Abstract: The non-linear wave equation (sometimes called the non-linear Klien-Gordon equation) is a much studied equation with applications in Josephson transmission lines, dislocations in crystals, DNA processes and much more.  It possess many types of solutions which may include stationary front solutions.  In this talk we shall consider what effect adding a step-like inhomogeneity has on the stability of such fronts and present some of the results we have derived over the last three years.

Abstract: Molecular monolayers, especially water monolayers, are playing a crucial role in modern science and technology. In order to derive simplified models of monolayer dynamics, we consider the set of rolling self-interacting particles on a plane with an off-set center of mass and a non-isotropic inertia tensor.  To connect with water monolayer dynamics, we assume the properties of the particles like mass, inertia tensor and dipole moment to be the same as water molecules. The perfect rolling constraint is considered as a simplified model of a very strong, but rapidly decaying bond with the surface. Since the rolling constraint is non-holonomic, it prevents the application of the standard tools of statistical mechanics: for example the system exhibits  two temperatures -- translational and rotational-- for some degrees of freedom, and no temperature can be defined for other degrees of freedom.

Abstract: The famous Laguerre polynomials are orthogonal over [0,\infty) with respect to a negative exponential weight function. They are thus natural  candidates for the efficient numerical approximation of such decaying exponential behaviour. We shall give a number of examples, including stable manifolds and fluid mechanics problems related to dynamical systems.

Abstract: We demonstrate existence of discrete solitons in Discrete Nonlinear Schrodinger equation (dlns) with saturable nonlinearity. We consider two types of solutions to (dlns) periodic and vanishing at infinity. In the second part of our talk, we prove the existence of periodic and solitary traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities. Calculus of variations and Nehari manifolds are employed to establish the existence of these solutions. We present some extensions of our results, combining the Nehari manifold approach and the Mountain Pass argument.

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Abstract: Growing plant cells undergo rapid axial elongation with negligible radial expansion: high internal turgor pressure causes viscous stretching of the cell wall. 

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Abstract: It has recently been argued that the seek for "consensus" and some form of "incompatibility" are basic mechanisms to explain the dynamics of cultural change and diversity [1]. Here, we will consider a basic, but mathematically amenable, three-state bounded compromise voter model (a constrained generalization of the classic two-state voter model [2]) that includes these ingredients. In this opinion dynamics model, a population of size N is composed of "leftists" and "rightists" that interact with "centrists" on a complete graph: a leftist and centrist can both become leftists with rate (1+q)/2 or centrists with rate (1-q)/2 (and similarly for rightists and centrists), where q denotes a selective bias towards extremism (q>0) or centrism (q<0). This system admits three absorbing fixed points and a "polarization" line along which a frozen mixture of leftists and rightists coexist. In the realm of the Fokker-Planck equation, and using a mapping onto a population genetics model, we compute the fixation probability of ending in every absorbing state and the mean times for these events. We therefore show, especially in the limit of weak bias and large population size (|q|~1/N, N>>1), how fluctuations alter the mean field predictions: polarization is likely when q>0, but there is always a finite probability to reach a consensus; the opposite happens when q<0. The findings are illustrated and corroborated by stochastic simulations. This presentation is based on the recent Ref.[3]"

Abstract: There is a growing realization among evolutionary biologists that heritable phenotypic variation is not always encoded in the DNA.

Abstract: Motivated by electrodynamic space tethers we consider the statics problem of a conducting rod in a uniform magnetic field. This problem has close analogies with that of a spinning top in rigid-body dynamics. We show that some cases are integrable while others are nonintegrable. For the latter we use a (Hamiltonian) Melnikov approach that highlights problems with similar Melnikov applications in rigid-body dynamics in the literature as well as ways around these problems.

Abstract: We consider the Lorentz process in the plane with periodic configuration of convex obstacles and with finite horizon. We define T(r) as the first return time of the flow to the r-neighbourhood of the initial position. We are interested in the behaviour of T(r) as r goes to zero. This is a joint work with Benoit Saussol.

Abstract: In this talk I will review my previous work on stability and mixing in two-dimensional vortices. This will include a threshold calculation for the existence of nonlinear cat's eye structures and an investigation into nonlinear vorticity staircase structures in a Gaussian vortex.

Abstract: We develop and test a mechanistic model of how diversity varies with body mass in marine ecosystems. The model predicts the form of the ``diversity spectrum,'' which quantifies the distribution of species' asymptotic body masses and is a species analogue of the classic size spectrum of individuals. 

Abstract: In this talk I will be presenting some of my work, in collaboration with many others, on the growth and regulation of bacterial biofilms; these are slimy colonies of non-motile bacteria on solid-fluid surfaces that have a number of implications in medicine and industry. The models to be discussed consist of nonlinear systems of PDEs and were analysed using asymptotic and computational methods. The main results and insights drawn from the work will be summarised.

Abstract: Food webs are the networks of who-eats-who in ecology. Despite being large and complex, the food webs observed in nature show relatively stable, stationary dynamics. Understanding this stability of food webs is a central challenge in ecology and could also inspire the design of more robust technical and organizational networks. Exploring food web stability is challenging because the food webs constitute high-dimensional and strongly nonlinear systems with dynamics on many different time scales. 

Abstract: Baker posed a question in the 1960's about when a rational map of degree d can be lacking periodic points of (minimum) period k. He gave the short list of pairs (d,k) that can occur.  In this talk we discuss the complete solution to this problem proved by Hawkins' Ph.D. student Rika Hagihara. 

Abstract: The propagation of long wavelength disturbances on the surface of a fluid layer of finite depth is considered. An arbitrary stress is applied at the surface with both tangential and normal components. 

Abstract: We consider (attracting) free semigroup actions (with two generators) on an interval. It is known that, if those two maps are sufficiently C2-close to the identity, then the (forward) minimal set. Namely, it must be an interval. (This statement is not accurate. I will give the precise statement in my talk.)

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In order to study the weakly coupled regime of theory we often make use of perturbative expansion of the physical quantities of interest. But such expansions are often divergent and defined only as asymptotic series. In fact, this divergence is connected to the existence of non-perturbative contributions, i.e. instanton effects that cannot be captured by a perturbative analysis. The theory of resurgence is a mathematical tool which allows us to effectively study this connection and its consequences. In this talk we will make use of this theory in order to show how to construct a full non-perturbative solution from perturbative data. To write this full non-perturbative solution we need to introduce generalised multi-instanton sectors, which can then be checked by precision tests on the asymptotic data. Finally, the study of this non-perturbative sectors in matrix models allows us to have a full picture of the phase diagram of this models.

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A G-strand is a map g :(t,s)\in RxR -> g(t,s)\in G into a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations. Some of these equations are completely integrable Hamiltonian systems that admit soliton solutions. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Various other examples will be discussed, including collisions of solutions with singular support (e.g. peakons) on Diff(R)-strands, in which Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions.

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The Wilson loop is one of the most important observables in gauge theories. It has long been suspected that its dynamics should have a natural description in terms of strings. The AdS/CFT correspondence is a realisation of this idea: the expectation value of Wilson loops in maximally supersymmetric Yang-Mills theory (MSYM) in four dimensions is captured by the partition function of type IIB superstring theory on AdS₅ x S⁵, satisfying suitable boundary conditions. Accordingly, the MSYM computation at strong coupling is equivalent to a semiclassical approximation on the string theory side, and amounts to finding certain minimal area surfaces in AdS₅ x S⁵.

In this talk, we deal with the correlator of two (supersymmetric) Wilson loops with contours lying on Hopf fibers of S³. A connected classical string surface, linking two different fibers, is presented. This string theory solution describes oppositely oriented fibers, and it is able to interpolate between a supersymmetric and non-supersymmetric configuration according to the fiber position on the Hopf base. We show that the system can be thought of as a deformation of the ordinary anti-parallel lines describing the static quark-antiquark potential, which is indeed correctly reproduced, at weak and strong coupling, as the fibers approach one another.

Abstract: Given a Lie group G acting smoothly on a Riemannian manifold X, let c0 ,c1 : D  → X be smooth functions, where D is also a smooth manifold (possibly with boundary). A geodesic relative to (c0 , c1) is a curve g : D → G with the property that gc0(s) = c1(s) for all sD, which minimises an energy integral over all such curves.

The talk will focus on G=SE(2), X=E2, D=[0,1] and applications to curve-matching. Another case, applicable to matching grayscale images, will also be mentioned if time permits.

Abstract: In this talk I report  recent work on the solution of the linear Schrödinger equation (LSE) by exponential splitting in a manner that separates different frequency scales. The main problem in discretizing LSE originates in the presence of a very small parameter, which generates exceedingly rapid oscillation in the solution. However, it is possible to exploit the features of the graded free Lie algebra spanned by the Laplacian and by multiplication with the interaction potential to split the evolution operator in a symmetric Zassenhaus splitting so that the arguments of consecutive exponentials constitute an asymptotic expansion in the small parameter. Once we replace the Laplacian by an appropriate differentiation matrix, this results in a high-order algorithm whose computational cost scales like O(N log N), where N is the number of degrees of freedom and whose error is uniform in the small parameter.

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In the context of gauge/string theory correspondences, the geometry of membranes and space-times must be seen as an emergent phenomenon arising from a strongly coupled quantum field theory with a large local symmetry or gauge group.

I shall discuss the dual description of membranes in terms of long, gauge-invariant, local operators in the Maldacena and ABJM correspondences, focusing primarily on a class of D-branes known as giant gravitons. These restricted Schur polynomial operators in the MSYM and ABJM theories form complete, exactly orthonormal bases with respect to the free field, two-point correlation functions and are built using the representation theory of the permutation group.

Various BPS, non-spherical, D3-brane and D4-brane giant graviton configurations, embedded into and moving on the compact spaces S⁵ and CP³, will be constructed in type IIB string theory on AdS₅xS⁵ and type IIA string theory on AdS₄xCP³ respectively. In the latter case, the D4-brane giants embedded into the complex projective space CP³ are descendants of M5-brane giant gravitons in S⁷ under the compactification on the S¹ Hopf fibre which takes S⁷ to CP³.

I shall finally discuss to what extent it has so far proven possible to view the non- trivial geometry of these membranes from the perspective of the dual operators.

Abstract:

Fish appear to swim by periodically moving their fins. Additionally, if they momentarily stop or swim through a tangle of seaweed they appear to recover speed very quickly soon after. This mixture of stability and oscillatory behavior suggests that swimming is a limit cycle. I will provide theoretical evidence which suggests the answer is ``yes''. This will be done by defining a transitive Lie groupoid which can be used for fluid structure interaction. The base of this groupoid will be a set of embeddings of a body into space. We will then define the Lagrangian on the Lie algebroid, and add a time periodic force to the shape of the body as well as a viscous dissipation force due to the non-zero viscosity of the fluid. We will then perform a reduction by SE(3) to obtain a Lie groupoid where the base is the shape space of the swimmer. We will go over some arguments which suggest that a limit cycle could exists in this reduced system. Assuming a limit cycle does exist, we can use reconstruction formulas to obtain the motion of the fish. However, many analytical issues remain to be discussed.

Abstract:

Einstein's theory of General Relativity beautifully realises gravity as a theory of geometry: matter causes space-time to curve. "Supergravity" is an very particular and intricate extension of Einstein's theory that appears as a low-energy limit of string theory. In this talk, I will describe how, despite its apparently complexities, supergravity also has an elegant geometrical description that unifies the different symmetries of the theory in a direct analogue of Einstein's theory. The appropriate extension of conventional differential geometry is a version of "generalised geometry" first introduced by Hitchin. This formulation gives hints as to what notions of geometry might underlie string theory, and hence perhaps our own space-time. It also gives a number of unexpected extensions to conventional complex, symplectic and Kähler geometries.

Abstract:

Motivated by an attempt to formulate (non chiral) conformal field theories of principal chiral models on manifolds of supergroups with vanishing Killing form as an OPE current algebras, we consider a related class of non semisimple affine (and non affine) (super)algebras, their representation theory and construct from them a chiral CFT. We address the questions of their representations, characters, fusion rules, and modular properties.

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Abstract:

I will discuss some recent developments and applications on the duality between gauge fields and strings.

Phase transitions for liquid crystals are typically characterized in terms of bifurcations of critical points for a free energy function. In a widely used model for the free energy due to Katriel et al. (1986) natural symmetries arise from the action of the group  SO(3) wr Z_2 (wreath product) acting on  5x5  matrices through left/right multiplication and transposition.  We investigate invariants for this group action, and also describe some bifurcation behaviour after a standard reduction is made to more tractable  2x2 matrices and the finite group  D_3 wr Z_2 .

Abstract:

Point vortices are singularities in the vorticity field of a perfect fluid. Physically, they show up as points in the fluid around which the fluid rotates, much like the fluid rotating around the core of a hurricane in the atmosphere.  From a fluid-dynamical point of view, the dynamics of point vortices forms a (finite-dimensional) Hamiltonian system, the solutions of which are weak solutions of Euler's equation, and there is consequently a great deal of interest in modeling and simulating the dynamics of point vortices.

Despite the fact that the equations of motion are Hamiltonian, however, this system has a number of special features which make direct numerical simulation hard: the vector field is not Lipschitz continuous, the symplectic form preserved is not the standard one, and there exists no Lagrangian formulation. The latter two problems especially make the construction of a symplectic integrator for this system a non-trivial task.

In this talk I will use the Hopf fibration to develop a new formulation of the point vortex equations on the sphere S^2. More specifically, I will show that there exists a nonlinear-Schrodinger-like equation on the Lie group SU(2), or alternatively the space of unit quaternions, whose solutions project down onto the trajectories of point vortices in S^2. Using the connection one-form of the Hopf fibration, I will also construct a Lagrangian for these equations, and we will investigate some symplectic integrators based on this formulation. We will illustrate the long-term qualitative features of these integrators with a number of examples from the recent literature, such as the spherical von Karman vortex street.

Abstract:

3d N=4 superconformal quiver theories arise as the low energy description of intersecting D3/D5/NS5 branes. I will discuss the backreacted solutions describing the near horizon of such configurations. The CFT partition function can be computed using localization technqiues and matched to the supergravity results.

Abstract:

We give a brief general review of the ADE classification problem. The survey includes simple Kleinian singularities, symmetries of Platonic solids, finite subgroups of SU(2), the Mckay correspondence, integer matrices of norm 2 and Brieskorn’s theory of subregular orbits. We conclude with some joint work with H. Sabourin and P. Vanhaecke on transverse Poisson structures to subregular orbits in semisimple Lie algebras. We show that the structure may be computed by means of a simple Jacobian formula, involving the restriction of the Chevalley invariants on the slice. In addition, using results of Brieskorn and Slodowy, the Poisson structure is reduced to a three dimensional Poisson bracket, intimately related to the simple rational singularity that corresponds to the subregular orbit.  Finally we present some recent results on the minimal orbit.

Abstract:

An interesting instance of the AdS/CFT correspondence is the case of AdS3/CFT2. Strings on AdS_3xS^3xS^3xS^1 can be completely understood when the background is purely NSNS thanks to 2-dimensional conformal symmetry. Unfortunately this symmetry is not as useful in the case of RR backgrounds. However, it appears that the integrability techniques that proved successful in understanding the planar limit of AdS5/CFT4 may be applicable also here, which sparked new interest and efforts in this field.

I will review such techniques and report on some recent developments (arXiv:1211.5119, 1212.0505) in the understanding of the string spectrum on AdS_3xS^3xS^3xS^1, exploring in particular the limit in which one S^3 blows up to a flat space.