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: 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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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.

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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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.

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 .