2012/13 semester 2 events in Mathematics

Many bacteria developed a possibility to recognise aspects of their environment or to com- municate with each other by chemical signals. The so-called Quorum sensing (QS) is a special case for such a communication, a regulatory system for gene expression. Nutrient availability does not only influence bacterial growth but also the gene regulatory system for QS. Further effects which cause e.g. detachment processes and heterogeneous behaviour within colonies appear. From a mathematical point of view, ODE, PDE and stochastic approaches are used to describe and analyse the qualitative behaviour of bacterial quorum sensing systems.

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If you zoom into the microstructure of materials called block copolymers you observe two phases forming a wide variety of patterns: lamellae, hexagonally-packed cylinders, gyroids and spheres. We studied the case where one phase has a much higher volume fraction than the other. Starting from a nonlocal energy, where the nonlocal term is a Wasserstein distance, we proved a crystallisation result showing the optimality of the triangular lattice. This agrees with experimental observations. This is joint work with Mark Peletier (Technische Universiteit Eindhoven), Florian Theil (University of Warwick) and Steven Roper (University of Glasgow).

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The solar corona is composed of a plasma at high temperature (above 1 MK) and is dom- inated by the magnetic field. To study in detail the solar corona, we need to consider a broad range of spatial scales and time scales: from tens of kilometer to above one solar radius, from seconds to days. During this seminar, I will present to opposite cases in term of scales and indeed show that the scales are coupled. The 11 July 2012 is a particular date owed to the possible combination of observations: the full solar disc observation from the Solar Dynamics Observatory (SDO), and the high spatial and time resolution instrument on a sounding rocket named Hi-C. I first study the structure and evolution of a pseudo- streamer, which is a large-scale structure from where the slow component of the solar wind is supposed to originate. Secondly, I will discuss the observation of small-scale brighten- ings, named EUV bright dots, which are one of the sources of the coronal heating. With the support of these observations, I discuss the coupling between the different spatial and time scales at play in the solar corona.

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This talk reviews the recent developments in the physics of quantum turbulence. Quantum turbulence was discovered in superfluid helium in the 1950s, and the research has tended toward a new direction since the mid 90s. The similarities and differences between quantum and classical turbulence have become an important area of research. quantum turbulence is comprised of quantized vortices that are definite topological defects, representing a ’skeleton’ of turbulence. In particular we shall discuss the energy spectrum of quantum turbulence at very low temperatures. At low wavenumbers, the energy is transferred through the Richardson cascade of quantized vortices, and the spectrum obeys the Kolmogorov law, which is the most important statistical law in turbulence; this classical region shows the similarity to conventional turbulence. At higher wavenumbers, the energy is transferred by the Kelvin-wave cascade on each vortex. This quantum regime depends strongly on the nature of each quantized vortex.

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A key emphasis of this seminar would be on the work which is going on within the Mathematics, Statistics and Operational Research discipline. I would also provide information on teaching development grants (of up to £60,000) and other funding opportunities which may be relevant to you and your colleagues.

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The origins of modern computing are dispersed over centuries, often in the work of pure mathematicians, who invented methods in order to understand mathematical objects and prove theorems. A typical example is the famous Schwarz method for parallel computing, whose origins lie in a problem in Riemann’s audacious proof of the Riemann mapping theorem. Another example is the finite element method, which has its origins in the variational calculus of Euler-Lagrange and in the thesis of Walther Ritz, who died just over 100 years ago at the age of 31 from tuberculosis. We will see in this talk that the path leading to modern computational methods and theory was a long struggle over three centuries requiring the efforts of many great mathematicians.

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Chebfun has become a remarkably powerful tool for solving differential equations and eigen- value problems in one dimension. This talk will demonstrate Chebfun and describe some of the algorithms underlying its ODE and eigenvalue capabilities, which are mainly due to Asgeir Birkisson, Toby Driscoll, and Nick Hale. In particular we will focus on:

• the “happiness iteration” underlying Chebfun adaptivity
• rectangular spectral collocation discretizations
• how EIGS selects its eigenfunctions
• Frechet derivative operators for nonlinear problems via AD
• the graphical user interface CHEBGUI

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Stochasticity is essential when modeling initiation of tumors, progression of tumors from benign to malignant states, or metastasis formation. Many aspects of these phenomena can be modeled by simple multi-type branching processes, and the results compare fairly well to experimental and clinical data. Models of developing resistance to chemotherapy, and modeling challenges related to spatial structure and sizes of tumors will also be discussed.

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Modeling magnetized plasmas often requires incorporating microscopic and macroscopic effects at the same time. This can be achieved by nonlinear multi-physics models coupling microscopic kinetic theories with the macroscopic effects of MHD fluid motion. Most often, these models are obtained by making physical assumptions on the equations of motion, although this operation may destroy fundamental properties such as exact energy conservation (even in the absence of dissipation). I will show how the use of symmetry in Hamiltonian/Lagrangian systems provides a unifying framework for coupling nonlinear kinetic and fluid theories in a consistent way. As a result, new Coriolis force terms appear in the equations of motion, thereby leading to the explicit form of exact helicity invariants.

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How does a mathematical physicist studying spin systems end up discussing leadership? I will given a short tour of how I have gone from looking at Ising models to modelling collective motion and on to social networks. Social networks are an area of active research in animals both in the wild and in domestic settings. Aided by novel theoretical models developed to capture details of the latest data from collective motion, myself and colleagues have been investigating the opposite question; how does social network structure affect collective motion? I will pose a series of questions regarding social structure and present theoretical predictions of how it can affect group level behaviours. I will conclude with a brief presentation of some recent work on the interactions of geese groups and wind farms.

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Graphs can be used in many ways to make the structure of a group visible. The best known construction is that of the Cayley-graph of a finitely generated group. A whole branch of group theory, geometric group theory, is founded on this concept. In this talk I want to describe related constructions that can be used in wider context, in particular the study of compactly generated locally compact groups. Various results, old and new, about concepts like accessibility, Willis's structure theory and hyperbolic groups and graphs will be used to show the uses of these constructions.

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I will present extensions to the "Functional Analysis Through Hidden Markov Models" (FATHMM) tool including cancer-associated variants, indels, predictions for variants in non-coding regions of the human genome and pre-computed results for all possible missense variants in human and other ENSEMBL genomes. FATHMM is a tool for variant analysis. The central component of FATHMM is the prediction of whether protein missense variants will have a significant impact or not. Also included is a prediction of the functional and phenotypic outcome of the variant. FATHMM performs very well compared to other popular methods such as SIFT and PolyPhen (and other less common methods). FATHMM works on all species and is very fast (high-throughput). · Shihab, H.A., Gough, J., Cooper, D.N., Barker, G.L.A., Edwards, K.J., Day, I.N.M., Gaunt, T. (2012) Predicting the consequences of amino acid substitutions: A Functional Analysis Through Hidden Markov Models (FATHMM). Human mutation 34(1), 57-65. · Shihab. H.A., Gough, J., Cooper, D.N., Day, I.N.M. and Gaunt, T.R. (2013) Predicting the functional consequences of cancer-associated amino acid substitutions. Bioinformatics in press.

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Possible collaboration with the Centre for Engery, Petroleum and Mineral Law and Policy.

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Even before the spaceflight era, radio signals from Jupiter revealed not only the fact that other planets are magnetised, too, but also that there is a magnetic interaction between the planet and its innermost Galilean satellite, Io. Today all planets of the Solar System except for Venus and Mars are known to be magnetised, forming magnetospheres as a result of the interaction of the planetary magnetic fields with the Solar wind. As shown, using the example of the weakly magnetised Mercury, the Solar wind with the embedded interplanetary magnetic field can directly interact with the magnetospheres. This interaction can lead to events like substorms known from the magnetosphere of the Earth. The magnetospheres of the gas planets, on the other hand, are dominated rather by internal processes like the rapid rotation of their planets. Moreover, their satellites move around their planets on much smaller orbits, if measured in planetary radii, leading to the above-mentioned possibility of a magnetic interaction between satellite and planet. In addition to a presentation of the magnetospheres of the gas planets, this interaction is exemplified in detail. The search for extrasolar planets has led to the detection of so-called "hot Jupiters", giant planets on extremely close orbits around their host stars. In contrast to the planets of the Solar system, some of these stars do magnetically interact with their stars. In the last part it will be shown how these observations can be explained qualitatively and quantitatively by transferring the model of the Jupiter-Io scenario to these systems.

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