Maxwell Institute for Mathematical Sciences

Analysis Seminar

The schedule includes talks in the Analysis seminar as well as closely related analysis events on Friday afternoon.

Unless noted otherwise, the seminar takes place in room CMS.01 of the Colin Maclaurin Building on the Riccarton Campus of Heriot-Watt University. Information on how to get there is available here.
In particular, the building has the number 22 on this campus map.

Beware! This is the more applied and PDE oriented of the two analysis seminars in Edinburgh. The second one (harmonic analysis oriented) is organized by Jonas Azzam on the King's Buildings Campus of the University of Edinburgh.

Date Speaker and title
16/3, 2.15pm Aram Karakhanyan (Edinburgh)
Singular perturbation problems and free boundaries
16/3, 3.15pm Marcelo Epstein (Calgary)
Geometry and continua
This presentation will survey some applications of Differential Geometry to Continuum Mechanics. The aim is to motivate the use of the geometric terminology and apparatus rather than to present technical details. Accordingly, the style will be informal and the scope as comprehensive as possible.
19/03 Maxwell Institute Mini-Symposium in PDEs
speakers include Xavier Ros-Oton (Zurich) and Peter Topping (Warwick)
26/3 Ph.D. student day in stochastic analysis
postponed Tom Leinster (Edinburgh), joint with algebra
but upcoming Title: TBA
Abstract: TBA
12-14/04, ICMS 50th anniversary meeting of the North British Functional Analysis seminar
21-22/05, ICMS Nonlinear Analysis and the Physical and Biological Sciences,
in honour of Jack Carr

Previous talks:
9/3, 2.15pm Frank Smith (UCL)
Shear flow over flexible in-wall patches
Shear flow over a finite compliant patch (bump or dip) in an otherwise fixed wall is considered here. This concerns unsteady flow in a channel, pipe or boundary layer, for two or three spatial dimensions. Applications in aerodynamics, sports, environment, biomedicine, drag reduction and flow-structure interactions form the background. Slowly evolving features are examined first to allow for variations in the incident flow. Linear and nonlinear analyses show that at certain parameter values (eigenvalues) resonances occur in which the interactive effect on flow and patch shape is enlarged by an order of magnitude. Similar findings apply to a boundary layer with several tiny patches present or to channel flows with patches of almost any length. These resonances lead on to fully nonlinear unsteady motion as a second stage, after some delay, combining with finite-time breakups to form a distinct path into transition of the flow.
9/3, 3.15pm Jacques Vanneste (Edinburgh), joint with applied maths
Geometric generalised Lagrangian mean theories
In fluid dynamics, it is often natural to separate flows between a mean and fluctuations (which often represent waves). It has long been recognised that the mean is best defined by Lagrangian averaging, that is, by averaging at fixed particle label rather than at fixed position in space. This is because the mean flow can then inherit key Lagrangian invariants, vorticity in particular, from the inviscid fluid equations. The best known theory of Lagrangian averaging is the generalised Lagrangian mean (GLM) theory of Andrews & McIntyre. The theory has some weaknesses which can be traced to its lack of geometrically intrinsic meaning. I will describe how this can be remedied, building on the geometric interpretation of incompressible fluid flows as trajectories in the group of volume-preserving diffeomorphisms. (Joint work with A D Gilbert.)
16/2 Susana Gutierrez (Birmingham)
The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions
The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe our work concerning the properties and dynamical behaviour of the family of self-similar solutions under the one-dimensional LLG-equation. Motivated by the properties of this family of self-similar solutions, in the second part of this talk we consider the Cauchy problem for the LLG-equation with Gilbert damping and provide a global well-posedness result provided that the BMO norm of the initial data is small.
9/2 Marcus Waurick (Strathclyde)
Fibre Homogenisation for n-dimensional systems of partial differential equations
In the talk we present a method to obtain order-sharp operator-norm resolvent estimates for a homogenisation problem related to a higher dimensional system of elliptic pdes. The method relies on the application of a suitable fibre decomposition and an appropriate scaling transform. With this one enters an abstract scheme, where the homogenised coefficients can be identified to be, in fact, Hilbert space projections applied to the coefficients. This approach leads to a set of 'asymptotically equivalent' homogenisation formulas; the standard effective coefficients being among these. The results have applications to the equations of linearised elasticity as well as to Maxwell's equations. This is joint work with Shane Cooper from the University of Durham. Details can be found in arXiv:1706.00645.
26/1 EMS meeting: Luis Vega (Basque country), ICMS
19/1 Mike Todd (St. Andrews)
Statistical stability in dynamical systems
If each member of a continuous family $(f_t)_t$ of dynamical systems possesses a `physical’ measure $\mu_t$ (that is, a measure describing the behaviour of Lebesgue-typical points), one can ask if the family of measures $(\mu_t)_t$ is also continuous in $t$: this is statistical stability, so called because the statistics (for example, in terms of Birkhoff averages for $(f_t, \mu_t)$) change continuously in $t$. I’ll discuss this problem for interval maps (eg tent maps, quadratic maps). Statistical stability can be destroyed by topological obstructions, or by a lack of uniform hyperbolicity. I’ll outline a general theory which guarantees statistical stability, giving examples to show the sharpness of our results. This is joint work with Neil Dobbs (UCD).
13/12 Mike Whittaker (Glasgow), joint with algebra
Fractal substitution tilings and applications to noncommutative geometry
Starting with a substitution tiling, such as the Penrose tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles typically have fractal boundary. As an application of fractal tilings, we construct an odd spectral triple on a C*-algebra associated with an aperiodic substitution tiling. Even though spectral triples on substitution tilings have been extremely well studied in the last 25 years, our construction produces the first truly noncommutative spectral triple associated with a tiling. My work on fractal substitution tilings is joint with Natalie Frank and Sam Webster, and my work on spectral triples is joint with Michael Mampusti.
01/12 Gilles Francfort (Paris)
Periodic homogenization in 2d linear elasticity redux
In 1993, Giuseppe Geymonat, Stefan Mueller and Nicolas Triantafyllidis demonstrated that, in the setting of linearized elasticity, a Gamma-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant over the whole space is zero while the corresponding coercivity constant on the torus remains positive. We revisit and complete this result. On the one hand we find sufficient (and essentially unique) conditions for such a situation to occur through a rigorous revisiting of a laminate construction given by Sergio Gutierrez in 1999. On the other hand, we demonstrate that the envisioned Gamma-convergence process actually delivers the correct homogenized energy when considering lower order terms, thereby alleviating concerns of irrelevance. This is joint work in part with Marc Briane (Rennes), and in part with Antoine Gloria (Paris).
15-17/11, ICMS Graduate School on Evolution Equations
courses by Jean-Frederic Gerbeau (Inria / Paris) and Hans Othmer (Minnesota)
10/11, 1.15pm, CMT.01 Luigi Berselli (Pisa)
On the regularity for nonlinear systems of the p-Stokes type
In this talk I will give an overview of recent results concerning the regularity up-to-the boundary for solutions of nonlinear systems of partial differential equations, as those arising in some problems related with non-Newtonian fluids, especially in the study of the rates convergence of numerical schemes. In particular, I will stress the delicate interplay between the boundary conditions and the pressure in terms of regularity. Together with -state of the art results, I will present also some open problems.
07/11, 2.15pm, CMT.01 Tobias Kuna (Reading)
The Truncated Moment Problem
Let K be a subset of the real numbers. The (one-dimensional) truncated moment problem on K is to find, for given numbers m_1,...,m_n, a random variable X which takes values in K and whose moments are given by the m_k: E[X^k]=m_k. More accurately, one wants to find necessary and sufficient conditions, in term of the m_k, for the existence of such a random variable. The multi-dimensional version of this problem, in which K is a subset of a Euclidean space of higher dimension, is surprisingly hard and is far from being resolved; we give a short introduction to the problem and to the state of the art. Finally, we describe a recent result concerning the truncated moment problem for a discrete set in one dimension; this is work in collaboration with M. Infusino, J. Lebowitz, and E. Speer.
03/11, 2.15pm Andrew Duncan (Sussex and Alan Turing Institute)
Measuring the quality of samples with diffusions
To improve the efficiency of Monte Carlo estimators, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. While a reduction in variance due to more rapid sampling can outweigh the bias introduced, the inexactness creates new challenges for parameter selection. In particular, standard measures of sample quality, such as effective sample size, do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that quantifies the maximum discrepancy between sample and target expectations over a large class of test functions. We demonstrate this tool by comparing exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.
01/11, 3.15pm, CMS.01 Roman Bessonov (St. Petersburg)
A spectral Szego theorem on the real line
One version of the classical Szego theorem describes probability measures on the unit circle with finite entropy integral in terms of their recurrence (or Verblunski) coefficients. I will discuss a version of this result for even measures supported on the real line. According to Krein - de Branges inverse spectral theory, with every nonzero measure on the real line that sums up Poisson kernels one can associate the unique canonical Hamiltonian system on the positive half-axis whose Hamiltonian has unit trace almost everywhere on the half-axis. The main subject of the talk is to give a characterization of the Hamiltonians arising from even measures on the real line with finite entropy integral. No background in canonical Hamiltonian systems is assumed. This is a joint work with Serguei Denissov (University of Wisconsin-Madison).
27/10, 2.15pm Lyuba Chumakova (UoE)
Effective Boundary Conditions for Semi-Open Dispersive Systems
In the classical linear dispersive wave theory it is shown that sinusoidal waves (i.e., $\propto e^i(k x -\omega t))$ carry energy with the group speed $c_g = d\omega/dk$. This concept is limited to the case where both the frequency $\omega(k)$ and the wavenumber $k$ must be real. On the other hand, semi-open dispersive systems allow more than just sinusoidal solutions: they can have exponentially blowing up and/or decaying solutions as well. In this talk I will address the questions of what is the direction and the speed of the energy propagation for these exponential waves. I will show how this can be implemented to develop radiation boundary conditions for semi-open dispersive systems and show an example of the an application of this theory to the dry atmosphere. While the real atmosphere does not have a definite top, it can be modelled as finite because the buoyancy frequency has a jump at the tropopause. This, effectively, acts as a ``leaky'' lid on the lower atmosphere. The leakage of gravity waves from the troposphere to the stratosphere is a significant physical effect that cannot be ignored -- as happens when a rigid lid is added at the tropopause.
27/10, 3.15pm Matthias Langer (Strathclyde)
Extensions of symmetric operators
In this talk I will consider a framework that can be used to describe extensions of symmetric operators in Hilbert spaces. In particular, I will consider elliptic operators with possibly non-self-adjoint and/or non-local boundary conditions and Schrödinger operators with delta-potentials supported on hypersurfaces and with complex coefficients. The abstract framework can be used to obtain enclosures for the spectrum or to prove that they are m-sectorial and hence generate analytic semigroups.
13/10 Lorenzo Foscolo (HW)
Degenerations of 4-dimensional Ricci-flat metrics
Einstein manifolds are the Riemannian analogues of solutions to Einstein’s equations of General Relativity. The Einstein condition is the most natural elliptic PDE for a Riemannian metric. One of the main objects of interest to differential geometers is the space of all Einstein metrics on a given smooth manifold. These "moduli spaces" in general are non-compact, as various degenerations can occur in sequences of Einstein metrics. In dimension 4 (the first interesting dimension for the Einstein equations), the possible degenerations are well understood only when the weak limit of a sequence of Einstein manifolds is itself 4-dimensional (the "non-collapsed" case). Almost nothing is known about the structure of the singularities that can arise in collapsing sequences of Einstein metrics, i.e. the case when the dimension of the limit is less than 4. In this talk I will describe the construction of large families of Einstein (Ricci-flat) metrics in dimension 4 that collapse to a 3-dimensional limit. For these families the formation of singularities is completely understood and is modelled on so-called ALF gravitational instantons. The construction is based on singular perturbation methods applied to a system of first-order PDEs (describing hyperkähler metrics) implying the Einstein condition.
06/10, CMT.01 Evolution Equations and Friends
4/10 Julia Bernatska (Kyiv)
4.15pm, CMT.01 On regularization of second kind integrals
We obtain expressions for second kind integrals on nonhyperelliptic (n, s)-curves. The curves possess a Weierstrass point at infinity which is a branch point where all sheets of the curve come together. The infinity of an (n, s)-curve serves as the base point for Abel’s map, and the base point in the definition of the second kind integrals. Since the second kind integrals introduced in this way are singular, we propose the regularization consistent with the structure of the field of Abelian functions on the Jacobian of the curve. We introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integrals over a local parameter in the vicinity of the infinity. This is a vector of dimension equal to genus of the curve, depending on parameters of the curve only. The presence of the term ensures correctness of all relations between Abelian functions on the Jacobian of an (n, s)-curve. We propose two methods of calculating the regularization constant, and obtain these constants for (3, 4), (3, 5), (3, 7), and (4, 5)-curves. Also we show how to extend the proposed regularization to the case when the pole of second kind integrals is moved from infinity to an arbitrary point. We show how to derive addition formulas, computation of which requires the second kind integrals, including correct regularization constants.
27-29/09, ICMS Joint Meeting of the Edinburgh and Catalan Mathematical Societies
21/09 Simon Chandler-Wilde (Reading)
Wave scattering by trapping obstacles: resolvent estimates and applications to boundary integral equations and their numerical solution
Obstacle scattering is an important applied problem in acoustics. It is also an important subject of study in mathematics, a place where PDE theory meets geometry and Hamiltonian dynamics. Important geometrical distinctions, these defined in terms of the high frequency/semiclassical ray-tracing/billiard-flow models of acoustics, are whether an obstacle is nontrapping or trapping and, if the geometry traps periodic billiard trajectories, whether these trajectories are elliptic (stable), hyperbolic, or parabolic (at the interface of the two regimes). We give an overview of what is known about the cut-off (Dirichlet) Laplacian resolvent across these cases, which just means studying outgoing solutions of the Helmholtz equation $\Delta u + k^2u = -f$ in the exterior of an obstacle, with $u=0$ on the boundary. In particular we present the first $k$-explicit resolvent estimate for parabolic trapping (an example of such a scenario is two cubes with parallel sides, the trapped billiard trajectories bouncing back and forth between the two sides). We also spell out applications of resolvent estimates, how they lead to bounds on: Dirichlet to Neumann maps, norms of inverses of boundary integral operators, lower bounds on inf-sup constants for standard variational formulations, and the first quasi-optimality results for finite element and boundary element methods for solving problems of scattering by trapping obstacles numerically. This is joint work with Euan Spence (Bath), Andrew Gibbs (Reading/Leuven), and Valery Smyshlyaev (UCL).
03/08, 2.15pm Jan Lang (Ohio State)
Spectral Theory on Banach Spaces
We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. We will try to find conditions under which the action of T is given by a series. This provides a Banach-space version of the well-known Hilbert-space result of E. Schmidt.
25/05 Jey Sivaloganathan (Bath)
3.15pm, CMT.01 Isoperimetric inequalities, Symmetrisation and Minimising Properties of Elastic Equilibria.
We first review some basic aspects of the variational theory of elasticity and the notion of quasiconvexity as a necessary condition for a strong local minimiser of the stored energy functional. We will then demonstrate how isoperimetric inequalities and symmetrisation arguments can be used to prove that various symmetric equilibria for elastic cylinders, shells and balls are energy minimising. (We will treat examples of both regular and singular equilibria using these methods.) Finally, we propose an approach to predicting the formation of singularities in energy minimisers using a novel variational derivative with respect to discontinuous deformations. (This is joint work with S.J. Spector (SIU) and P. Negron (UPRH).)
07/04 Javier Gomez-Serrano (Princeton)
Uniformly rotating smooth solutions for active scalars
Motivated by our previous results of global existence for active scalars in the patch setting, we are able to construct the first nontrivial family of global smooth solutions for the surface quasi-geostrophic (SQG) equations. These solutions rotate with uniform angular velocity both in time and space. We will outline the basic ingredients of the proof: bifurcation theory and computer-assisted estimates. Moreover, we will also discuss the case of uniformly rotating smooth solutions to the 2D incompressible Euler equations. Joint work with Angel Castro and Diego Cordoba.
24-25/3 Scottish Operator Algebras Research meeting (U Glasgow)
24/03 Houry Melkonian (Heriot-Watt)
Computable criteria for Schauder basis of dilated functions
17/03, 2.15pm Mark Wilkinson (Heriot-Watt)
Non-uniqueness of Classical Solutions to Euler's Equations of Rigid Body Mechanics
15/03, 2pm Maxwell Institute Mini-Symposium in Partial Differential Equations
ICMS Luigi Ambrosio (SNS Pisa), Massimiliano Gubinelli (Bonn),
Clement Mouhot (Cambridge)
14/03, 3pm Afternoon on geophysical fluid dynamics
ICMS Bin Cheng (Surrey), Ton S. van den Bremer (Edinburgh),
Michael J.P. Cullen (Met Office)
10/03, 2.15pm Michael Grinfeld (Strathclyde)
Rate equations and other mathematical challenges in submonolayer deposition
Submonolayer deposition (SD) is a blanket term used to describe the initial stages of processes such as molecular beam epitaxy, in which material is deposited onto a surface, diffuses and forms large-scale structures. It is easy to simulate using Monte Carlo methods, but theoretical results are few and far between. I will discuss rate equations, where centre manifold methods are very useful, and our recent attempts to develop a general 2-dimensional theory for SD.
This is mainly joint work with Paul Mulheran.
7/03, 3.15pm John Curtis (AWE / UCL)
CMT.01 Modelling high explosive violent reaction
Explosives enable the storing and release of huge amounts of energy when used as intended by deliberate detonation. They are employed over a wide range of applications, including e.g. oil well perforators and explosive welding as well as the obvious military uses. There is an ongoing need to store and handle them safely, as the unintended release of this stored energy can have disastrous consequences, examples of which will be shown. This can happen as a result of accidental impacts on the explosive or as a result of accidental thermal loading as occurs in fires. It is often not possible for cost or other reasons to test experimentally what the consequences of an unintended ignition (commencement of reaction) will be for the exact system of which the explosive forms part. Prediction of these consequences is the prime driver for having validated models of explosive reaction. The modelling of detonics addressing the explosive behaviour in its intended manner is long established and mature. However, the modelling of initially less severe reactions resulting from accidents is far less well established but vital to assess the safety of explosive systems, as such reactions can grow e.g under circumstances where there is strong confinement. A suite of tests is available to assess the response of an explosive to low speed impacts and thermal loadings, examples of which will be presented. These both give an immediate indicator of the likely sensitivity of a particular explosive, and also serve to provide validation data for the models under development, which will be described. It will be shown that even with relatively simple experimental geometry the modelled response of the explosive can be highly complex. In particular these simulations have revealed the key roles of friction and shear. While there is still a great deal of work to do there has been encouraging progress in capturing observed effects of the explosive response in several cases.
03/03, 2.15pm Epifanio Virga (Pavia)
Towards Onsager's density functional via Penrose's tree identity
Onsager's celebrated theory for liquid crystals, put forward in 1949, showed that purely steric, repulsive interactions between molecules can explain the ordering transition that underpins the formation of the nematic phase. Often Onsager's theory is considered as the first successful instance of modern density functional theory. It was however a theory rooted in its time, in the theory that Mayer had proposed in the late 1930's with the hope of explaining condensation of real gases. Despite its undeniable success, Onsager's theory lacks rigour at its onset. This lecture will review the conceptual basis of Onsager's theory and it will show how this theory can be made rigorous by use of Penrose's tree identity, a powerful technical tool already exploited to ensure convergence to Mayer's cluster expansion.
22/02, 4.00pm Alexander Pushnitski (Kings College), joint with UoE
ICMS An inverse spectral problem for positive Hankel operators
Hankel operators are infinite matrices with entries a_{n+m} depending on the sum of indices. I will discuss an inverse spectral problem for a certain class of positive Hankel operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a step towards description of evolution in a model integrable non-dispersive equation. Several features of this inverse problem make it strikingly (and somewhat mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will describe the available results for Hankel operators, emphasizing this similarity. This is joint work with Patrick Gerard (Orsay).
17/02, 1.15pm Paul Glendinning (Manchester and ICMS)
Attractors of piecewise smooth maps
I will review some of the methods and results available to describe attractors of piecewise smooth maps. The structure of non-wandering sets, the existence of invariant measures, and the dimension of bifurcating attractors will be addressed with varying levels of generality. I believe that the methods available to prove these results are just as important as the results themselves, and these techniques (and hopes for extensions of these techniques) will be the main driver of the talk.
10/02 MIGSAA recruitment day
03/02 Maxwell Institute Mini-Symposium in Partial Differential Equations
Daniel Coutand (Heriot-Watt), Alessio Figalli (ETH Zurich), Patrick Gerard (Paris)
27/01, 1.15pm Jonas Azzam (Edinburgh)
Harmonic measure, absolute continuity, and rectifiability
For reasonable domains $\Omega\subseteq\mathbb{R}^{d+1}$, and given some boundary data $f\in C(\partial\Omega)$, we can solve the Dirichlet problem and find a harmonic function $u_{f}$ that agrees with $f$ on $\partial\Omega$. For $x_{0}\in \Omega$, the association $f\rightarrow u_{f}(x_{0})$. is a linear functional, so the Riesz Representation gives us a measure $\omega_{\Omega}^{x_{0}}$ on $\partial\Omega$ called the harmonic measure with pole at $x_{0}$. One can also think of the harmonic measure of a set $E\subseteq \partial\Omega$ as the probability that a Brownian motion of starting at $x_{0}$ will first hit the boundary in $E$. In this talk, we will survey some very recent results about the relationship between the measure theoretic behavior of harmonic measure and the geometry of the boundary of its domain. In particular, we will study how absolute continuity of harmonic measure with respect to $d$-dimensional Hausdorff measure implies rectifiability of the boundary and vice versa.
20/01, 1.15pm Ernesto Estrada (Strathclyde)
k-path Laplacians, super-diffusion and super-fast random walks on graphs
I will start by a short introduction to the problem of diffusion on graphs, defining the graph Laplacian and some applications in areas ranging from autonomous robots to diffusion of innovations. Then, I will motivate the necessity of incorporating long-range interactions to account for certain physical diffusive processes. I will then introduce the k-path Laplacians as operators in l_2 Hilbert space and prove a few of their properties (boundedness, self-adjointnes). At this point I will introduce a generalisation of the diffusion equation on graphs by using Mellin- and Laplace-transformed k-path Laplacians. I will prove the existence of super-diffusive regimes for certain values of the parameter in the Mellin-transformed k-path Laplacian in one-dimension and will indicate our progress in extending the results to the 2D-case. Finally, I will introduce a multi-hopper model, that generalises the random walk model on graphs, by allowing non-nearest neighbours jumps. I will show the differences between this model and the random walk with Levy flights, which is valid only in the continuous space. I will prove that for certain asymptotic value of the parameters in the transforms of the k-path Laplacians, the multi-hopper reaches the minimum hitting and commute times in graphs of any topology. I will illustrate the results in certain classes of graphs and real-world networks.
2-3/12 Scottish Operator Algebras Research meeting (U Glasgow)
30/11, 12.30pm Xiaoyu Luo (Glasgow), joint Numerical Analysis and IIE seminar
Soft tissue mechanics and fluid-structure interaction in the heart
This talk will start with an overview of the invariant-based continuum mechanics approach for anisotropic soft tissues that undergo nonlinear large deformation. I will then report how we model the left ventricle and the mitral valve using the invariant-based constitutive laws. Fluid-structure interaction will be modelled using a hybrid version of immersed boundary and finite element methods. All the models are derived from in vivo clinical magnetic resonance images, with material parameters determined using an inverse approach so that the model results agree with in vivo observations. We model the cardiac function both in diastole and in systole, and some preliminary results of an integrated model of a mitral valve coupled with a left ventricle will also be reported. Finally, I will briefly introduce the newly funded EPSRC Maths for Healthcare Centre and the ongoing research themes in the Centre.
25/11 Dialogue on Cancer (IB3, Heriot-Watt)
18/11, 2.15pm Franz Gmeineder (Oxford)
A unifying approach to Korn-type inequalities
As an important tool in various applications, so for instance in mathematical fluid mechanics, Korn-type inequalities allow to bound the p-norms of the full gradients against the p-norms of specific combinations of derivatives. In this talk we give a survey of known results and applications, finally ending up with a complete characterisation of differential operators allowing for Korn-type inequalities.
11/11, 2.15pm Federica Dragoni (Cardiff)
Stochastic homogenisation for degenerate Hamilton-Jacobi equations
In the talk I investigate the limit behaviour for a family of Cauchy problems for Hamilton-Jacobi equations describing a stochastic microscopic model. The Hamiltonian considered is not coercive in the total gradient. The Hamiltonian depends on a lower dimensional gradient variable which is associated to a Carnot group structure. The rescaling is adapted to the Carnot group structure, therefore it is anisotropic. Under suitable stationary-ergodic assumptions on the Hamiltonian, the solutions of the stochastic microscopic models will converge to a function independent of the random variable: the limit function can be characterised as the unique viscosity solution of a deterministic PDE. The key step will be to introduce suitable lower-dimensional constrained variational problems. In collaboration with Nicolas Dirr, Claudio Marchi and Paola Mannucci.
4/11 LMS Harmonic Analysis Network Meeting
28/10 50 Years of Heriot-Watt University - Mathematical Sciences
21/10, 2.15pm Jan Kristensen (Oxford)
Morse-Sard type results for Sobolev mappings
The Morse-Sard theorem, and the generalizations by Dubovitskii and Federer, have numerous applications and belong to the core results of multivariate calculus for smooth mappings. In this talk we discuss extensions of these results to suitable classes of Sobolev mappings. The quest for optimal versions of the results leads one to consider possibly nondifferentiable mappings that in turn warrants new interpretations. A key point of the proofs is to show that the considered Sobolev mappings enjoy Luzin N type properties with respect to lower dimensional Hausdorff contents. The talk is based on joint work with Jean Bourgain, Piotr Hajlasz and Mikhail Korobkov.
19-21/10, 4pm Tim Candy (Bielefeld)
Teviot Pl. MIGSAA Mini-Course: Bilinear Restriction Estimates and Applications
14/10, 1.15pm Ramon Quintanilla (Universitat Politecnica de Catalunya)
room CMT.01 Decay of solutions for non-simple elasticity with voids
In this talk we consider the non-simple theory of elastic material with voids and we investigate how the coupling of these two aspects of the material affects the behavior of the solutions. We analyze only two kind of different behavior, slow or exponential decay. We introduce four different dissipation mechanisms in the system and we study, in each case, the effect of this mechanism in the behavior of the solutions.
7/10 MIGSAA Industrial Sandpit
30/09, 4.00pm Julian Tugaut (Saint-Etienne), joint with UoE Probability seminar
Appleton Tower Exit-time of a self-stabilizing diffusion
In this talk, we will briefly present some results of the Freidlin and Wentzell theory then we will give a Kramers type law for the McKean-Vlasov diffusion when the confining potential is uniformly strictly convex. We will give briefly two previous proofs of this result before providing a third proof which is simpler, more intuitive and less technical.
22/09, 4.15pm Beatrice Pelloni (HW) and Matthias Fahrenwaldt (HW)
Pelloni: "Boundary value problems and integrability / Analysis of a geophysical fluid dynamics model"
I will describe my recent and current work, which is in two rather different areas: (1) boundary value problems and integrability, and (2) analysis of a specific geophysical fluid dynamics model. (1) I have worked several years now on understanding the disturbance introduced by boundary conditions to the integrability of a PDE (say in 2 variables). A PDE is integrable when it is linearisable in the sense that it can be written as the compatibility condition of a pair of linear equations (the Lax pair). For these very special models, which include linear constant coefficient PDEs, all sorts of nice properties hold - but what happens when you pose them on a bounded domain, and prescribe boundary conditions? Thinking about this has produced many results in several directions, and I will highlight the most important ones, and illustrate (in pictures only!) some work in progress. (2) the semigeostrophic system is a reduction of the Euler equations that models the dynamics of large-scale atmospheric flows. The interest from a mathematical point of view is that it can be reformulated in such a way that it decouples into a an optimal transport problem for a certain measure, coupled with a simple time evolution. I will sketch why this problem is so interesting mathematically, and what are the open questions.
Fahrenwaldt: Some nonlinear differential equations in mathematical finance
In this talk we will present recent examples of nonlinear (partial) differential equations arising in the context of finance and economics. The first example treats the pricing of financial derivatives in illiquid markets where the derivative price can be characterised by a semilinear diffusion equation. The PDE, whose quadratic error term reflects the lack of liquidity in the market, has a weak solution and one can study the asymptotics as the market becomes perfectly liquid. The second example addresses the issue of optimal consumption/investment: consumers-investors maximize a (global) forward looking non separable expected utility. This leads to a nonlinear Bellman equation and a corresponding verification theorem. If time permits, I will also present a third example which covers the relatively new topic of cyber insurance. We model the spread of a cyber threat (e.g., a computer virus) along a graph and derive mean-field approximations for the moments of the infection probabilities in the form of a system of nonlinear ODEs. This allows the pricing of insurance contracts.
01/08 MIGSAA Evolution Equations and Friends
(on leave in autumn 2015 and spring 2016)
29/04 Maxwell Mini-Symposium Analysis and its Applications
Zdzislaw Brzezniak (York), Nicolas Burq (Paris), Nicolai Krylov (Minnesota)
16-17/03 Scottish Operator Algebras Research meeting (U Aberdeen)
21/10, 3.15pm Filip Rindler (Warwick)
On the structure of PDE-constrained measures and applications
Vector-valued measures satisfying a PDE constraint appear in various areas of nonlinear PDE theory and the calculus of variations. Often, the shape of singularities that may be contained in these measures, such as jumps or fractal parts, is of particular interest. In this talk, I will first motivate how variational problems in crystal plasticity naturally lead to such PDE-constrained measures and how their shape is physically relevant. Then, I will present a recent general structure theorem for the singular part of any vector-valued measure satisfying a linear PDE constraint. As applications, we obtain a simple new proof of Alberti's seminal Rank-One Theorem on the shape of derivatives of functions with bounded variation (BV), an extension of this theorem to functions of bounded deformation (BD), and a structure theorem for families of normal currents. Further, our structure result for currents implies the solution to the conjecture that if every Lipschitz function is differentiable almost everywhere with respect to some positive measure (i.e. the Rademacher theorem holds with respect to that measure), then this measure has to be absolutely continuous relative to Lebesgue measure. This is joint work with Guido De Philippis (SISSA Trieste).
22/1/16, 4:30pm Laure Saint-Raymond (ENS Paris), EMS meeting at Heriot-Watt
2015 (on leave in autumn 2015 and spring 2016)
13/11 Scottish Operator Algebras Research meeting (U Glasgow)
24/09, 5.15pm Lars Diening (Munich)
Finite elements for electro-rheological fluids
(Joint work with L. C. Berselli, D. Breit and S. Schwarzacher) Electrorheological fluids have the property that their viscosity changes when an electric field is applied. The friction term of the model behaves like the p-Laplacian, where the exponent p additionally depends on the electrical field. We study the finite element approximation of the steady flow.
16-18/09 2nd Maxwell Institute Graduate School on Evolution Equations
Arnulf Jentzen (ETH) and Alessandra Lunardi (Parma)
13/05, 3.15pm Sebastian Schwarzacher (Prague)
Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems
The work I wish to present establishes existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are sufficient and in many cases also necessary for building such a theory. We provide a unified approach that leads qualitatively to the same theory as that one available for linear elliptic problems with continuous coefficients, e.g. the Poisson equation.
29/04 Maxwell Mini-Symposium Analysis and its Applications
Laszlo Erdös (IST, Vienna), Joachim Krieger (EPFL, Lausanne), Tony Lelievre (ParisTech)
09/04, 2.15pm Martina Hofmanova (TU Berlin)
Stochastic Navier-Stokes equations for compressible fluids
We study the Navier-Stokes equations governing the motion of isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density and establish existence of the so-called finite energy weak martingale solution under the condition that the adiabatic constant satisfies $\gamma>3/2$. The proof is based on a four layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.
09/04, 3.15pm Grigori Rozenblioum (Chalmers)
Toeplitz operators with distributional symbols and related problems in classical analysis
The Fock space $B^2$ is the subspace in $L^2(C^1)$ with respect to the Gaussian measure, consisting of analytical functions. For a given function $F$ on $C^1$ the Toeplitz operator $T_F$ in $B^2$ acts as $T_F u=PFu$ where $P$ is the orthogonal projector from $L^2(C^1)$ to $B^2$. These operators, as well as their generalizations for $F$ being a distribution, play an important role in Mathematical Physics. In the talk we discuss certain recent result about the spectral properties of such Toeplitz operators. These results are based upon some new or obscure facts in complex analysis (estimates for entire functions) and real analysis (a generalization of the Stone-Weierstrass theorem and global estimates for solutions of the $\bar{\partial}$ in spaces of distributions). The presentation is meant to address a general audience with background in Analysis.
27/03 Juan Bonachela (Strathclyde), CM
20/03 Mariya Ptashnyk (Dundee), CM
13/03 Kristof Cools (Nottingham), CM
13-14/03 North British Functional Analysis Seminar
Aline Bonami (Orleans) and Uffe Haagerup (Odense)
11/03, 4.15pm Helmut Abels (Regensburg)
Diffuse Interface Models for Viscous Fluids With and Without Surfactants
We discuss so-called diffuse interface models for the flow of two viscous incompressible Newtonian fluids in a bounded domain. Such models were introduced to describe the flow when singularities in the interface, which separates the fluids, (droplet formation/coalescence) occur. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interface region is assumed in the model. Moreover, diffusion of both components is taken into account. We discuss analytic results concerning well-posedness in the case of fluids with different densities. Furthermore, we will present a recent diffuse interface model for a two-phase flow with a soluble surfactant, which effects the surface tension and discuss the existence of weak solutions for it.
06/03, 2.15pm Charalambos Makridakis (Sussex), CM
06/03, 3.15pm Sascha Trostorff (Dresden)
On Non-autonomous Evolutionary Problems
27/02 Maxwell Mini-Symposium Analysis and its Applications
Pierre Degond (Imperial College), Martin Dindos (Edinburgh), Ben Goddard (Edinburgh)
26/02, 2pm, ICMS Herbert Koch (Bonn)
Analyticity of level sets of solutions to elliptic equations and applications
20/02 (Tadashi Tokieda (Cambridge), EMS popular lecture)
20/02 Kevin Painter and Nikola Popovic (Edinburgh), joint with CM
Painter: Non local models for interaction driven cell movement
Cell movement plays a key role in many important biological processes, both essential (such as wound healing and immune cell guidance) and pathological (e.g. cancer). In many instances cells communicate directly, via connections at their cell surfaces leading to responses ranging from attractive (e.g. adhesion) to repelling (e.g. contact inhibition). In this talk I will describe some nonlocal models based on integro-PDE equations to model contact based cell movement, consider some specific applications and raise some of the mathematical and numerical challenges that they present.
13/02 Antoine Choffrut (Edinburgh)
On the global structure of the set of steady-states to the 2D incompressible Euler equations
In this talk I will present some recent results with Herbert Koch on a natural extension of shear flows when the channel is bounded by two graphs. More specifically, we construct 2D stationary flows whose vorticity have "arbitrary" distribution function, in the sense that it satisfies a compatibility condition. This is a global version of a local result with Vladimir Sverak. One crucial ingredient is to derive sufficiently strong a priori estimates. I will also discuss other interesting aspects of the proof.
06/02 Michela Ottobre and Dominic Breit
Ottobre: Diffusion processes, collective dynamics and applications

In this talk I will give a broad overview of my research activity. I will touch in particular on the following aspects of my work
1. Where I come from. Ergodic theory for Markov processes and exponentially fast convergence to equilibrium for hypoelliptic/hypocoercive diffusions.
2. What am I moving towards. Collective dynamics
Regarding 1: Hypoelliptic diffusions are very popular processes in non-equilibrium statistical mechanics Such processes are, under mild assumptions, typically ergodic, i.e. they have only one invariant measure. I will show how the issue of exponentially fast convergence to equilibrium for such dynamics can be regarded by at least three different standpoints: classic ergodic theory for Markov processes, spectral theory or using the more functional analytic hypocoercivity theory. (time allowing, I might also mention applications of this framework to the optimal design of MCMC algorithms)
Regarding 2: while the dynamics at point 1 exhibit only one equilibrium state, it is the case that many processes in nature will have many possible limiting behaviours, i.e. they will present multiple equilibria. I will show in particular two examples of equations displaying this phenomenon, both of them used in the modelling of so called collective dynamics: McKean-Vlasov equation in non-convex potential and a non-linear Fokker-Planck equation.

Breit: Existence theory for generalized Newtonian fluids

The time evolution of an incompressible and viscous fluid is governed by the Navier-Stokes system of partial differential equations describing the balance of mass and momentum respectively. The unknowns are the velocity field and the pressure. In order to prescribe specific material properties one needs a constitutive law which relates the stress deviator and the symmetric gradient of the velocity field. Linear relations are too restrictive to describe fluids with a more complex molecular structure which leads to the considerationof generalized Newtonian fluids. Here the viscosity is assumed to be a function of the shear rate. The most popular approach is the power-law model where the viscosity is proportional to a power of the shear rate. In contrast to the classical Navier-Stokes equations a nonlinear p-Laplacian type operator appears as the main part of the equations. If p<2 even the existence of weak solutionsis highly non-trivial. However, this situation is quite interesting from the physical point of view as it models shear-thinning fluids like ketchup, paint and blood.
We discuss the existence of weak solutions for stationary, non-stationary and stochastic problems connected with the motion of generalized Newtonian fluids. A main tool for this theory is the Lipschitz truncation method.
30/01, 3:00pm, ICMS Norbert Peyerimhoff (Durham)
Expanders, lifts, and Ramanujan graphs
Expanders are finite graphs which have strong connectivity properties and are at the same time sparse. Because of these competing properties, expanders are not not only interesting mathematical objects but also of importance in theoretical computer science and networks. In this talk, we will discuss an explicit example of 4-regular expander graphs which form an infinite tower of 2-fold lifts. An equivalent definition of expanders can be given via spectral gaps. Expander families with optimal asymptotic spectral gaps are families of Ramanujan graphs. A general conjecture of Bilu-Linial states that every Ramanujan graph has a 2-fold lift which is, again, Ramanujan. In the special case of bipartite graphs, this conjecture was proved in a breakthrough work by Marcus, Spielman and Srivastava in 2013. We will briefly discuss this method and explain that it can be extended to 3-fold lifts. The talk covers joint work with Shiping Liu and Alina Vdovina.
30/01, 4:15pm, ICMS North British Differential Equations Seminar
Paul A. Martin (Colorado School of Mines)
23/01, 2:15pm Wolfgang König (WIAS and TU Berlin)
Eigenvalue order statistics and mass concentration in the parabolic Anderson model
We consider random Schroedinger operators of the form $\Delta+\xi$, where $\Delta$ is the lattice Laplacian on $\Z^d$ and $\xi$ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of $\Z^d$. We show that, for $\xi$ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class, and the corresponding eigenfunctions are exponentially localized in regions where~$\xi$ takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Our proofs are new and self-contained and permit a rather explicit description of the shape of the potential and the eigenfunctions. We use this to prove a concentration property of the corresponding time-dependent problem, the heat equation with random potential, called the parabolic Anderson model. Here we show that the total mass of the solution to this PDE with random potential is asymptotically concentrated in the local area with the best relation between the size of the principal eigenvalue of $\Delta+\xi$ and the distance to the origin.
(joint work with Marek Biskup and Renato dos Santos)
16/01, 3:15pm Arghir Zarnescu (Sussex)
Thermodynamics, energetics and regularity for liquid crystal models
We consider several evolutionary models for liquid crystals that use De Gennes' Q-tensor theory and present a few existence and regularity results for these models. The models appear as the coupling of the incompressible Navier-Stokes equations with matrix-valued parabolic equations (modelling the orientation of liquid crystal molecules), and a (roughly speaking) transport-type equation modelling the temperature. We focus on the interplay between the physical features of the models and the well-posedness and regularity analysis.
14/01, 4:15pm Erik Wahlen (Lund)
Solitary water waves in three dimensions
I will discuss some existence results for solitary waves with surface tension on a three-dimensional layer of water of finite depth. The waves are fully localised in the sense that they converge to the undisturbed state of the water in every horizontal direction. The existence proofs are of variational nature and different methods are used depending on whether the surface tension is weak or strong. In the case of strong surface tension, the existence proof also gives some information about the stability of the waves. The solutions are to leading order described by the Kadomtsev-Petviashvili I equation (for strong surface tension) or the Davey-Stewartson equation (for weak surface tension). These model equations play an important role in the theory. This is joint work with B. Buffoni, M. Groves and S.-M. Sun.
12/12, 4:30pm Martin Hairer (Warwick), Taming infinities, EMS meeting at Heriot-Watt
21/11, 3:15pm Daniele Avitabile (Nottingham), CM
20/11, 4:15pm Sebastian Schwarzacher (Prague)
Pointwise gradient bounds for nonlinear elliptic systems
17/11, 11:00am Chris Budd (Bath), CM
14/11, 3:15pm Nadia Ansini (Rome)
A variational approach to non-symmetric linear operators. An overview and recent results.
We consider a sequence of Dirichlet problems for second order linear operator in divergence form where the matrices are uniformly elliptic and possibly non-symmetric. It is well known that if the matrices are symmetric, the equations have a variational structure since they can be seen as the Euler-Lagrange equations associated with a suitable sequence of functionals and the convergence of the solutions can be equivalently studied by means of the Gamma-convergence of the associated functionals or in terms of the G-convergence of the uniformly elliptic, symmetric matrices. In this seminar, we show that also in the non-symmetric case it is possible to give a variational structure to the equations of the aforementioned Dirichlet problems. We give a general overview of the main techniques: G-convergence, Gamma-convergence and H-convergence. We discuss some recents results obtained in collaboration with G. Dal Maso (SISSA, Italy) and C.I. Zeppieri (University of Munster, Germany).
07/11, 3:15pm Mark Chaplain (Dundee)
Hopf Bifurcation in a Gene Regulatory Network Model: Molecular Movement Causes Oscillations
Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this talk we present and analyse a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The initial model consists of a system of partial differential equations describing the spatio-temporal interactions between the Hes1 protein and its mRNA in a 2-dimensional domain representing the interior of a cell. Such intracellular negative feedback systems are known to exhibit oscillatory behaviour and this is the case for our model, shown initially via computational simulations. Computational results for a spatial stochastic version of the model will also be shown. We next present a simplified version of the model consisting of two partial differential equations describing the key interactions of the protein and mRNA in a 1-dimensional domain. In order to investigate the oscillatory behaviour more deeply, we undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. Our result has implications for transcription factors such as p53, NF-?B and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer.
31/10, 2:15pm Enrico Valdinoci (WIAS Berlin and Milano)
Dislocation dynamics and fractional equations
We consider a 1D evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of the dislocation function using analytic techniques of fractional Laplace type.
We show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium.
These dislocation points evolve according to the external stress and an interior potential. The potential is either attractive or repulsive, according to the different orientations of the dislocations (in particular, collisions, and even multiple collisions, of the atom dislocation may occur).
The results that will be presented have been obtained in some papers in collaboration with S. Dipierro, A. Figalli, G. Palatucci, S. Patrizi and extend previous works of R. Monneau and M.d.M. Gonzalez.
31/10, 3:15pm Andrea Cangiani (Leicester), CM
24/10, 3:30pm Des Higham (Strathclyde) and Peter Markowich (Cambridge, CANCELLED), joint Analysis/CM
Twitter Dynamics
Digital records of human interactions produce large-scale and rapidly changing data sets. Information such as who phoned whom and who tweeted whom provides a fascinating insight into our behaviour that can be of great value to social scientists, commercial organisations and governments. I will discuss some recent mathematical models that describe the evolution of these interactions and quantify the central players in the network. I will show some results on Twitter data.
17/10 (Trevor Wooley (Bristol), EMS meeting at University of Edinburgh)
8-10/10 Maxwell Institute Graduate School on Evolution Equations
Erwan Faou (Inria/ENS) and Christian Lubich (Tuebingen)
report in LMS Newsletter
02/10, 4:15pm Serena Dipierro (Edinburgh)
Nonlocal problems with Neumann boundary conditions
We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model.
We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass, decreasing energy, and convergence to a constant as $t\to\infty$. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions.
This is a joint work with X. Ros-Oton and E. Valdinoci.
26/09, 3:00pm Michael Dreher and Markus Schmuck, joint Analysis/CM
M. Schmuck: Upscaling of Nonlinear Transport Problems for Complex Materials
M. Dreher: The Csiszar - Kullback inequality and an application to a semiconductor model
27/06 Monika Winklmeier (Bogota)
On the spectral decomposition of dichotomous operators
Let S be a linear operator on a Banach space X such that the imaginary axis belongs to the resolvent set of S. I will show conditions which guarantee the decomposition of X in S-invariant subspaces such that the spectrum of the restricted operator belongs either to the left or right complex half plane.
11/06 Magnus Goffeng (Hannover), joint Analysis/Math Physics
Noncommutative geometry and "dimensions"
This is intended as a quick introduction to noncommutative geometry from the perspective of analysis. Starting from the Weyl law, we will discuss applications and drawback of these techniques by means of examples.