The Leeds Logic group
Mathematical logic, a young subject, has developed over the last 30 years into an amalgam of fast-moving disciplines, each with its own sophisticated techniques. These are linked by profound common concerns, around definability, decidability, effectiveness and computability, the nature of the continuum, and foundations. Some branches are highly multidisciplinary and force the researcher to be fully conversant with other fields (e.g. algebra, computer science).
The Logic Group at the University of Leeds is one of the largest and most active worldwide, with a long uninterrupted tradition dating back to 1951, when its founder Martin Löb moved to Leeds. It has an international reputation for research in most of the main areas of mathematical logic - computability theory, model theory, set theory, proof theory, and in applications to algebra, analysis, number theory and theoretical computer science.
Adele Padgett (McMaster University)
Title: Regular solutions of systems of transexponential-polynomial equations
It is unknown whether there are o-minimal fields that are transexponential, i.e., that define functions which eventually grow faster than any tower of exponential functions. In past work, I constructed a Hardy field closed under a transexponential function $E$ which satisfies $E(x+1) = \exp E(x)$. Since the germs at infinity of unary functions definable in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by $E$ could be o-minimal. To prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach using a criterion for o-minimality due to Lion. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.
Impromptu Leeds-McMaster model theory days
An informal miniworkshop of early career model theorists in McMaster and (possibly formerly) Leeds. Carefully non-organised by Pantelis Eleftheriou and Vincenzo Mantova.
Elliot Kaplan (McMaster University)
Title: Hilbert polynomials for finitary matroids
Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets $A$ and $B$ of a commutative semigroup, the size of the sumset $A+tB$ is eventually polynomial in $t$. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks. This is joint work with Antongiulio Fornasiero.