 | Lunch | |
14.30-15.30 | Luck Darnière (Angers) | Lattices of closed semi-algebraic sets. Abstract: Let K be a p-adically closed field, X a semi-algebraic set of dimension d defined over K and L(X) the lattice of semi-algebraic subsets of X which are closed in X. We prove that the complete theory of L(X) is decidable (contrary to what happens over a real closed field) and eliminates the quantifiers in a certain language Lasc, the Lasc-structure on L(X) being an extension by definition of the lattice structure. We classify these structures up to elementary equivalence, and get in particular that the complete theory of L(K^d) only depends on d, not on K nor even on p. |
15.30-16.00 | Coffee | |
16.00-17.00 | Andrew Brooke-Taylor (Leeds) | Generalised model theory from a category theory perspective Abstract: Shelah introduced the framework of Abstract Elementary Classes to generalise model theory beyond the first order. Parallel to this, in category theory the notion of accessible categories was developed with a similar aim. It has recently emerged that the two approaches are intimately related to each other. In this talk I will give an overview of this connection, with a particular eye to how set theory can contribute to the mix. |
17.00- | Pub and Dinner |
10.30-11.00 | Arrival and coffee | |
11.00-12.00 | Sam Dean (Glasgow) | Positive primitive formulas for sheaves Abstract: Others have defined what it should mean for a sheaf to sit inside another sheaf as a pure substructure. This is done in an algebraic fashion. But in the model theory of modules, we know well that this condition can be said in terms of pp formulas. Sheaves, not usually being 1st-order structures, can't obviously be approached like this. Even when we do get a nice class of sheaves which are (secretly) 1st-order structures, the answer we get for what the usual notion of purity means is geometrically wrong. I will give a notion of a pp formula for sheaves which fits with the geometric notion of purity, and explain what the remaining questions are. |
12.00-13.00 | Sasha Borovik (Manchester) | Permutation groups of finite Morley rank Abstract: I will introduce some basic concepts and ideas of this theory, and will survey latest results by Altinel, Berkman, Borovik, Deloro, and Wiscons (in various combinations of authors). |
13.00-14.30 | Lunch | |
14.30-15.30 | Gabor Elek (Lancaster) | Limits of finite graphs via ultraproducts Abstract: I will show how to obtain the Lovasz-Szegedy resp. Benjamini-Schramm graph limits of dense resp. sparse graphs using ultraproducts and ultralimits. |
15.30-16.00 | Coffee | |
16.00-17.00 | Gareth Jones (Manchester) | Pfaffian functions and elliptic functions Abstract: I will discuss work with Harry Schmidt in which we give a pfaffian definition of Weierstrass elliptic functions, refining a result due to Macintyre. The complexity of our definition is bounded by an effective absolute constant. As an application we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of Manin-Mumford for non-split extensions of elliptic curves by the additive group. We also give a higher dimensional version of their result. |
17.00- | Pub and Dinner |
10:30-11:00 | Arrival and Coffee |
11:00-12:00 | Talk by Martin Hils. Title: Model theory of compact complex manifolds with an automorphism Abstract: One may develop the model theory of compact complex manifolds (CCM) with a generic automorphism in rather close analogy to what has been done for existentially closed difference fields, in important work by Chatzidakis and Hrushovski, among others. The corresponding first order theory CCMA is supersimple, and the Zilber trichotomy holds for "finite-dimensional" types of SU-rank 1. In the talk, I will present some results in CCMA in the spirit of geometric simplicity. Moreover, I will discuss the issue whether a given sort of CCM is stably embedded in CCMA. This is joint work with Martin Bays and Rahiim Moosa. |
12:00-13:00 | Question proposal session |
13:00-14:30 | Lunch |
14:30-17:00 | Question answer session and discussion |
17:00- | Pub and dinner |
10.30 - 11.00 |
Arrival and coffee | |
11.00 - 12.00 |
Isolde Adler(Leeds) | Testing logically defined properties on structures of bounded degree
Abstract: Property testing (for a property P) asks for a given input, whether it has property P, or is "far" from having that property. A "testing algorithm" is a probabilistic algorithm that answers this question with high probability correctly, by only looking at small parts of the input. Testing algorithms are thought of as "extremely efficient", making them relevant in the context of big data. We extend the bounded degree model of property testing from graphs to relational structures, and we discuss testability of first-order logic and monadic second-order logic in this model. This is joint work with Frederik Harwath. |
12.00 - 13.00 |
Omar León Sánchez (Manchester) | Conditions for finite-rank types to be isolated in omega-stable theories (and applications) Abstract: In omega-stable theories, isolated types play a crucial role; for instance, they are known to be dense in the (Stone) type space over any set of parameters. It is thus important to understand characterizations of such types in terms of more "algebraic" conditions. One potential condition is that of weak-orthogonality. We will see that weak-orthogonality together with analisability (to a given definable set) imply isolation. While this does not characterize isolated types, it does yield interesting applications in the theory of differential Hopf algebras and, more generally, Hopf-Ore extensions (these notions will be explained). This is joint work with J. Bell and R. Moosa. |
13.00 - 14.30 |
Lunch | |
14.30 - 15.30 |
Thomas Quinn-Gregson (York) | Homogeneity of Inverse Semigroups Abstract: The concept of homogeneity of relational structures has connections to model theory, permutation groups and combinatorics. A number of complete classifications have been obtained, including those for graphs, semilattices and posets. We may extend this definition by naming an arbitrary structure homogeneous if every isomorphism between finitely generated sub-substructures extends to an automorphism. The key to this extension is that connections with model theoretic properties such as quantifier elimination and $\aleph_0$-categoricity remain. An inverse semigroup $S$ is a semigroup in which every element has a unique inverse, that is, if $a \in S$ then there exists a unique $b \in S$ such that $a = aba$ and $b = bab$, which we denote as $a^{-1}$. It is clear that groups are inverse semigroups, as indeed are semilattices with binary operation of meet. Since an inverse semigroup can be viewed as either a semigroup or as a unary semigroup (a semigroup equipped with a basic unary operation), we obtain two concepts of homogeneity; homogeneous semigroups and homogeneous inverse semigroups. We discuss how the two concepts of homogeneity differ, and how the homogeneity of an inverse semigroup effects its substructure, in particular its semilattice of idempotents and maximal subgroups. We also consider the following question: Given a homogeneous group $G$, which homogeneous inverse semigroups contains $G$ as a maximal subgroup? This will be completely answered for the case where $G$ is finite and where $G$ is Hall's universal locally finite group. |
15.30 - 16.00 |
Tea/Coffee | |
16.00 - 17.00 |
Jan Dobrowolski (Leeds) | Polish structures
Abstract: The notion of a Polish structure is a purely topological concept (it can be thought of as a Hausdorff topological space X equipped with a continuous action of a Polish group G), which is, however, inspired by model theory. After introducing the basic concepts and explaining in what way some model-theoretic intuitions can be transferred to this setting, I will discuss the main directions of research and open problems related to that subject. |
17.00- | Pub and Dinner |
10.30-11.00 | Arrival and coffee | |
11.00-12.00 | Sonia L'Innocente (Camerino) | Irreducible generalised power series Abstract: A classical tool in the study of real closed fields are the fields K[[G]] of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. A fundamental result by Berarducci ensures the existence of irreducible series in the subring of K[[G]] consisting of the generalised power series with non-positive exponents. In this work, we are able to prove that all series in this subring can be factorized as a product of irreducibles and a "small" series, first in the case when the group is the additive group of real numbers and then, in the case of arbitrary groups. |
12.00-13.00 | Daoud Siniora (Leeds) | Ample Generics and Fraisse limits Abstract: The automorphism group of a countably infinite first order structure becomes a topological group when endowed with the pointwise convergence topology. We then can ask whether the automorphism group has 'ample generics'? By the work of Hodges, Hodkinson, Lascar, and Shelah, and later by Kechris and Rosendal, we can answer this question in the setting of homogeneous structures, or Fraisse limits, by examining the age of the structure. In this talk I will introduce ample generics, and an approach to show their existence for homogeneous structures. Moreover, I will discuss some group theoretic consequences of the existence of ample generics including the small index property, uncountable cofinality, and the Bergman property. |
13.00-14.30 | Lunch | |
14.30-15.30 | Davide Penazzi (UCLan) | Topological dynamics in the p-adic world Abstract: I will provide a brief introduction on topological dynamics and model theoretic applications, i.e. when the flow is that of a group G acting on its type space S_G(M). I will then focus on the study of the case of SL_2(Q_p). In a joint work with Pillay and Yao we determine its minimal flows, the Ellis group and the universal minimal flow. |
15.30-16.00 | Coffee | |
16.00-17.00 | Arno Fehm (Manchester) | Existentially definable henselian valuation rings with p-adic residue fields Abstract: Earlier joint work with Sylvy Anscombe gave us an abstract valuation theoretic condition characterizing those fields F for which F[[t]] is existentially 0-definable in F((t)). In this talk I will report on joint work with Sylvy Anscombe and Philip Dittmann in which the study of this condition leads us to some beautiful results on the border of number theory and model theory. In particular, I will present and apply a p-adic analogue of Lagrange's Four Squares Theorem. |
17.00- | Pub and Dinner |
10.30 - 11.00 |
Arrival and coffee | |
11.00 - 12.00 |
Dario Garcia (Leeds) | Unimodularity unified
Abstract: Unimodularity was defined by Hrushovski, in his proof that a unimodular strongly minimal set is locally modular, thus generalising Zilber’s result thata locally finite strongly minimal set is locally modular. It was claimed in the same paper that unimodularity was equivalent to an a priori weaker notion known later as functional unimodularity. In an attempt to clarify the situation, Pillay and Kestner distinguished two types of functional unimodularity -one for definable sets and one for type-definable sets- and studied their relationship in the context of strongly minimal structures. In this talk, I will present joint with Wagner where we introduce yet another variant called correspondence unimodularity (for types and for definable sets) and present several results describing the relationship between the different concepts. For instance, we show the variants of unimodularity for types coincide in omega-stable theories, and all variants coincide for non-multidimensional theories where the dimension is associated to strongly minimal types (e.g. strongly minimal theories or groups of finite Morley rank). |
12.00 - 13.00 |
Franziska Jahnke (Münster) | Henselianity in the language of rings Abstract: (Joint work with Sylvy Anscombe) We consider four properties of a field K related to the existence of (definable) henselian valuations on K and on elementarily equivalent fields and study the implications between them. Surprisingly, the full pictures look very different in equicharacteristic and mixed characteristic. |
13.00 - 14.30 |
Lunch | |
14.30 - 15.30 |
Erick Garcia Ramirez (Leeds) | Tangent cones and stratifications in RCVF Abstract: I will talk about tangent cones of definable sets in real closed valued fields. A notion of 't-stratification' will be introduce too and I will then explain how a t-stratification of a definable set induces t-stratifications on tangent cones. I will also discuss further interests on this subject. |
15.30 - 16.00 |
Tea/Coffee | |
16.00 - 17.00 |
Antongiulio Fornasiero (Parma) | Non-elementary lovely pairs
Abstract: We present Lovely Pairs: expansions of a structure M with a predicate P for a "small" set (satisfying certain additional properties). Lovely pairs (a generalization of Poizat's "Belle paires") have been studied (in several contexts and under various names) for a long time. The prototypical cases are the real field R with P denoting the real algebraic numbers, or the complex field C with P a proper algebraically closed subfield. In the classical cases, P has always been an elementary substructure of M (the "elementary" lovely pairs). However, more recent works have considered other kind of structures that resemble lovely pairs, but where P is not an elementary substructure (e.g.: P a dense transcendence basis of R, or P a transcendence basis of C, or P a dense multiplicative subgroup of R* of finite rank). We will show that such "non-elementary" lovely pairs have much in common with the elementary ones. |
17.00- | Pub and Dinner |
10:30-11:00 | Arrival and Coffee | |
11:00-12:00 | Fabrizio Barroero (Manchester) |
Unlikely intersections in families of powers of elliptic curves Abstract: Let E_t be the Legendre elliptic curve of equation Y^2=X(X-1)(X-t). In 2010 Masser and Zannier proved that, given two points on E_t with coordinates algebraic over Q(t), there are at most finitely many specializations of t such that the two points become simultaneously torsion on the specialized elliptic curve, unless they were already generically linearly dependent. One of the main ingredients of the proof is a result of Pila about counting rational points of bounded height on subanalytic surfaces, which is a special case and predates the celebrated Pila-Wilkie theorem. As a natural higher-dimensional analogue, we considered the case of n generically independent points on E_t with coordinates algebraic over Q(t). Then there are at most finitely many specializations of t such that two independent relations hold between the specialized points. Here one needs a more sophisticated counting theorem: relying on results of Pila, we estimate the number of points on some subanalytic surfaces lying on certain linear affine varieties defined by equations with rational coefficients of bounded height. This is joint work with L. Capuano. |
12:00-13:00 | Davide Penazzi (Preston) | Existence Theorems for Differential Equations Abstract: We build on the article "Existence Theorems for Systems of Implicit Differential Equations" of Grill, Knebusch and Tressl; where it was shown that given a polynomial differential ideal of R{X_1,...X_n} which is semireal, then there exists an analytic map c from an interval I in R to R^N such that c solves the differential equations of the ideal (i.e. f(c(t))=0 for all f in the ideal and t in I). Our work aims at obtaining similar results for differential equations with initial value conditions (IVPs) and in a more general context: when R is the ring of convergent power series in one variable, i.e. for differential equations with power series coefficients. I will outline the results we have obtained so far and some of the ideas behind them. |
13:00-14:30 | Lunch | |
14:30-15:30 | Ivo Herzog (Ohio State) | Universal *-regular rings Abstract: Using the model theory of modules, we prove that if (R,i) is a ring with involution, then there exists a morphism $u: (R,i) \to (R',i'),$ with R' a *-regular ring, that is universal, in the sense that any such morphism factors in a unique way through u. Recall that an involution i of a ring R is an anti-automorphism of order 2 and that a von Neumann regular ring R with involution is called *-regular if for all r in R, $i(r)r \neq 0.$ For a commutative ring equipped with the identity involution, the existence of a universal *-regular ring was proved by Olivier. Suppose that L is a split semisimple Lie algebra over a field k of characteristic 0. We will use a result from J.C. Jantzen's thesis together with a theorem of Harish-Chandra to prove that the universal enveloping algebra U(L) may be equipped with an involution i in such a way that the morphism of (U(L),i) into its universal *-regular ring is an embedding. |
15:30-16:00 | Coffee | |
16:00-17:00 | Vincenzo Mantova (Leeds) | Towards a composition on surreal numbers Abstract: In a recent work with Alessandro Berarducci, we have shown that surreal numbers admit the structure of a field of transseries with a compatible "simplest" derivation. This raises the question whether surreal numbers can also be interpreted as differentiable functions, forming in fact a non-standard Hardy fields closed under composition. I will present the early partial results on this problem, with both positive and negative answers. This is joint work with Alessandro Berarducci. |
17:00- | Pub and dinner |
10.30-11.00 | Arrival and coffee | |
11.00-12.00 | Daniel Wolf (Leeds) | R-macs and Lie coordinatisation Abstract: I will present the notion of an R-mac, a generalisation of the definition of an N-dimensional asymptotic class given by Elwes, Macpherson and Steinhorn in 2007. I will then go over my current efforts to try to adapt the work of Cherlin and Hrushovski on Lie Coordinatisation to the R-mac setting. Joint work with Sylvy Anscombe (UCLan), Dugald Macpherson (Leeds) and Charles Steinhorn (Vassar) |
12.00-13.00 | Rosie Laking (Manchester) | Pointed morphisms and the lattice of pp formulas |
13.00-14.30 | Lunch | |
14.30-15.30 | Edith Vargas-Garcia (Leeds) | An introduction to the reconstruction of the topology on monoids of the rationals Abstract |
15.30-16.00 | Coffee | |
16.00-17.00 | Alessandro Berarducci (Pisa) | Compact domination, o-minimal homotopy and Pillay's conjectures Abstract: I will report on work Hrushovski, Peterzil, Pillay and Simon on NIP theories and compact domination and develop it further, yielding a new proof of Pillay's conjectures via an o-minimal "nerve theorem". This is joint work with Alessandro Achille |
17.00- | Pub and Dinner |
10:30-11:00 | Arrival and Coffee |
11:00-12:00 | Talk by Amador Martin-Pizarro. Title: Definable and interpretable groups in pairs of algebraically closed fields. Abstract: We will provide a characterisation of definable groups in a beautiful pair (K, E) of algebraically closed fields: every definable group projects, up to isogeny, onto the subgroup of E-rational points of some algebraic group defined over E with kernel an algebraic group. If time permits, we will discuss the characterisation of interpretable groups. |
12:00-13:00 | Question proposal session |
13:00-14:30 | Lunch |
14:30-17:00 | Question answer session and discussion |
17:00- | Pub and dinner |
10.30 - 11.00 |
Arrival and coffee | |
11.00 - 12.00 |
Dugald Macpherson (Leeds) | Pseudofinite dimension and pseudofinite structures
Abstract: I will discuss recent joint work with Garcia and Steinhorn on a notion of pseudofinite dimension for definable sets in pseudofinite structures, introduced by Hrushovski and Wagner and developed further by Hrushovski. In particular, I will discuss conditions on pseudofinite dimension which ensure that a structure is simple, or supersimple, or stable, or that forking can be characterised by dimension-drop. I will discuss examples, and some possible applications. |
12.00 - 13.00 |
Lovkush Agarwal (Leeds) | The 11 Reducts of the Generic Digraph Abstract: Given two structures M and N, we say that N is a reduct of M if, intuitively speaking, N is a less detailed version of M or if N is obtained from M by discarding information. In this talk, I will describe what the reducts of the generic digraph are and time permitting will describe some aspects of the proof. |
13.00 - 14.30 |
Lunch | |
14.30 - 15.30 |
Lorna Gregory (Manchester) | Interpretation functors, wild algebras and undecidability Abstract: In this talk I will present results about uniform interpretations between module categories over finite dimensional algebras. In particular, I will focus on attempts to prove a conjecture of Prest which says that if a finite-dimensional $k$-algebra is of wild representation type, a notion coming from representation theory, then it uniformly interprets $ Mod-k\langle x,y\rangle$ and hence has undecidable theory of modules. |
15.30 - 16.00 |
Tea/Coffee | |
16.00 - 17.00 |
Charlotte Kestner (UCLan) | Some model theory of bilinear forms
Abstract: I will give a short introduction to geometric stability theory and independence relations, focussing on the tree properties. I will then introduce one of the main examples for general measureable structures, the two sorted structure of a vector space over a field with a bilinear form. I will state some results for this structure, and give some open questions. |
17.00- | Pub and Dinner |
10:30-11:00 | Arrival and Coffee |
11:00-12:00 | Talk by Pierre Simon |
12:00-13:00 | Question proposal session |
13:00-14:30 | Lunch |
14:30-17:00 | Question answer session and discussion |
17:00- | Pub and dinner |
10.30-11.00 | Arrival and coffee | |
11.00-12.00 | Mike Prest (Manchester) | TBA |
12.00-13.00 | Immanuel Halupczok (Leeds) | Families of definable sets in the ordered group $\mathbb{Z}$ Abstract |
13.00-14.30 | Lunch | |
14.30-15.30 | Ronnie Nagloo (Leeds) | On Transformations in the Painlevé family Abstract |
15.30-16.00 | Coffee | |
16.00-17.00 | Ivan Tomašić (Queen Mary) | Applications of the twisted theorem of Chebotarev Abstract |
17.00- | Pub and Dinner |
10.30-11.00 | Arrival and coffee | |
11.00-12.00 | Charlotte Kestner (Preston) | NIP categories |
12.00-13.00 | Marcus Tressl (Manchester) | Externally definable sets in real closed fields |
13.00-15.00 | Lunch | |
15.00-16.00 | Tamara Servi (Lisbon) | Quantifier elimination for generalised quasianalytic classes. Abstract |
16.00-17.00 | Andres Aranda-Lopez (Leeds) | Supersimple homogeneous 3-graphs |
17.00- | Pub and Dinner |