Abstract: A successive  averaging method is developed for explaining the regularization mechanism in the periodic Korteweg- deVries (KdV) equation in the homogeneous Sobolev spaces  H^s for s>0.  Specifically, a proof is given of global existence existence, uniqueness, and Lipschitz continuous dependence on the initial data of the solutions of the periodic KdV. For the case  where the initial data is in L_2 we also show the Lipschitz continuous dependence of these solutions with respect to the initial data as maps from H^s to H^s for -1 < s < 0.

Abstract: A big problem in malaria control is the rapidity with which mosquitoes can develop resistance to insecticides. The possibility of creating evolution-proof insecticides is therefore of considerable interest. Biologists have suggested that effective malaria control, with only weak selection for insecticide resistance, could be achieved if insecticides target only old mosquitoes that have already laid most of their eggs. The strategy aims to exploit the fact that most malarial mosquitoes do not live long enough to transmit the disease. 

Abstract: Abstract:  Linkages in the plane are know to have geometrically interesting configuration spaces M.  
One can describe the dynamics of some simple idealized  linkages in terms of the geodesic flow on M, for an  appropriate Riemannian metric. We describe how to associate to the linkage the curvature of this metric, which has a direct bearing on the dynamics of geodesic  flow, and thus of the linkage.

Abstract: In this talk we will discuss a unifying approach to the smoothing estimates for dispersive equations based on the canonical transforms from the microlocal analysis, as well as on the comparison principles for evolution equations. The talk will be based on the joint work with Mitsuru Sugimoto (Nagoya).

Abstract: Insects have evolved diverse and delicate morphological structures in order to capture the inherently low energy of a propagating sound wave. In mosquitoes, the capture of acoustic energy, and its transduction into neuronal signals, is assisted by the active mechanical participation of the scolopidia.

Abstract: Abstract: The vector nonlinear Schroedinger equation (VNLS) models the propagation of an ultra-short light pulse along an optical fibre. The equation is integrable, affords multi-soliton soliton solutions, and also finite-gap solutions where the underlying algebraic curve is trigonal. The vector nature of the complex-valued dependent variable contains information about the polarisation state of the electromagnetic field.

Abstract: We present recent results regarding escape rates and conditionally invariant measures for a periodic Lorentz gas with holes.  We then derive a variational principle connecting the escape rate to the pressure on the survivor set, the set of points which never enters the hole.  This relation generalizes to a broad class of systems with holes and requires only weak assumptions on the size and boundary of the hole.

Abstract: Magnetic liquid or ferrofluids are complex fluids which interact with external magnetic field. Many effects can be observed.

Abstract: Air cushioning in droplet impacts will be investigated. A model will be presented, which couples an inviscid liquid droplet to a gas film satisfying a lubricating squeeze film equation. As the droplet approaches impact a high pressure is generated in the gas film, which causes the droplet free-surface to deform and ultimately trap a gas bubble. Predictions of the size of this trapped bubble will be made and compared to experimental data.

Abstract: TBA

Abstract: In the weakly nonlinear long wave regime, solitary waves are often modeled by the Korteweg-de Vries equation, which is well-known to support an exact solitary wave solution. However, when the effect of background rotation is taken into account, the resulting relevant nonlinear wave equation, the Ostrovsky equation, does not support an exact solitary wave solution. Instead an initial solitary like disturbance decays into radiating oscillatory waves. In this talk, we will demonstrate through a combination of theoretical analyses, numerical simulations and laboratory experiments that the long time outcome of this radiation is a nonlinear wave packet, whose carrier wavenumber is determined by an extremum in the group velocity.

Abstract: TBA

Abstract: Plants defend themselves using chemical toxins the concentration of which often varies with the age of twig segments. In boreal forests, woody internodes of the youngest segments of the twigs of the deciduous angiosperm species that herbivores such as hares prefer to eat are more defended by toxins than the woody internodes of the older segments that subtend and support the younger segments. Thus, the per capita daily intake of the biomass of the older segments of twigs by herbivores is much higher than their intake of the biomass of the younger segments of twigs. 

Abstract: TBA

(N.B. Seminar at 2 pm in LTB)

Abstract: TBA

Abstract: Any triangle ABC has a pedal triangle (whose vertices are the feet of the perpendiculars from each vertex to the opposite side).  We study how the shape changes as we repeatedly take the pedal triangle.  Does the sequence of triangles converge?  (Background material on Euclidean Geometry will be included for those too young to have studied this.)

Abstract: Here we give a pedagogical introduction to Euler-Poincar\'e flows on the (extended) Virasoro orbit, divided into three parts. In the first part, we introduce some basic notions of Euler-Poincar\'e flows, Virasoro orbit etc. The second part describes an Euler-Poincar\'e formulation of a new class of peakon equations with cubic nonlinearity found by Z. Qiao and V. Novikov. We show that the Hamiltonian structures obtained by Qiao and Hone and Wang can be obtained by our method. The last part introduces integrable ODEs associated to stabilizer set of Virasoro orbit.

Abstract: Biochemical pathways involve the interaction of large number of species, a property which can make their simulations computationally expensive and their analysis prohibitively complicated. To reduce this complexity, it is common practice to group a bunch of elementary reactions together as a single complex reaction and to do the analysis / simulation with this simplified model. But is this justified? Are the statistics at least qualitatively correct if one does such a procedure? In this talk I'll address these questions, illustrate the problems of such approaches and a new method which we developed to obtain a statistically correct coarse-grained description of the chemical master equation.

Abstract: The recently developed method of control based continuation allows the continuation of periodic solutions directly in physical experiments. The key ideas of this approach are (1) to introduce a control scheme that locally stabilizes periodic solutions without perturbing them, and (2) to use continuation to guide the experiment around folds and through bifurcation points. The experiment runs asynchronously from the actual continuation method, which communicates with the experiment by setting a control target and by taking measurements. In this talk we will present results obtained in a lab experiment of an impact oscillator that exhibits a large hysteresis loop. We will indicate current challenges with this method and how we intend to tackle them.

Abstract: Isostatic mounts are used in applications like telescopes and robotics to move and hold part of a structure in a desired pose relative to the rest, by driving some controls rather than driving the subsystem directly. To achieve this successfully requires an understanding of the coupled space of configurations and controls, and of the singularities of the mapping from the coupled space to the space of controls. It is crucial to avoid such singularities because generically they lead to large constraint forces and internal stresses which can cause distortion. In this paper we outline design principles for isostatic mount systems for dynamic structures, with particular emphasis on robots.

Abstract: The economist Paul Krugman was awarded the Nobel prize for his work in economic geography. In his book "The Self-Organizing Economy" he promotes a Santa Fe-style approach to modelling complex economic systems. Krugman's arguments are illustrated with some simple mathematical models. In this talk, I explain how one such model has exact solutions with very similar properties to the peaked solitons ("peakons") that appearing in an integrable partial differential derived by Camassa and Holm in shallow water theory. The exact solutions in Krugman's integro-differential model turn out explain the numerical behaviour he observed. This is joint work with J. Kelsey and F. Medda.

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.

Abstract: TBA

Abstract: TBA

Abstract: TBA

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

Abstract: TBA

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: 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.)

Abstract: TBA