Thu, 29 Apr 2021

12:00 - 13:00
Virtual

Bubble propagation in modified Hele-Shaw channels

Alice Thompson
(Manchester)
Abstract

The propagation of a deformable air finger or bubble into a fluid-filled channel with an imposed pressure gradient was first studied by Saffman and Taylor. Assuming large aspect ratio channels, the flow can be depth-averaged and the free-boundary problem for steady propagation solved by conformal mapping. Famously, at zero surface tension, fingers of any width may exist, but the inclusion of vanishingly small surface tension selects symmetric fingers of discrete finger widths. At finite surface tension, Vanden-Broeck later showed that other families of 'exotic' states exist, but these states are all linearly unstable.

In this talk, I will discuss the related problem of air bubble propagation into rigid channels with axially-uniform, but non-rectangular, cross-sections. By including a centred constriction in the channel, multiple modes of propagation can be stabilised, including symmetric, asymmetric and oscillatory states, with a correspondingly rich bifurcation structure. These phenomena can be predicted via depth-averaged modelling, and also observed in our experiments, with quantitative agreement between the two in appropriate parameter regimes. This agreement provides insight into the physical mechanisms underlying the observed behaviour. I will outline our efforts to understand how the system dynamics is affected by the presence of nearby unstable solution branches acting as edge states. Finally, I will discuss how feedback control and control-based continuation could be used for direct experimental observation of stable or unstable modes.

Thu, 16 May 2019

16:00 - 17:00
L6

A quantitative bound in the nonlinear Roth theorem

Sean Prendiville
(Manchester)
Abstract

We discuss a nonlinear variant of Roth’s theorem on the existence of three-term progressions in dense sets of integers, focusing on an effective version of such a result. This is joint work with Sarah Peluse.
 

Thu, 09 Nov 2017
16:00
L6

Probabilistic arithmetic geometry

Daniel Loughran
(Manchester)
Abstract

A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos.

Thu, 08 Jun 2017
17:30
L6

On the differential Dixmier-Moeglin equivalence.

Omar Leon Sanchez
(Manchester)
Abstract

Motivated by the Dixmier-Moeglin equivalence, which belongs to the realm of algebra representations, we look at a differential version of this equivalence for algebraic D-groups, which belong to the realm of finite Morley rank groups in differentially closed fields. We will see how the proof of this equivalence reduces to a standard model-theoretic fact (on binding groups). Time permitting we will present an application to Hopf-Ore extensions. This is joint work with J. Bell and R. Moosa.

Thu, 26 Jan 2017
17:30
L6

Existentially definable henselian valuation rings with p-adic residue fields

Arno Fehm
(Manchester)
Abstract

In joint work with Sylvy Anscombe we had found an abstract
valuation theoretic condition characterizing those fields F for which
the power series ring F[[t]] is existentially 0-definable in its
quotient field F((t)). In this talk I will report on recent 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 suggest and apply a
p-adic analogue of Lagrange's Four Squares Theorem.

Thu, 01 Dec 2016
17:30
L6

Pfaffian functions and elliptic functions

Gareth Jones
(Manchester)
Abstract

After giving some motivation, I will discuss work in progress 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.

Thu, 19 May 2016
17:30
L6

Interpreting formulas of divisible abelian l-groups in lattices of zero sets

Marcus Tressl
(Manchester)
Abstract

An abelian l-group G is essentially a partially ordered subgroup of functions from a set to a totally ordered abelian group such

that G is closed under taking finite infima and suprema. For example, G could be the continuous semi-linear functions defined on the open
unit square, or, G could be the continuous semi-algebraic functions defined in the plane with values in (0,\infty), where the group
operation is multiplication. I will show how G, under natural geometric assumptions, can be interpreted (in a weak sense) in its lattice of
zero sets. This will then be applied to the model theory of natural divisible abelian l-groups. For example we will see that the
aforementioned examples are elementary equivalent. (Parts of the results have been announced in a preliminary report from 1987 by F. Shen
and V. Weispfenning.)

Tue, 23 Feb 2016

14:15 - 15:30
L4

Discrete triangulated categories

David Pauksztello
(Manchester)
Abstract
This is a report on joint work with Nathan Broomhead and David Ploog.
 
The notion of a discrete derived category was first introduced by Vossieck, who classified the algebras admitting such a derived category. Due to their tangible nature, discrete derived categories provide a natural laboratory in which to study concretely many aspects of homological algebra. Unfortunately, Vossieck’s definition hinges on the existence of a bounded t-structure, which some triangulated categories do not possess. Examples include triangulated categories generated by ‘negative spherical objects’, which occur in the context of higher cluster categories of type A infinity. In this talk, we compare and contrast different aspects of discrete triangulated categories with a view toward a good working definition of such a category.
 

 
Thu, 18 Jun 2015

17:30 - 18:30
L6

On the Consistency Problem for Quine's New Foundations, NF

Peter Aczel
(Manchester)
Abstract

In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short.  Since then the consistency of NF has been a problem that remains open today.  But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have  solved the problem by showing that TST^+ is indeed consistent.  But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.

 
In my talk I will introduce TST^+ and its possible models and discuss some of the interesting ideas, that I have understood, that Holmes uses in one of his possible proofs.  If there is time in my talk I will also mention a more recent approach of Jamie Gabbay who is taking a nominal sets approach to the problem.
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