Past

In this talk I will present two recent findings concerning the preconditioning of elliptic problems. The first result concerns preconditioning of elliptic problems with variable coefficient K by an inverse Laplacian. Here we show that there is a close relationship between the eigenvalues of the preconditioned system and K. 
The second results concern the problem on mixed form where K approaches zero. Here, we show a uniform inf-sup condition and corresponding robust preconditioning. 

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

I will discuss the basics of normal surface theory, and how they were used to give an algorithm for deciding whether a given diagram represents the unknot. This version is primarily based on Haken's work, with simplifications from Schubert and Jaco-Oertel.
 

What does abstract nonsense (category theory) have to do with the apparently opposite proverbial concreteness of physics? In this talk I will try to convey the importance of understanding physical theories from a compositional and structural perspective, where the fundamental logic of interaction between systems becomes the real protagonist. Firstly, we will see how different classes of symmetric monoidal categories can be used to model physical processes in a very natural and intuitive way. We will then explore the claim that category theory is not only useful in providing a unified framework, but it can be also used to perfect and modify preexistent models. In this direction, I will show how the introduction of a trace in the symmetric monoidal category describing QIT can be used to talk about quantum interactions induced by cyclic causal relationships.

I will discuss joint work with Gismatullin, Halupczok, and Simonetta on the following problem: given a henselian valued field of characteristic 0, possibly equipped with analytic structure (in the sense stemming originally from Denef and van den Dries), describe the possibilities for a definable group G in the valued field sort which is definably almost simple, that is, has no proper infinite definable normal subgroups. We also have results for an algebraically closed valued field K in characteristic p, but assuming also that the group is a definable subgroup of GL(n, K).

The (total) Steenrod square is a ring homomorphism from the cohomology of a topological space to the Z/2-equivariant cohomology of this space, with the trivial Z/2-action. Given a closed monotone symplectic manifold, one can define a deformed notion of the Steenrod square for quantum cohomology, which will not in general be a ring homomorphism, and prove some properties of this operation that are analogous to properties of the classical Steenrod square. We will then link this, in a more general setting, to a definition by Seidel of a similar operation on Floer cohomology.
 

In this talk I will introduce a new mathematical model for the computational modelling of the active contraction of cardiac tissue using stress-assisted conductivity as the main mechanism for mechanoelectrical feedback. The coupling variable is the Kirchhoff stress and so the equations of hyperelasticity are written in mixed form and a suitable finite element formulation is proposed. Next I will introduce a simplified version of the coupled system, focusing on its analysis in terms of solvability and stability of continuous and discrete mixed-primal formulations, and the convergence of these methods will be illustrated through two numerical tests.

The extremal number ${\rm ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [0,2]$ is realisable if there exists a graph $F$ with ${\rm ex}(n , F) = \Theta(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$.

We discuss some progress on the conjecture of Erdős and Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

The celebrated localisation theorem of Beilinson-Bernstein asserts that there is an equivalence between representations of a Lie algebra and modules over the sheaf of differential operators on the corresponding flag variety. In this talk we discuss certain analogues of this result in various contexts. Namely, there is a localisation theorem for quantum groups due to Backelin and Kremnizer and, more recently, Ardakov and Wadsley also proved a localisation theorem working with certain completed enveloping algebras of p-adic Lie algebras. We then explain how to combine the ideas involved in these results to construct
a p-adic analytic quantum flag variety and a category of D-modules on it, and we show that the global section functor on these D-modules yields an equivalence of categories.

In petroleum reservoir simulation, the standard preconditioner is a two-stage process which involves solving a restricted pressure system with AMG. Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not incorporate heat diffusion or the effects of temperature changes on fluid flow through viscosity and density. We seek to develop preconditioners which consider this cross-coupling between pressure and temperature. In order to study the effects of both pressure and temperature on fluid and heat flow, we first consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Then, we extend this method for multiphase flow as a two-stage preconditioner.

We present a self-contained construction of the Euclidean $\Phi^4$ quantum
field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we
consider an approximation of the stochastic quantization equation on
$\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and
side length $M$. We introduce an energy method and prove tightness of the
corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow
\infty$. We show that every limit point satisfies reflection positivity,
translation invariance and nontriviality (i.e. non-Gaussianity). Our
argument applies to arbitrary positive coupling constant and also to
multicomponent models with $O(N)$ symmetry. Joint work with Massimiliano
Gubinelli.

Disease transmission and rumour spreading are ubiquitous in social and technological networks. In this talk, we will present our last results on the modelling of rumour and disease spreading in multilayer networks.  We will derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks in a multiplex network. Moreover, we will introduce a model of epidemic spreading with awareness, where the disease and information are propagated in different layers with different time scales. We will show that the time scale determines whether the information awareness is beneficial or not to the disease spreading. 

In recent years it has been discovered that also non-linear, degenerate equations like the $p$-Poisson equation $$ -\mathrm{div}(A(\nabla u))= - \mathrm{div} (|\nabla u|^{{p-2}}\nabla u)= -{\rm div} F$$ allow for optimal regularity. This equation has similarities to the one of power-law fluids. In particular, the non-linear mapping $F \mapsto A(\nabla u)$ satisfies surprisingly the linear, optimal estimate $\|A(\nabla u)\|_X \le c\, \|F\|_X$ for several choices of spaces $X$. In particular, this estimate holds for Lebesgue spaces $L^q$ (with $q \geq p'$), spaces of bounded mean oscillations and Holder spaces$C^{0,\alpha}$ (for some $\alpha>0$).

In this talk we show that we can extend this theory to Sobolev and Besov spaces of (almost) one derivative. Our result are restricted to the case of the plane, since we use complex analysis in our proof. Moreover, we are restricted to the super-linear case $p \geq 2$, since the result fails $p < 2$. Joint work with Anna Kh. Balci, Markus Weimar.

I will present a construction of large families of singularity-free stationary solutions of Einstein equations, for a large class of matter models including vacuum, with a negative cosmological constant. The solutions, which are of course real-valued Lorentzian metrics, are determined by a set of free data at conformal infinity, and the construction proceeds through elliptic equations for complex-valued tensor fields. One thus obtains infinite dimensional families of both strictly stationary spacetimes and black hole spacetimes.

In this talk I will discuss recent joint work with Dominik Gruber where 
we find a reasonable model for random (infinite) Burnside groups, 
building on earlier tools developed by Coulon and Coulon-Gruber.

The free Burnside group with rank r and exponent n is defined to be the 
quotient of a free group of rank r by the normal subgroup generated by 
all elements of the form g^n; quotients of such groups are called 
Burnside groups.  In 1902, Burnside asked whether any such groups could 
be infinite, but it wasn't until the 1960s that Novikov and Adian showed 
that indeed this was the case for all large enough odd n, with later 
important developments by Ol'shanski, Ivanov, Lysenok and others.

In a different direction, when Gromov developed the theory of hyperbolic 
groups in the 1980s and 90s, he observed that random quotients of free 
groups have interesting properties: depending on exactly how one chooses 
the number and length of relations one can typically gets hyperbolic 
groups, and these groups are infinite as long as not too many relations 
are chosen, and exhibit other interesting behaviour.  But one could 
equally well consider what happens if one takes random quotients of 
other free objects, such as free Burnside groups, and that is what we 
will discuss.
 

I will talk about R^n valued random processes driven by a "noise", which is generated by a deterministic dynamical system, randomness coming from the choice of the initial condition.

Such processes were considered by D.Kelly and I.Melbourne.I will present our joint work with I.Chevyrev, P.Friz, I.Melbourne and H.Zhang, where we consider the noise with long term memory. We prove convergence to solution of a stochastic differential equation which is, depending on the noise, driven by either a Brownian motion (optimizing the assumptions of Kelly-Melbourne) or a Lévy process.Our work is made possible by recent progress in rough path theory for càdlàg paths in p-variation topology.

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

We consider the problem of optimally hedging a portfolio of derivatives in a scenario based discrete-time market with transaction costs. Risk-preferences are specified in terms of a convex risk-measure. Such a framework has suffered from numerical intractability up until recently, but this has changed thanks to technological advances: using hedging strategies built from neural networks and machine learning optimization techniques, optimal hedging strategies can be approximated efficiently, as shown by the numerical study and some theoretical results presented in this talk (based on joint work with Hans Bühler, Ben Wood and Josef Teichmann).

This week is a women's only week in which we are joined by the Mirzakhani society, the society for undergraduate women in maths, for lunch.

I will review some recent progress on D=4, N=2 superconformal field theories in what has come to be known as "Class-S". This is a huge class of (mostly non-Lagrangian) SCFTs, whose properties are encoded in the data of a punctured Riemann surface and a collection (one per puncture) of nilpotent orbits in an ADE Lie algebra.

How much do you know actually about the research that is going on across the department? The SIAM Student Chapter brings you a 3 minute thesis competition challenging a group of DPhil students to go head to head to explain their research in just 3 minutes with the aid of a single slide. This is the perfect opportunity to hear about a wide range of topics within applied mathematics, and to gain insight into the impact that mathematical research can have. The winner will be decided by a judging panel comprising Professors Helen Byrne, Jon Chapman, Patrick Farrell, and Christina Goldschmidt.
 

A matrix M of real numbers is called totally positive if every minor of M is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis - notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions - as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix M of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients. Also, a sequence (an)n≥0  of real numbers (or polynomials) will be called (coefficientwise) Hankel-totally positive if the Hankel matrix H = (ai+j)i,j ≥= 0 associated to (an) is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.

The magma chamber - an underground vat of fluid magma that is tapped during volcanic eruptions - has been the foundation of models of volcanic eruptions for many decades and successfully explains many geological observations.  However, geophysics has failed to image the postulated large magma chambers, and the chemistry and ages of crystals in erupted magmas indicate a more complicated history.  New conceptual models depict subsurface magmatic systems as dominantly uneruptible crystalline networks with interstitial melt (mushes) extending deep into the Earth's crust to the mantle, containing lenses of potentially eruptible (low-crystallinity) magma.  These lenses would commonly be less dense than the overlying mush and so Rayleigh Taylor instabilities should develop leading to ascent of blobs of magma unless the growth rate is sufficiently slow that other processes (e.g. solidification) dominate.  The viscosity contrast between a buoyant layer and mush is typically extremely large; a consequence is that the horizontal dimension of a magma reservoir is commonly much less than the theoretical fastest growing wavelength assuming an infinite horizontal layer.  

I will present laboratory experiments and linear stability analysis for low Reynolds number, laterally confined Rayleigh Taylor instabilities involving one layer that is much thinner and much less viscous than the other.  I will then apply the results to magmatic systems, comparing timescales for development of the instability and the volumes of packets of rising melt generated, with the frequencies and sizes of volcanic eruptions.  I will then discuss limitations of this work and outstanding fluid dynamical problems in this new paradigm of trans-crustal magma mush systems.

The Goldbach conjecture is a famous unsolved problem in mathematics. It asks whether every even number greater than or equal to 4 is the sum of two primes. I will discuss some of the history of the problem, explaining among other things why the answer is surely yes, and also why this appears to be very hard to prove.

Despite progress in understanding many aspects of malignancy, resistance to therapy is still a frequent occurrence. Recognised causes of this resistance include 1) intra-tumour heterogeneity resulting in selection of resistant clones, 2) redundancy and adaptability of gene signalling networks, and 3) a dynamic and protective microenvironment. I will discuss how these aspects influence each other, and then focus on the tumour microenvironment.

The tumour microenvironment comprises a heterogeneous, dynamic and highly interactive system of cancer and stromal cells. One of the key physiological and micro-environmental differences between tumour and normal tissues is the presence of hypoxia, which not only alters cell metabolism but also affects DNA damage repair and induces genomic instability. Moreover, emerging evidence is uncovering the potential role of multiple stroma cell types in protecting the tumour primary niche.

I will discuss our work on in silico cancer models, which is using genomic data from large clinical cohorts of individuals to provide new insights into the role of the tumour microenvironment in cancer progression and response to treatment. I will then discuss how this information can help to improve patient stratification and develop novel therapeutic strategies.

Harmonic map equations are an elliptic PDE system arising from the  
minimisation problem of Dirichlet energies between two manifolds. In  
this talk we present some some recent works concerning the symmetry  
and stability of harmonic maps. We construct a new family of  
''twisting'' examples of harmonic maps and discuss the existence,  
uniqueness and regularity issues. In particular, we characterise of  
singularities of minimising general axially symmetric harmonic maps,  
and construct non-minimising general axially symmetric harmonic maps  
with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.  
Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$  
to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for  
$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and  
Mazowiecka--Strzelecki.

(Joint work with Prof. Robert M. Hardt.)

Over the past decade, and particularly over the past five years, research at the interface of topology and neuroscience has grown remarkably fast.  In this talk I will briefly survey a few quite different applications of topology to neuroscience in which members of my lab have been involved over the past four years: the algebraic topology of brain structure and function, topological characterization and classification of neuron morphologies, and (if time allows) topological detection of network dynamics.

The Oxford-Princeton Workshops on Financial Mathematics & Stochastic Analysis have been held approximately every eighteen months since 2002, alternately in Princeton and Oxford. They bring together leading groups of researchers in, primarily, mathematical and computational finance from Oxford University and Princeton University to collaborate and interact. The series is organized by the Oxford Mathematical and Computational Finance Group, and at Princeton by the Department of Operations Research and Financial Engineering and the Bendheim Center for Finance.

Oxford Mathematics Public Lectures
Hooke Lecture

Michael Berry - Chasing the dragon: tidal bores in the UK and elsewhere
15 November 2018 - 5.15pm

In some of the world’s rivers, an incoming high tide can arrive as a smooth jump decorated by undulations, or as a breaking wave. The river reverses direction and flows upstream.

Understanding tidal bores involves

· analogies with tsunamis, rainbows, horizons in relativity, and ideas from  quantum physics;

· the concept of a ‘minimal model’ in mathematical explanation;

· different ways in which different cultures describe the same thing;

· the first unification in fundamental physics.

Michael Berry is Emeritus Professor of Physics, H H Wills Physics Laboratory, University of Bristol

5.15pm, Mathematical Institute, Oxford

Please email @email to register.

Watch live:

https://www.facebook.com/OxfordMathematics
https://livestream.com/oxuni/Berry

Oxford Mathematics Public Lectures are generously supported by XTX Markets.

I will give an overview of some recent progress on potential automorphy results over CM fields, that is joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne. I will focus on explaining an application to the generalized Ramanujan-Petersson conjecture. 

Lagrangian Floer homology has been used by Ozsvath and Szabo to define a package of three-manifold invariants known as Heegaard Floer homology. I will give an introduction to the topic.

Boundary layers control the transport of momentum, heat, solutes and other quantities between walls and the bulk of a flow. The Prandtl-Blasius boundary layer was the first quantitative example of a flow profile near a wall and could be derived by an asymptotic expansion of the Navier-Stokes equation. For higher flow speeds we have scaling arguments and models, but no derivation from the Navier-Stokes equation. The analysis of exact coherent structures in plane Couette flow reveals ingredients of such a more rigorous description of boundary layers. I will describe how exact coherent structures can be scaled to obtain self-similar structures on ever smaller scales as the Reynolds number increases.

A quasilinear approximation allows to combine the structures self-consistently to form boundary layers. Going beyond the quasilinear approximation will then open up new approaches for controlling and manipulating boundary layers.

Abstract: Akshay Venkatesh and his coauthors (Galatius, Harris, Prasanna) have recently introduced a derived Hecke algebra and a derived Galois deformation ring acting on the homology of an arithmetic group, say with p-adic coefficients. These actions account for the presence of the same system of eigenvalues simultaneously in various degrees. They have also formulated a conjecture describing a finer action of a motivic group which should preserve the rational structure $H^i(\Gamma,\Q)$. In this lecture we focus in the setting of classical modular forms of weight one, where the same systems of eigenvalues appear both in degree 0 and 1 of coherent cohomology of a modular curve, and the motivic group referred to above is generated by a Stark unit. In joint work with Darmon, Harris and Venkatesh, we exploit the Theta correspondence and higher Eisenstein elements to prove the conjecture for dihedral forms.

In many applications requiring the solution of a linear system Ax=b, the matrix A has been shown to have a low-rank property: its off-diagonal blocks have low numerical rank, i.e., they can be well approximated by matrices of small rank. Several matrix formats have been proposed to exploit this property depending on how the block partitioning of the matrix is computed.
In this talk, I will discuss the block low-rank (BLR) format, which partitions the matrix with a simple, flat 2D blocking. I will present the main characteristics of BLR matrices, in particular in terms of asymptotic complexity and parallel performance. I will then discuss some recent advances and ongoing research on BLR matrices: their multilevel extension, their use as preconditioners for iterative solvers, the error analysis of their factorization, and finally the use of fast matrix arithmetic to accelerate BLR matrix operations.

I will prove the 2d Biot-Savart law for the vorticity being an unbounded measure $\mu$, i.e. such that $\mu(\mathbb{R}^2)=\infty$, and show how can one infer some useful information concerning Kaden's spirals using it. Vorticities being unbounded measures appear naturally in the engineering literature as self-similar approximations of 2d Euler flows, see for instance Kaden's or Prandtl's spirals. Mathematicians are interested in such objects since they seem to be related to the questions of well-posedness of Delort's solutions of the 2d vortex sheet problem for the Euler equation. My talk is based on a common paper with K.Oleszkiewicz, M. Preisner and M. Szumanska.

Polycyclic groups either have polynomial growth, in which case they are virtually nilpotent, or exponential growth. I will give two interesting examples of "small" polycyclic groups which are extensions of $\mathbb{R}^2$ and the Heisenberg group by the integers, and attempt to justify the claim that they are small by sketching an argument that every exponential growth polycyclic group contains one of these.

Nets of lines are line arrangements satisfying very strict intersection conditions. We will see that nets can be defined in a very natural way in algebraic geometry, and, thanks to the strict intersection properties they satisfy, we will see that a lot can be said about classifying them over the complex numbers. Despite this, there are still basic unanswered questions about nets, which we will discuss. 
 

A lot of physical processes are modelled by conservation laws (mass, momentum, energy, charge, ...) Because of natural symmetries, these conservation laws express often that some symmetric tensor is divergence-free, in the space-time variables. We extract from this structure a non-trivial information, whenever the tensor takes positive semi-definite values. The qualitative part is called Compensated Integrability, while the quantitative part is a generalized Gagliardo inequality.

In the first part, we shall present the theoretical analysis. The proofs of various versions involve deep results from the optimal transportation theory. Then we shall deduce new fundamental estimates for gases (Euler system, Boltzmann equation, Vlaov-Poisson equation).

One of the theorems will have been used before, during the Monday seminar (PDE Seminar 4pm Monday 12 November).

All graduate students, post-docs faculty and visitors are welcome to come to the lectures. If you aren't a member of the CDT please email @email to confirm that you will be attending.

During the talk I will present some new computational technique based on excursion theory for Markov processes. Some new results for classical processes like Bessel processes and reflected Brownian Motion will be shown. The most important point of presented applications will be the new insight into Hartman-Watson (HW) distributions. It turns out that excursion theory will enable us to deduce the simple connections of HW with a hyperbolic cosine of Brownian Motion.

I will explain how complex varieties which have asymptotically large intersections with finite grids can be seen to correspond to projective geometries, exploiting ideas of Hrushovski. I will describe how this leads to a precise characterisation of such varieties. Time permitting, I will discuss consequences for generalised sum-product estimates and connections to diophantine problems. This is joint work with Emmanuel Breuillard.

The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

This work determines an aim point selection strategy for players in order to improve their chances of winning at the classic darts game of 501. Although many previous studies have considered the problem of aim point selection in order to maximise the expected score a player can achieve, few have considered the more general strategical question of minimising the expected number of turns required for a player to finish. By casting the problem as a Markov decision process, a framework is derived for the identification of the optimal aim point for a player in an arbitrary game scenario.

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all intersection sizes larger than $2^{k-1}$ (the “large” sizes) are of the form $2^{k-1} + 2^i$. We show that this is almost true: the large intersection sizes are either of this form or of the form $35\cdot2^{k-6}$ . We also disprove a second conjecture of Melo and Winter by proving that a positive fraction of the “small” values is missing.

Polyfree groups are defined as groups having a series of normal
subgroups such that each sucessive quotient is free. This property
imples locally indicability and therefore also right orderability. Right
angled Artin groups are known to be polyfree (a result shown
independently by Duchamp-Krob, Howie and Hermiller-Sunic). Here we show
that Artin FC-groups for which all the defining relation are of even
type  are also polyfree. This is a joint work with Ruven Blasco and Luis
Paris.

The talk introduces the problem of completing a partially observed matrix whose columns obey a nonlinear structure. This is an extension of classical low-rank matrix completion where the structure is linear. Such matrices are in general full rank, but it is often possible to exhibit a low rank structure when the data is lifted to a higher dimensional space of features. The presence of a nonlinear lifting makes it impossible to write the problem using common low-rank matrix completion formulations. We investigate formulations as a nonconvex optimisation problem and optimisation on Riemannian manifolds.

In gas-liquid two-phase pipe flows, flow regime transition is associated with changes in the micro-scale geometry of the flow. In particular, the bubbly-slug transition is associated with the coalescence and break-up of bubbles in a turbulent pipe flow. We consider a sequence of models designed to facilitate an understanding of this process. The simplest such model is a classical coalescence model in one spatial dimension. This is formulated as a stochastic process involving nucleation and subsequent growth of ‘seeds’, which coalesce as they grow. We study the evolution of the bubble size distribution both analytically and numerically. We also present some ideas concerning ways in which the model can be extended to more realistic two- and three-dimensional geometries.

Quantum Yang-Mills theory is an important part of the Standard model built
by physicists to describe elementary particles and their interactions. One
approach to this theory consists in constructing a probability measure on an
infinite-dimensional space of connections on a principal bundle over
space-time. However, in the physically realistic 4-dimensional situation,
the construction of this measure is still an open mathematical problem. The
subject of this talk will be the physically less realistic 2-dimensional
situation, in which the construction of the measure is possible, and fairly
well understood.

In probabilistic terms, the 2-dimensional Yang-Mills measure is the
distribution of a stochastic process with values in a compact Lie group (for

example the unitary group U(N)) indexed by the set of continuous closed
curves with finite length on a compact surface (for example a disk, a sphere
or a torus) on which one can measure areas. It can be seen as a Brownian
motion (or a Brownian bridge) on the chosen compact Lie group indexed by
closed curves, the role of time being played in a sense by area.

In this talk, I will describe the physical context in which the Yang-Mills
measure is constructed, and describe it without assuming any prior
familiarity with the subject. I will then present a set of results obtained
in the last few years by Antoine Dahlqvist, Bruce Driver, Franck Gabriel,
Brian Hall, Todd Kemp, James Norris and myself concerning the limit as N
tends to infinity of the Yang-Mills measure constructed with the unitary
group U(N).