
 
 Previous talks: 
 
 
 
 
 
1/2/19  Ben Sharp (Leeds) 
 Geometry and topology of freeboundary minimal hypersurfaces 
 
 Freeboundary minimal hypersurfaces (FBMH) lie inside some ambient Riemannian manifold with boundary; they are critical points of the area functional under the sole constraint that their boundary stays within the boundary of the ambient space. Thus the mean curvature vanishes on its interior and it meets the ambient boundary orthogonally. Their study goes back (at least) to Courant and is a very active area of research even in the case where the ambient manifold is a threedimensional Euclidean ball.
After looking at lots of pictures of concrete examples, we will discuss the 'bubbling analysis' for degenerating sequences of embedded FBMH which, in particular, will lead to qualitative relationships between the variational, topological and geometric properties of these objects. I will present joint works with L. Ambrozio, A. Carlotto and R. Buzano. 
 
 
24/1/19  Florian Theil (Warwick) 
 Application of Hodge theory to discrete
dislocation models with randomness 
 
 I will present novel results on discrete models for discrete
dislocation models in three dimensions at finite temperature. Thanks
to powerful ideas from pde Analysis it is possible to show that with
high probability the atoms form a lattice with small fluctuations if
the temperature is low. 
 
 
18/1/19  Anton Savostianov (Durham) 
 Homogenisation with error estimates of attractors for damped semilinear anisotropic wave equations 
 
 It is well known that long time behaviour of a dissipative dynamical system generated by an evolutionary PDE can be described in terms of attractor, an attracting set which is essentially thinner than a ball of the corresponding phase space of the system. In this talk we compare long time behaviour of damped anisotropic wave equations with the corresponding homogenised limit in terms of their attractors. Firstly, we will formulate order sharp estimates between the trajectories of the corresponding systems and will see that the hyperbolic nature of the problem results in extra correction comparing with parabolic equations. Then, after a brief review on the previous results on the homogenisation of attractors, we will discuss the new results. This is joint work with Shane Cooper. 
 
 
7/12  EMS meeting hosted by HW 
ICMS  speaker: Jeremy Quastel (Toronto).

 
 
3/12, 3.30pm  Jose Carrillo (Imperial College London), joint with UoE 
ICMS  Nonlinear AggregationDiffusion Equations in the DiffusionDominated and Fair Competitions Regimes 
 
 We analyse under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction, occurs. I will discuss several regimes that appear in aggregation diffusion problems with homogeneous kernels. I will first concentrate in the fair competition case distinguishing among porous medium like cases and fast diffusion like ones. I will discuss the main qualitative properties in terms of stationary states and minimizers of the free energies. In particular, all the porous medium cases are critical while the fast diffusion are not. In the second part, I will discuss the diffusion dominated case in which this balance
leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregationdiffusion equations towards this unique equilibrium profile up to translations as time tends to infinity. This talk is based on works in collaboration with S. Hittmeir, B. Volzone and Y. Yao and with V. Calvez and F. Hoffmann. 
 
 
23/11  Thomas Hudson (Warwick) 
 Modelling dislocation motion via Discrete Dislocation Dynamics

 
 Dislocations are line defects found in crystals, and act as the carriers of irreversible (or plastic) deformation for these materials. Understanding and accurately modelling the complex collective evolution of dislocations is therefore viewed as a key challenge in obtaining predictive models of plasticity. Since the 1960s, a wide variety of models to describe dislocation motion have been proposed, and with the growth of computer power in the 1990s, Materials Scientists began using these models computationally. In this talk, I will present mathematical results which link a particular class of dislocation evolution model (Discrete Dislocation Dynamics) to microscopic principles, and discuss the precise mathematical formulation and wellposedness of the relevant evolution problem in three dimensions. 
 
 
22/11  Guido De Philippis (SISSA, Trieste) 
 (Boundary) Regularity for solutions of the Plateau problem 
 
 Plateau problem consists in finding the surface of minima area spanning a given boundary. Since the beginning of the 50’s the study of this problem led to the development of fundamental tools in Geometric Analysis and in the Calculus of Variations. Aim of the talk is to give an overview of the problem and of the technques used to solve it. In the end I will also present some recent results concerning boundary regularity. 
 
 
7/11  Evolution Equations and Friends 
 talks by Ph.D. students, pizza and more 
 
 
2/11  Ivan Moyano (Cambridge) 
 Spectral inequalities for the Schrödinger operator \Delta_x + V(x) in Rd 
 
 In this talk, we will first review some classical results on the
socalled ’spectral inequalities’, which yield a sharp quantification of the
unique continuation of the spectral family associated with the LaplaceBeltrami
operator in a compact manifold. In a second part, we will discuss how to
obtain the spectral inequalities associated to the Schrodinger operator
\Delta_x + V(x), in \mathbb{R}^d, in any dimension $d\geq 1$, where
V=V(x) is a real analytic potential. In particular, we can handle some
longrange potentials. This is a joint work with Prof G. Lebeau (Université
de NiceCôte d'Azur, France) 
 
 
31/10, 3.15pm  John M. Ball 
 Generalized Hadamard jump conditions and polycrystal microstructure 
 
 The talk will describe various generalizations of the Hadamard jump
condition, and how
they can lead to information about polycrystal microstructure arising from
martensitic phase
transformations. Joint work with Carsten Carstensen (Humboldt University,
Berlin). 
 
 
26/10, 1.15pm  Giacomo Canevari (BCAM) 
 Variational models for nematic liquid crystals with subquadratic growth 
 
 Nematic liquid crystals are matter in an intermediate phase between solids and liquid. The constituent molecules, while isotropically distributed in space, retain longrange orientational order. The classical variational theories for nematic liquid crystals are quadratic in the gradient and as a consequence, configurations with a singular line have infinite energy within these theories. On the other hand, line defects are commonly observed in these materials. Based on this observation, Ball and Bedford (2014) have proposed models with subquadratic growth in the gradient. In this talk, we consider a modified Landaude Gennes model where the elastic modulus has subquadratic growth, but otherwise is quite general and may not behave as a power at infinity. We study minimisers on threedimensional domains and discuss their asymptotic behaviour in a singular limit of the model, proving convergence to manifoldvalued phiharmonic maps away from a singular set. The talk is based on a joint work with Apala Majumdar (University of Bath) and Bianca Stroffolini (Universita Federico II, Naples, Italy). 
 
 
19/10, 2.15pm  Siran Li (Rice / Centre de Recherches Mathematiques) 
 General Axially Symmetric Harmonic Maps 
 
 Harmonic map equations are an elliptic PDE system arising as the
EulerLagrange equation for the minimisation problem of Dirichlet
energies between two manifolds. In this talk, we consider the harmonic
maps from the unit 3ball to the 2sphere with a generalised type of
axial symmetry. Examples include certain ''twisting'' maps. We discuss
the existence, uniqueness and regularity issues of this family of
harmonic maps. In particular, we characterise of singularities of
minimising general axially symmetric harmonic maps, and construct
nonminimising general axially symmetric harmonic maps with arbitrary
0 or 1dimensional singular sets on the symmetry axis. Problems about
numerical studies of harmonic maps shall also be discussed. (Joint
work with Prof. Robert M. Hardt.) 
 
 
26/09  Tom Leinster (Edinburgh), joint with algebra 
 Magnitude 
 
 The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, produces a family of invariants in different contexts across mathematics, thanks to a very general categorical definition. All of them can be loosely understood as "size".
For example, one can speak of the magnitude of a metric space. This is a newish invariant from which one can recover the volume and dimension of a subset of R^n, and (at least under hypotheses) other classical geometric measures such as surface area, perimeter, etc. Proving this has required serious analysis, some of which has been provided by local talent (Carbery and Gimperlein).
There are other manifestations of magnitude in algebra (with a relation to Cartan matrices and the Euler form), graph theory, topology, and the quantification of biological diversity. I will give an overview. 
 
 
45/06  Prediction and Data Assimilation for Nonlocal Diffusions 
Appleton Tower 

 
 
28/5, 3.15pm  Connor Mooney (ETH Zurich) 
EM1.82  Singular solutions to parabolic systems 
 

A classical result of Morrey shows that solutions to linear, uniformly elliptic systems with measurable coefficients are continuous in two dimensions. The parabolic analogue of this result remained elusive. We will discuss some recent examples which show not only that discontinuity from smooth data happens in the plane, but also that the singularities are as bad as parabolic energy estimates allow. In particular, we will make a connection between the regularity problems for parabolic systems in dimension $n$, and elliptic systems in dimension $n+2$. 
 
 
28/5, 4.15pm  Costas Dafermos (Brown), North British Differential Equations Seminar 
EM1.82  Progress and challenges in the theory of hyperbolic conservation laws 
 

The lecture will provide a survey of the state of the art in the theory of hyperbolic conservation laws, emphasizing, both, recent achievements and future challenges. 
 
 
2122/05, ICMS  Nonlinear Analysis and the Physical and Biological Sciences, 
 in honour of Jack Carr

 
 
9/5, 1.15pm  Fernando P. da Costa (Lisbon) 
CMS.01  Bifurcation problems in liquid crystal cells 
 
 Liquid crystal cells are technological devices of huge practical importance. Their use in liquid crystal displays is presently ubiquitous and it is based on the change of the liquid crystal’s optical properties in response to applied electromagnetic fields: a bifurcation phenomenon known as Freedericksz transition. In this talk I shall present some examples of its occurrence in simple models of liquid crystal cells.
In the cases I consider the models for the stationary configurations of the liquid crystal reduce to a nonlinear pendulum equation coupled with several types of nonhomogeneous boundary conditions. The effect of changing an applied magnetic field is mathematically translated into a corresponding bifurcation problem, the mathematical analysis of which, based on phase plane tools and ``time maps,’’ will be presented. I also briefly present ongoing work on a similar boundary value problem where a minor change in the boundary condition led to an unexpectedly more difficult analysis. (This talk will be based on joint works with E.C. Gartland Jr., M. Grinfeld, M.I. Mendez, N. Mottram, J. Pinto, and K. Xayxanadasy.) 
 
 
9/5, 2.15pm  Peter Pang (Surrey) 
CMS.01  Invariant Measures of Stochastic Conservation Laws 
 
 In this talk I will discuss some new results on the existence and uniqueness of invariant measures to stochastically driven degenerate parabolic equations, and techniques used in deriving them. I will also discuss some associated open problems. This was work done under the supervision of GuiQiang Chen. 
 
 
27/4, 4.00pm  Byeon Jaeyoung 
ICMS  Effect of large interaction forces to the structure of solutions
in an elliptic system 
 
 For an elliptic system coming from a nonlinear Schroedinger
system, the structure of solutions depends very subtly on the interaction
parameters between components. We are interested in the case that
intraspecies interaction forces are fixed and interspecies forces are
very large. In this case, depending on the kinds of forces, repulsive or
attractive, several different types of solutions we can see. Moreover, the
formation of different types of solutions strongly depends on the ratios
between large repulsive or attractive forces. I would like to introduce an
overview on recent studies on such phenomena. 
 
 
1214/04, ICMS  50th anniversary meeting of the North British Functional Analysis seminar 
 
 
26/3  Ph.D. student day in stochastic analysis 
 
 
 
19/03  Maxwell Institute MiniSymposium in PDEs 
 speakers include Xavier RosOton (Zurich) and Peter Topping (Warwick) 
 
 
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. 
 
 
9/3, 2.15pm  Frank Smith (UCL) 
 Shear flow over flexible inwall 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 flowstructure 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 finitetime 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 volumepreserving diffeomorphisms. (Joint work with A D Gilbert.) 
 
 
16/2  Susana Gutierrez (Birmingham) 
 The Cauchy problem for the LandauLifshitzGilbert equation in BMO and selfsimilar solutions 
 
 The LandauLifshitzGilbert 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 selfsimilar solutions under the onedimensional LLGequation. Motivated by the properties of this family of selfsimilar solutions, in the second part of this talk we consider the Cauchy problem for the LLGequation with Gilbert damping and provide a global wellposedness result provided that the BMO norm of the initial data is small. 
 
 
9/2  Marcus Waurick (Strathclyde) 
 Fibre Homogenisation for ndimensional systems of partial differential equations 
 
 In the talk we present a method to obtain ordersharp operatornorm 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 Lebesguetypical 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). 
 
 
 
 
 
2017  
 
 
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 Gammaconvergence 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 Gammaconvergence 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). 
 
 
1517/11, ICMS  Graduate School on Evolution Equations 
 courses by JeanFrederic Gerbeau (Inria / Paris) and Hans Othmer (Minnesota) 
 
 
10/11, 1.15pm, CMT.01  Luigi Berselli (Pisa) 
 On the regularity for nonlinear systems of the pStokes type 
 
 In this talk I will give an overview of recent results
concerning the regularity uptothe boundary for solutions of nonlinear
systems of partial differential equations, as those arising in some
problems related with nonNewtonian 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 (onedimensional) 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 multidimensional 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 biasvariance 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 halfaxis whose Hamiltonian has unit trace almost everywhere on the halfaxis. 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 WisconsinMadison). 
 
 
27/10, 2.15pm  Lyuba Chumakova (UoE) 
 Effective Boundary Conditions for SemiOpen 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, semiopen 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 semiopen 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 nonselfadjoint and/or nonlocal boundary conditions and Schrödinger operators with deltapotentials 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 msectorial and hence generate analytic semigroups.

 
13/10  Lorenzo Foscolo (HW) 
 Degenerations of 4dimensional Ricciflat 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 noncompact, 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 4dimensional (the "noncollapsed" 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 (Ricciflat) metrics in dimension 4 that collapse to a 3dimensional limit. For these families the formation of singularities is completely understood and is modelled on socalled ALF gravitational instantons. The construction is based on singular perturbation methods applied to a system of firstorder 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.

 
 
2729/09, ICMS  Joint Meeting of the Edinburgh and Catalan Mathematical Societies 
 
 
21/09  Simon ChandlerWilde (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 raytracing/billiardflow 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 cutoff (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 infsup constants for standard variational formulations, and the first quasioptimality 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 Banachspace version of the wellknown Hilbertspace 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 GomezSerrano (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 quasigeostrophic (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 computerassisted 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. 
 
 
2425/3  Scottish Operator Algebras Research meeting (U Glasgow) 
 
 
24/03  Houry Melkonian (HeriotWatt) 
 Computable criteria for Schauder basis of dilated functions 
 
 
17/03, 2.15pm  Mark Wilkinson (HeriotWatt) 
 Nonuniqueness of Classical Solutions to Euler's Equations of Rigid Body Mechanics 
 
 
15/03, 2pm  Maxwell Institute MiniSymposium 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 largescale
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 2dimensional 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 nondispersive equation.
Several features of this inverse problem make it strikingly (and
somewhat mysteriously) similar to an inverse problem for
SturmLiouville 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 nonwandering 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 MiniSymposium in Partial Differential Equations 
 Daniel Coutand (HeriotWatt), 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) 
 kpath Laplacians, superdiffusion and superfast 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 longrange interactions to account for certain physical diffusive processes. I will then introduce the kpath Laplacians as operators in l_2 Hilbert space and prove a few of their properties (boundedness, selfadjointnes). At this point I will introduce a generalisation of the diffusion equation on graphs by using Mellin and Laplacetransformed kpath Laplacians. I will prove the existence of superdiffusive regimes for certain values of the parameter in the Mellintransformed kpath Laplacian in onedimension and will indicate our progress in extending the results to the 2Dcase. Finally, I will introduce a multihopper model, that generalises the random walk model on graphs, by allowing nonnearest 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 kpath Laplacians, the multihopper reaches the minimum hitting and commute times in graphs of any topology. I will illustrate the results in certain classes of graphs and realworld networks.

 
 
2016  
 
 
23/12  Scottish Operator Algebras Research meeting (U Glasgow) 
 
 
30/11, 12.30pm  Xiaoyu Luo (Glasgow), joint Numerical Analysis and IIE seminar 
 Soft tissue mechanics and fluidstructure interaction in the heart 
 
 This talk will start with an overview of the invariantbased 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 invariantbased constitutive laws.
Fluidstructure 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, HeriotWatt) 
 
 
18/11, 2.15pm  Franz Gmeineder (Oxford) 
 A unifying approach to Korntype inequalities 
 
 As an important tool in various applications, so for instance in mathematical fluid mechanics, Korntype inequalities allow to bound the
pnorms of the full gradients against the pnorms 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 Korntype inequalities. 
 
 
11/11, 2.15pm  Federica Dragoni (Cardiff) 
 Stochastic homogenisation for degenerate HamiltonJacobi equations 
 
 In the talk I investigate the limit behaviour for a family of Cauchy
problems for HamiltonJacobi 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 stationaryergodic
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 lowerdimensional
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 HeriotWatt University  Mathematical Sciences 
 
 
21/10, 2.15pm  Jan Kristensen (Oxford) 
 MorseSard type results for Sobolev mappings 
 
 The MorseSard 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. 
 
 
1921/10, 4pm  Tim Candy (Bielefeld) 
 
Teviot Pl.  MIGSAA MiniCourse: Bilinear Restriction Estimates and Applications 
 
 
14/10, 1.15pm  Ramon Quintanilla (Universitat Politecnica de Catalunya) 
room CMT.01  Decay of solutions for nonsimple elasticity with voids 
 
 In this talk we consider the nonsimple 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 (SaintEtienne), joint with UoE Probability seminar 
Appleton Tower  Exittime of a selfstabilizing 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
McKeanVlasov 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 largescale 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: consumersinvestors 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 meanfield 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 MiniSymposium Analysis and its Applications 
 Zdzislaw Brzezniak (York), Nicolas Burq (Paris), Nicolai Krylov (Minnesota) 
 
 
1617/03  Scottish Operator Algebras Research meeting (U Aberdeen) 
 
 
21/10, 3.15pm  Filip Rindler (Warwick) 
 On the structure of PDEconstrained measures and applications 
 
 Vectorvalued 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 PDEconstrained measures and how their shape is physically relevant. Then, I will present a recent general structure theorem for the singular part of any vectorvalued measure satisfying a linear PDE constraint. As applications, we obtain a simple new proof of Alberti's seminal RankOne 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 SaintRaymond (ENS Paris), EMS meeting at HeriotWatt 
 
 
 
 
 
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 electrorheological 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 pLaplacian, where the exponent p additionally depends on the electrical field. We study the finite element approximation of the steady flow. 
 
1618/09  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 MiniSymposium Analysis and its Applications 
 Laszlo Erdös (IST, Vienna), Joachim Krieger (EPFL, Lausanne), Tony Lelievre (ParisTech) 
 
09/04, 2.15pm  Martina Hofmanova (TU Berlin) 
 Stochastic NavierStokes equations for compressible fluids 


We study the NavierStokes 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 socalled 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 StoneWeierstrass 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 
 
 
1314/03  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 socalled 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 wellposedness in the case of
fluids with different densities. Furthermore, we will present a recent
diffuse interface model for a twophase 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 Nonautonomous Evolutionary Problems

 
 
27/02  Maxwell MiniSymposium 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 integroPDE 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 steadystates 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 nonequilibrium 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: McKeanVlasov equation in nonconvex potential and a nonlinear FokkerPlanck equation.

 Breit: Existence theory for generalized Newtonian fluids
The time evolution of an incompressible and viscous fluid is governed by the NavierStokes 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 powerlaw model where the viscosity is proportional to a power of the shear rate. In contrast to the classical NavierStokes equations a nonlinear pLaplacian type operator appears as the main part of the equations. If p<2 even the existence of weak solutionsis highly nontrivial. However, this situation is quite interesting from the physical point of view as it models shearthinning fluids like ketchup, paint and blood.
We discuss the existence of weak solutions for stationary, nonstationary 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 4regular expander
graphs which form an infinite tower of 2fold 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 BiluLinial states that every Ramanujan
graph has a 2fold 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 3fold 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 doublyexponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel maxorder 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 selfcontained 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 timedependent 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' Qtensor theory and present a few existence and regularity results for these models. The models appear as the coupling of the incompressible NavierStokes equations with matrixvalued parabolic equations (modelling the orientation of liquid crystal molecules), and a (roughly speaking) transporttype equation modelling the temperature. We focus on the interplay between the physical features of the models and the wellposedness 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 threedimensional 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 KadomtsevPetviashvili I equation (for strong surface tension) or the DaveyStewartson 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 HeriotWatt 
 
 
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 nonsymmetric 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 nonsymmetric. It is well known
that if the matrices are symmetric, the equations have a variational structure since they can
be seen as the EulerLagrange equations associated with a suitable sequence of functionals and
the convergence of the solutions can be equivalently studied by means of the Gammaconvergence of
the associated functionals or in terms of the Gconvergence of the uniformly elliptic, symmetric
matrices.
In this seminar, we show that also in the nonsymmetric 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: Gconvergence, Gammaconvergence and Hconvergence. 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 spatiotemporal interactions between the Hes1 protein and its mRNA in a 2dimensional 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 1dimensional 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 PeierlsNabarro 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 largescale 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) 
 
810/10  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. RosOton 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 Sinvariant 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. 