Logic Workshop

All talks in Room G.107 - Alan Turing Building

15:30 Lorna Gregory
University of East Anglia
Decidability for theories of modules over finite-dimensional algebras
The representation type of a finite-dimensional k-algebra is an algebraic measure of how hard it is to classify its finite-dimensional indecomposable modules.

Intuitively, a finite-dimensional k-algebra is of tame representation type if we can classify its finite-dimensional modules and wild representation type if its module category contains a copy of the category of finite-dimensional modules of all other finite-dimensional k-algebras. An archetypical (although not finite-dimensional) tame algebra is k[x]. The structure theorem for finitely generated modules over a PID describes its finite-dimensional modules. Drozd’s famous dichotomy theorem states that all finite-dimensional algebras are either wild or tame.

The tame/wild dividing line is not seen by standard model theoretic invariants or even the more specialised invariants coming from model theory of modules. A long-standing conjecture of Mike Prest claims that a finite-dimensional algebra has decidable theory of modules if and only if it is of tame representation type. More recently, I conjectured that a finite-dimensional algebra has decidable theory of (pseudo)finite dimensional modules if and only if it is of tame representation type. I will give an overview of results around these conjectures.

16:10 Sebastian Eterović
University of Leeds
Solutions of equations involving transcendental functions
Suppose V is a complex algebraic variety and that f(z) is a transcendental holomorphic function. When does V intersect the graph of the function? This is both a natural question in complex analysis, and a necessary step in understanding the model-theoretic properties of the function. 

In this talk we will look at various famous examples of transcendental functions (exponentiation, modular functions, the Gamma function) and describe a general method using complex analysis for proving the existence of intersections with a special family of algebraic varieties. This method has been developed with various collaborators.

16:50 Anna Dmitrieva
University of East Anglia
Generic functions and quasiminimality
In 2002 Zilber introduced the theory of a generic function on a field, coinciding with the limit theory of generic polynomials developed by Koiran. Axiomatized in first-order logic by a version of Schanuel property and existential closedness, this theory turns out to be ω-stable. As shown by Wilkie and Koiran, one can explicitly construct such a generic function on the complex plane in a form of a Taylor series, using the ideas behind Liouville numbers. In this talk we look further into the properties of the theory of generic functions. As the main result, we show that adding any of these generic functions to the complex field gives an isomorphic structure, which ought to be quasiminimal, i.e. any definable subset has to be countable or cocountable. Thus we obtain a non-trivial example of an entire function which keeps the complex field quasiminimal.

11:30 Rosario Mennuni
University of Pisa
Positive model theory and automorphisms of ordered structures
Positive logic (also known as coherent logic) is a fragment of first-order logic that, via a construction known as Morleyisation, may also be regarded as a generalisation of the latter. From the points of view of algebras of definable sets and type spaces it corresponds to generalising from Boolean algebras/Stone spaces to distributive lattices/spectral spaces.

This framework allows for a generalisation of (neo-)stability theory techniques to certain non-elementary classes.

After introducing the framework, I will talk about a recent joint work with Jan Dobrowolski, where we develop the notion of NIP for positive logic, and apply this to the study of generic ordered abelian groups with an automorphism, and about work in progress of Jan Dobrowolski, Francesco Gallinaro, and myself, concerning automorphisms of valued fields.

13:45 Mark Kamsma
Queen Mary University of London
Unstable independence from the categorical point of view
Independence relations, such as linear independence, are a central tool in classification theory, an important branch of model theory. Through them, classical model-theoretic results have been vastly generalised to classes of structures that are not first-order axiomatisable. Recent developments in a category-theoretic approach have pushed this even further. In this talk we will look at one such development: a category-theoretic construction of independence relations in the context of locally presentable and accessible categories. Such a construction was already given by Lieberman, Vasey and Rosický for stable independence relations, the most well-behaved class of independence relations. We generalise this construction by extending it to the more general classes of simple and NSOP1-like independence relations. Throughout, we will consider many examples to make the talk accessible to a broader audience. This is joint work with J. Rosický.

14:25 Giacomo Tendas
University of Manchester
Enriched universal algebra
Universal algebra, introduced by Birkhoff, studies classes of algebras that satisfy a set of equations on a certain signature. In the framework of category theory, a treatment of universal algebra was given by Lawvere through his concept of algebraic theory. This categorical approach was later developed in the context of enriched categories, with the aim of “doing” universal algebra, not just over sets, but inside a base category with a suitable notion of tensor product.

In this talk, based on joint work with Rosický, we provide a logical counterpart to this story by introducing appropriate notions of enriched signatures and terms, and by characterizing the classes of algebras satisfying a given set of enriched equations. Our running examples of base categories will include those of posets and of metric spaces, where in addition to usual equalities between terms, enriched universal algebra allows to also compute inequalities and equalities up to ε>0.

11:30 Alberto Miguel Gómez
Imperial College London
A strictly NSOP_4 structure without stationarity

11:50 Pablo Andujar Guerrero
University of Leeds
A brief story of model-theoretic (p,q)-theorems

12:10 Calum Hughes
University of Manchester
Homotopy theory and Logic

14:00 Ricardo Palomino (Online)
University of Manchester
Local real closed SV-rings of finite rank

14:20 Cas Burton
University of Manchester
TBA

14:40 Shezad Mohamed
University of Manchester
TBA