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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.

More details about the current grants, PhD students, and research interests are on our Research page.

Upcoming events

Andrew Brooke-Taylor (University of Leeds)

Location: MALL
Title: A free 2-generator shelf from large cardinals

One of the strongest known large cardinal axioms is I3, positing the existence of a non-trivial elementary embedding $j$ from $V_λ$ to $V_λ$ for some $λ$.  Given two such embeddings $j$ and $k$ for the same lambda, there is a natural "application" operation to yield a third, $j*k$, and elementarity shows that this operation is left self-distributive: $j*(k*l)=(j*k)*(j*l)$. Structures with such an operation are called LD-algebras or shelves. Laver showed that the algebra of embeddings generated by a single such $j$ under $*$ is in fact the free LD-algebra on 1 generator; and the set-theoretic context around this concrete (once you've assumed I3) instantiation of the free LD-algebra gives rise to various theorems about LD-algebras that are only known under this very strong large cardinal assumption.  Given I3, there will be many other embeddings from $V_λ$ to $V_λ$, and it is natural to ask if one can obtain from amongst them a free LD-algebra on more than one generator.  In joint work with Scott Cramer and Sheila Miller, we show that the answer is positive if one assumes a little more: from I2 we get a free 2-generator LD-algebra of embeddings.  This talk will focus on set-theoretic aspects of the proof; a week later I will be giving a talk in the ARTIN conference on the same topic, focusing more on the algebraic aspects.

Francesco Gallinaro (University of Pisa)

Location: MALL
Title: Valued difference fields: amalgamation and existential closedness

A valued difference field is a valued field equipped with an automorphism which fixes the valuation ring setwise. I will discuss various properties of the existentially closed valued difference fields, both from the algebraic and the model-theoretic perspective, and I will highlight the role of tropical geometry in some of the proofs. This is joint work with Jan Dobrowolski and Rosario Mennuni.

Jonathan Kirby (UEA)

Location: MALL
Title: Integration in finite terms and exponentially algebraic functions

The problem of integration in finite terms is the problem of finding exact closed forms for antiderivatives of functions, within a given class of functions. Liouville introduced his elementary functions (built from polynomials, exponentials, logarithms and trigonometric functions) and gave a solution to the problem for that class, nearly 200 years ago. The same problem was shown to be decidable and an algorithm given by Risch in 1969.

We introduce the class of exponentially algebraic functions, generalising the elementary functions and much more robust than them, and give characterisations of them both in terms of o-minimal local definability and in terms of definability in a reduct of the theory of differentially closed fields.

We then prove the analogue of Liouville's theorem for these exponentially-algebraic functions.

This is joint work with Rémi Jaoui.

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