Landscapes in Mathematical Sciences

This is our Department Colloquium, with distinguished speakers giving an overview of a topic of general mathematical interest. The talks are aimed at a level to be accessible to all postgraduate students and staff in the department. Advanced undergraduates and members of other departments are also welcome to attend.

Talks last a full hour and, unless otherwise stated, take place in the Wolfson Theatre (4W1.7), beginning at 3:15pm. Tea is available from 30 minutes before the talk in the foyer outside the lecture theatre.

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Next Landscapes Seminar

Date Speaker Title/Abstract
8 March 2019 Anne Juel
Manchester
Unstable front propagation
Morphological growth ranging from tumours and bacterial colonies, to electrodeposition is susceptibleto interfacial instabilities which can lead to pattern formation and the emergence of disordered dynamics. Investigation of their complex dynamics benefits from a multi-pronged approach of experiments and modelling. We focus on the canonical viscous fingering instability, which arises when a more viscous fluid is displaced by a less viscous one between two closely spaced plates known, a geometry known as a Hele-Shaw cell. If the displacement is radial, the unstable interface deforms into a set of growing fingers which evolve continuously in time through a sequence of tip-splitting events and subsequent competition of the newly formed fingers -- an example of disordered front propagation. By contrast, if the displacement flow is driven in a channel, a steadily propagating finger forms which is linearly stable for all values of the driving parameter but which can also undergo a subcritical transition to an unsteady, disordered front -- a scenario reminiscent of the transition to turbulence in shear flows identified more than a century ago, where significant progress in understanding has been achieved in the last 25 years. In that system, weakly unstable solutions play a key role in the transition.

In this talk, we ally experiments and numerical modelling to explore viscous fingering from a dynamical systems point of view and examine the role of unstable modes of propagation in two related geometries:
  1. A rigid channel with a small depth perturbation, where we show that the evolution of the steady state bifurcation diagram as the perturbation height is increased, leads to the stabilisation of previously unstable modes.
  2. A channel where the top boundary has been replaced with an elastic sheet. This bench top model of pulmonary airway reopening exhibits a wealth of novel instabilities including two distinct stable 'peeling' modes, which are separated by a parameter region where the front evolves transiently via an unstable 'pushing' mode towards either stable mode.

Autumn 2018

Date Speaker Title/Abstract
5 October 2018 Philip Maini
Oxford
Does mathematics have anything to do with biology?
In this talk, I will review a number of interdisciplinary collaborations in which I have been involved over the years that have coupled mathematical modelling with experimental studies to try to advance our understanding of processes in biology and medicine. Examples will include somatic evolution in tumours, collective cell movement in epithelial sheets, and pattern formation in slime mould. These are examples where verbal reasoning models are misleading and insufficient, while mathematical models can enhance our intuition.
19 October 2018 Nigel Hitchin
Oxford
Integrable systems and algebraic geometry
Completely integrable Hamiltonian systems form an important concept in many areas of mathematics and include classical examples like the equations for a spinning top or the geodesics on an ellipsoid. A huge range of examples comes from considering the algebraic geometry of moduli spaces of Higgs bundles on a Riemann surface. The talk will focus on the geometry of the singular locus for these systems and how certain constructions in algebraic geometry help to understand the structure of this locus.
9 November 2018 Eugene Shargorodsky
KCL
Some open problems related to Stokes waves
A Stokes wave is a steady periodic wave, propagating under gravity with constant speed on the surface of an infinitely deep irrotational flow. Its free surface is determined by Laplace's equation, kinematic and periodic boundary conditions and by a dynamic boundary condition given by the requirement that pressure in the flow at the surface should be constant (Bernoulli's theorem). The modern theory of Stokes waves has been shaped by the works of L.E. Fraenkel, J.F. Toland, and their collaborators. This part of nonlinear analysis has surprising connections to many other fields ranging from quantum mechanics to number theory. The aim of the talk is to discuss some of those connections and some related open problems.
23 November 2018 Richard Elwes
Leeds
Concrete Incompleteness
In 1931, Kurt Göel proved his famous incompleteness theorems, establishing that any attempt to axiomatize the arithmetic of the natural numbers would necessarily leave gaps: statements which can be neither proved nor disproved.

Gödel's original unprovable statements have a strongly meta-mathematical flavour, with little apparent relevance to mainstream mathematical concerns. In this talk, we will look at various examples of concrete incompleteness: unprovable statements which are also interesting and natural from a broader mathematical perspective. Particularly striking are some simple combinatorial statements, found recently by Harvey Friedman, which can only be resolved with large cardinal axioms.
7 December 2018 Ruth Gregory
Durham
The Decay of the Universe
Phase transitions are part of everyday life, yet are also believed to be part of the history of our universe, where the nature of particle interactions change as the universe settles into its vacuum state. The recent discovery of the Higgs and its mass suggests that our vacuum may not be entirely stable, and that a further phase transition could take place. My talk will review how we find the probability of these phase transitions and I will discuss my recent work on how black holes can dramatically change the result!
14 December 2018 Alejandro Adem
UBC Vancouver
The Topology of Commuting Matrices
In this talk we will discuss the structure of spaces of commuting elements in a compact Lie group. Their connected components and other basic topological properties will be discussed. We will also explain how they can be assembled to produce a space which classifies certain bundles and represents an interesting cohomology theory. A number of explicit examples will be provided for orthogonal, unitary and projective unitary groups.

Spring 2019

Date Speaker Title/Abstract
8 February 2019 Martin Hairer
Imperial College London
Taming Infinities
Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of "renormalisation", have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will dip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used to study equations whose meaning was not even clear until now.
1 March 2019 Juan Carlos Pardo
CIMAT Guanajuato
Population dynamics with applications to mathematical biology
In this talk I will survey some recent developments on the topic of population dynamics. In particular, I will focus on the long term behaviour of two families of branching processes: branching processes with interactions and branching processes in random environment. For each family of processes, I will explain how these results can be applied to some specific models in biology.
8 March 2019 Anne Juel
Manchester
Unstable front propagation
Morphological growth ranging from tumours and bacterial colonies, to electrodeposition is susceptibleto interfacial instabilities which can lead to pattern formation and the emergence of disordered dynamics. Investigation of their complex dynamics benefits from a multi-pronged approach of experiments and modelling. We focus on the canonical viscous fingering instability, which arises when a more viscous fluid is displaced by a less viscous one between two closely spaced plates known, a geometry known as a Hele-Shaw cell. If the displacement is radial, the unstable interface deforms into a set of growing fingers which evolve continuously in time through a sequence of tip-splitting events and subsequent competition of the newly formed fingers -- an example of disordered front propagation. By contrast, if the displacement flow is driven in a channel, a steadily propagating finger forms which is linearly stable for all values of the driving parameter but which can also undergo a subcritical transition to an unsteady, disordered front -- a scenario reminiscent of the transition to turbulence in shear flows identified more than a century ago, where significant progress in understanding has been achieved in the last 25 years. In that system, weakly unstable solutions play a key role in the transition.

In this talk, we ally experiments and numerical modelling to explore viscous fingering from a dynamical systems point of view and examine the role of unstable modes of propagation in two related geometries:
  1. A rigid channel with a small depth perturbation, where we show that the evolution of the steady state bifurcation diagram as the perturbation height is increased, leads to the stabilisation of previously unstable modes.
  2. A channel where the top boundary has been replaced with an elastic sheet. This bench top model of pulmonary airway reopening exhibits a wealth of novel instabilities including two distinct stable 'peeling' modes, which are separated by a parameter region where the front evolves transiently via an unstable 'pushing' mode towards either stable mode.

A list of talks in recent past years can be found here.