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



Program of the 10th meeting on December 10th, 2016 in Preston

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

Venue: Foster Building, lecture theatre 2, UCLAN. Directions to the campus can be found here. The campus map can be found here.



Previous meetings



Program of the 9th meeting on Wednesday, June 15th, 2016 in Leeds

Venue: MALL, School of Mathematics, University of Leeds, Leeds. Campus Map; Walking from Leeds railway station to MALL, School of Mathematics

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




Program of the 8th meeting on Saturday, March 12th, 2016 in Manchester

Venue: Frank Adams 1 in the Alan Turing building, The University of Manchester.

10:30-11:00Arrival and Coffee
11:00-12:00Fabrizio 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.
12:00-13:00Davide 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
14:30-15:30Ivo 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
16:00-17:00Vincenzo 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



Program of the 7th meeting on December 5th, 2015 in Preston

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

Venue: Foster Building, lecture theatre 2, UCLAN. Directions to the campus can be found here. The campus map can be found here.


Program of the 6th meeting on Friday, June 26th, 2015 in Manchester

Venue: Frank Adams 1 in the Alan Turing building, The University of Manchester.

10:30-11:00Arrival 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




Program of the 5th meeting on Tuesday, January 13th, 2015 in Leeds

Venue: Roger Stevens Lecture Theatre 16 -- the Roger Stevens lecture theatre block (adjacent to the School of Mathematics), is building 89 on the Campus Map. Please note that the School of Mathematics is currently being refurbished. We propose meeting initially for coffee at 10.30 in the Cafe above the School of Mathematics (building 84a on the campus map), which is linked to the Roger Stevens building by an overhead walkway and remains open.

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
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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




Program of the 3rd meeting on Wednesday, June 18th, 2014 in Manchester


The format of the day was a bit different than usually.

10:30-11:00Arrival 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

Venue: Alan Turing building, University of Manchester.


Program of the 2nd meeting on March 29th, 2014 in Leeds

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

Venue: MALL, School of Mathematics, University of Leeds, Leeds. Directions to the school can be found here. The campus map can be found here.



Program of the 1st meeting on December 7th, 2013 in Preston

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

Venue: Foster Building, lecture theatre 2, UCLAN. Directions to the campus can be found here. The campus map can be found here.


Manchester-Leeds meetings



Links

Model Theory Preprint Server
Logic Seminar Leeds
Logic Seminar Manchester

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