Jon Bevan will be giving a talk in the PDE seminar at Oxford on Monday 14th January on “N-covering stationary points and constrained variational problems”.  The talk will show how degree N maps of the form u_N(z) = \frac{z^N}{|z|^{N-1}}} arise naturally as stationary points of functionals like the Dirichlet energy. Moreover it is shown that functions of this form are minimizers of related variational problems, including one whose associated Euler-Lagrange equation bears a striking resemblance to a system studied by N Meyer in the 1960s, and another where the constraint det{\nabla u}=1 a.e. plays a prominent role.

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Professor Guido Gentile from the Universita di Roma Tre will be visiting the department in the new year.  He will be visiting from 11-16 January and 6-17 February.  His hosts are Michele Bartuccelli and Jonathan Deane.

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Kristian Kristiansen of the Danish Technical University in Copenhagen is visiting the department from 12-21 December.  His host is Claudia Wulff.  He gave a talk in the Geometric Mechanics Seminar on Wednesday 12th December on “Iterative methods for computing canards“.  Claudia and Kristian are working on a project to prove exponential asymptotic estimates of slow manifolds in analytic systems.  A preprint is available here .

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Anastasios Rossides passed the PhD confirmation examination today, 17 December.  His project title is “An efficient numerical scheme for computing multi-pulse solutions in reaction-diffusion equations“.  His supervisors are David Lloyd and Sergey Zelik.  The examiner was Tom Bridges.

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Sergey Zelik is on a ten-day (10-20 December) research visit to the Université de Cergy-Pontoise.   He is visiting  Armen Shirikyan in the Département de Mathématiques, as part of a project to study the Navier-Stokes equations with stochastic forcing.  The visit is supported by an UK-France collaboration grant from the Royal Society of London.

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Allesandro Torrielli’s project on “Exotic quantum groups, Lie superalgebras and integrable systems” has been funded by EPSRC for a period of 12 months.  The £120K grant funds research time for Allesandro, travel and a workshop in 2014 (with the provisional workshop title: New trends in quantum groups and integrable systems).  A summary of the project for a general audience follows.

In recent years, motivated by the study of a particular class of integrable systems, new remarkable mathematical structures have been discovered which had not been investigated before.  These exotic algebraic constructions extend the standard framework of quantum groups to situations where new exciting effects manifest themselves.

Integrable systems have the property that their evolution equation admit an exact solution via reduction to linear problem.  When these systems are combined with Lie superalgebras – namely Lie algebras for which a grading exists identifying even and odd generators – unconventional features emerge.  This has been established in part through the work of the PI.  The Hopf algebra describing tensor products of these algebras, for instance, acquires non-trivial deformations, whose consequences have not been fully understood.  Furthermore, the systems in question exhibit a symmetry enhancement which is not manifest from the Hamiltonian formulation.  This “secret” symmetry results in novel higher-level quantities being conserved during the time evolution.  A complete mathematical formulation of these phenomena has yet to be developed, and it is very much sought for in order to understand potential implications for branches of mathematics such as algebra, geometry, the theory of knots and link invariants and integrable systems.

The aim of this research is to understand such exotic structures, and use this new understanding to attack challenging problems at the interface between algebra and integrable systems.  One of these problems is the so-called non-ultralocality of Poisson structures, governing the formulation of integrable systems in their semiclassical approximation.  Non-ultralocality makes the algebraic interpretation of the solution to these systems dramatically more obscure, and it is a difficult problem which has challenged mathematicians for years.  We believe that the key to significant progress in this direction is a rigorous understanding of the underlying exotic algebras.  Any progress in this direction will have a major long-term impact on the mathematical community, and on the scientific environment in the UK and internationally.

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Michele Bartuccelli has been invited to speak at the International Winter School on Mathematical Fluid Dynamics .  The conference will be held at the Bellavista Relax Hotel in Levico Terme (Trento) Italy from 16 to 21 December.  Michele will speak on “Explicit estimates for the L-infinity norm of solutions of the two-dimensional Navier-Stokes equations on the torus“.

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Cesare Tronci’s paper on “A Lagrangian kinetic model for collisionless magnetic reconnection” has been accepted for publication in the journal Plasma Physics and Controlled Fusion.  This journal has an impact factor of 2.731. In the paper, a new fully kinetic system is proposed for modeling collisionless magnetic reconnection. The formulation relies on fundamental principles in Lagrangian dynamics, in which the inertia of the electron mean flow is neglected in the expression of the Lagrangian, rather then enforcing a zero electron mass in the equations of motion. This is done upon splitting the electron velocity into its mean and fluctuating parts, so that the latter naturally produce the corresponding pressure tensor. The model exhibits a new Coriolis force term, which emerges from a change of frame in the electron dynamics. Then, if the electron heat flux is neglected, the strong electron magnetization limit yields a hybrid model, in which the electron pressure tensor is frozen into the electron mean velocity.  A preprint of the paper is available on the arXiv

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Ian Morris gave an invited talk at the One Day Ergodic Theory Meeting, held at the School of Mathematical Sciences, Queen Mary College, University of London, on Wednesday 7th November.  The talk was on “Unavoidable structures in positive-density subsets of Euclidean space; or, how I learned to stop worrying and love the weak-* topology on L-infinity“.  Inspired by a question of L. A. Szekely, in the early 1980s Furstenberg, Katznelson and Weiss proved the following theorem: if a measurable subset of the plane has positive density in a certain natural sense, then for all sufficiently large real numbers D we may find a pair of points in the set which are separated by Euclidean distance D. Extensions and alternative proofs were subsequently given by various authors including Bourgain, Falconer-Marstrand, Quas, and Bukh. We prove a general sufficient condition for a property to hold at all sufficiently large scales in all positive-density subsets of ℝd, improving a theorem of B. Bukh. Our proof uses a characterization of the density of a measurable set in terms of the translation orbit of its characteristic function, which we view as an element of L(ℝd) equipped with the weak-* topology. The result is then deduced using a pointwise ergodic theorem for ℝd-actions. Using this method we give new proofs of theorems of Furstenberg-Katznelson-Weiss, Marstrand, and R. L. Graham.

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The Mathematics Department at Surrey is part of a EU team that has been awarded an International Research Staff Exchange Scheme (IRSES) by the Marie Curie EU FP7 program.  The project title is “Brazilian-European Partnership in Dynamical Systems“.  The other EU partners are Imperial, Warwick, Augsburg, Dresden, ICTP, Pisa, CNRS, Porto, St Andrews, Paris, IM-Poland, Sevilla, Queen Mary, Porto, IST  & UBI (Portugal),  KTH (Sweden), Amsterdam, and Liverpool.  The coordinator for the Surrey node is Mark Roberts.  The grant provides 40 mission-months of funding for collaborative visits between nodes, as well as funding for focused meetings and workshops. The principal areas of research collaboration are nonautonomous dynamical systems, ergodic theory, low-dimensional dynamics, Hamiltonian systems and bifurcation theory.  The grant runs for four years and starts in January 2013.

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