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2016/17 semester 2 events in Mathematics

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Mon 16/01/17 15:00
Fulton J20
Mathematics Seminar
Dr. Alison Pease (University of Dundee)
The role of the machine in collaborative mathematics
abstract

Abstract

Computer support for mathematics, such as computer algebra or computational mathematics, has typically been for the polished and public frontstage. A second approach is to focus on the messy, fallible, and speculative backstage and to try to extract principles which are sufficiently clear as to allow an algorithmic interpretation. The study of mathematical practice provides a starting point for this work. We discuss our investigations into backstage mathematical research - in particular into what is discussed and how it is communicated and explained - in the context of the new challenges for that these raise for artificial intelligence and computational mathematics.

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Mon 23/01/17 14:00
Fulton G20
Mathematics Seminar
Prof. Philip Maini (University of Oxford)
Modelling cell population movement
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Abstract

The coordinated movement of cell populations is of vital importance in biology, for example during normal development, wound healing and disease (such as cancer). In this talk, I will consider some applications of mathematical modelling to this phenomenon. In the embryo, cells from the neural crest have to move from the neural tube to the branchial arches - how this is controlled is not known. We will show how a discrete-based cellular automaton model, combined with an experimental program, has led to new insights into this problem. I will then consider the problem of epithelial sheet movement, with application to the colon crypt and to rosette formation.

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Mon 30/01/17 14:00
Fulton G20
Mathematics Seminar
Dr. Masoud Hayatdavoodi (University of Dundee)
A Direct Approach to Water Wave Problems
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Abstract

Beginning from the last three decades of the twentieth century, an attempt was made to approach the subject of wave propagation in shallow waters using an approach different from the perturbation methods, namely by use of the theory of directed fluid sheets. Unlike the classical approximations, this method does not require a priori assumptions on any scaling parameter, nor does it depend on a perturbation expansion, and irrotationality of the flow is not necessary. The theory has its roots in the theory of plates and shells in structural mechanics, and in the general form, it is applicable to any type of medium. The direct theory is based on a continuum model, namely directed or Cosserat surface. A Cosserat surface is a deformable surface embedded in a Euclidean three-dimensional space to every point of which a deformable vector, called a director, is assigned. The Cosserat surface, although three-dimensional in character, only depends on two space dimensions x1, x3 and time t, and it includes K directors. The number of the directors, K, defines the Level of the theory. In water wave applications of the theory, namely the Green-Naghdi (GN) Theory, these directors prescribe the variation of the vertical component of the three-dimensional velocity along the water column. In Level I, K=1, and the distribution of the vertical velocity (u2) in the vertical direction (x2) is linear, i.e., u2(x1,x2,x3,t)=C(x1,x3,t) x2. This assumption, along with the incompressibility condition, results in constant horizontal velocity along the water column, i.e., u1(x1,x2,x3,t)=u1(x1,x3,t). This condition is applicable to propagation of long waves in shallow water. In higher level GN equations, higher degree polynomials are used to construct the velocity field. Adopting the Cosserat surfaces, the mass, momentum and angular momentum of the deformable sheet-like (or shell-like) body are expressed in such a way that a general set of equations of motion of the medium can be obtained. The resulting GN equations satisfy the nonlinear boundary conditions exactly, identically conserve mass, and exactly satisfy the integrated momentum equations in an average sense. Unlike the KdV equations, the GN equations are invariant under Galilean transformation.

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Mon 06/02/17 14:00
Fulton G20
Mathematics Seminar
Dr. Nikolaos Sfakianakis (Johannes Gutenberg University)
Chemotaxis and haptotaxis from a cellular level
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Abstract

The cytoskeleton is a cellular skeleton inside the cytoplasm of living cells. The front of the cytoskeleton, also known as lamellipodium, is the driving mechanism of cell motility and is comprised by long double helix polymers of actin protein termed actin-filaments. The actin-filaments polymerize/depolymerize and exhibit a series of physical properties like elasticity, friction with the substrate, crosslink binding, repulsion, myosin-drive contractility, nucleation, fragmentation, capping and more. In this talk we address the FBLM that describes the above (microspcopic) dynamics of the actin-filaments and results to the (macroscopic) movement of the cell, and introduce the Finite Element Method (FEM) used to simulate this system. We embed the resulting cell on an environment of variable chemical and adhesive properties, and study its response to the corresponding chemotaxis and haptotaxis scenarios. We finally reproduce particular (bio) experiments and compare with our findings. The material for this talk is from joint works with: Chr. Schmeiser, D. Oelz, A. Manhart, A. Brunk, N. Kolbe, and V. Small

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Mon 13/02/17 15:00
Fulton G20
Mathematics Seminar
Dr. Jonathan Hodgson (University of St Andrews)
Finding distribution functions via metaheuristic methods.
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Abstract

In order to find a particle distribution function from a given macroscopic description of the magnetic field, one must solve a difficult inverse problem. To make progress analytically, a specific representation of the pressure must be chosen, which restricts the form of the resulting distribution function. In this talk we consider the numerical approach to solving the inverse problem. Specifically, we will give an overview of some interesting metaheuristic algorithms, and show how they may be applied to find an approximate solution to the inverse problem at hand.

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Wed 15/02/17 14:30
Fulton J20
Mathematics Seminar
Dr. Miguel Bustamante (University College Dublin)
Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows
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Abstract

We revisit numerically and analytically the finite-time blowup of an infinite-energy solution of 3D Euler equations [Gibbon et al. (1999)]. By employing the method of mapping to regular systems [Bustamante (2011), Mulungye et al. (2015)], we establish a curious property of this solution that was not observed previously: near singularity time T*, a fast transient is followed by a slower late-time blowup regime that is well resolved spectrally at mid-resolutions (512^2), with a Gaussian wavenumber spectrum. The analyticity-strip width decays `slowly' to zero at t = T*, remaining above the collocation-point scale for all simulation times t < T* - 10^{-9000}. Reaching such a proximity to singularity time is not possible in the original temporal variable, because of the floating-point double-precision barrier (10^{-16}). Due to this limitation on the original variables, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to singularity time: T* - t = 10^{-140}.

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Mon 20/02/17 14:00
Fulton G20
Mathematics Seminar
Dr. Ben Goddard (University of Edinburgh)
Photo-dissociation of molecules: mathematics meets quantum chemistry
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Abstract

Photo-dissociation is a chemical reaction in which molecules are broken down into atoms or smaller molecules by interaction with light (photons). Important examples include the formation and removal of the ozone layer, the break down of CFCs in the atmosphere, and part of photosynthesis in plants. More fundamentally, the photo-dissociation of diatomic molecules is one of the paradigmatic chemical reactions of quantum chemistry, typically used to benchmark new methods. The associated mathematical problem is the study of transitions in a two-level partial differential equation, with one effective spatial degree of freedom - the internuclear separation. Given a wavepacket that travels on the upper level, the challenge is to determine the size and shape of the part of the wavepacket transmitted to the lower level at large times. Such problems are highly multi-scale; the transmitted wavepacket is typically very small with rapid oscillations. This leads to great difficulty in performing accurate numerical calculations, and an alternative method is required. Fortunately, there exists a small parameter $\epsilon$ which is the square root of the the ratio of the electron and nuclear masses. In the standard adiabatic representation, widely used in chemistry, the transmitted wavepacket is of order $\epsilon$ globally in time but exponentially small (order $\exp(-1/\epsilon)$) for large times. This strongly suggests that the adiabatic representation is not the optimal one in which to study the problem. Using the more general superadiabatic representations, and an approximation of the dynamics in the region where the two energy levels become close but do not cross, we obtain an explicit formula for the transmitted wavepacket. Our results agree extremely well with high precision ab-initio calculations, in particular for the real-world sodium iodide molecule. Joint work with Volker Betz (TU Darmstadt), Uwe Manthe (Bielefeld) and Stefan Teufel (Tuebingen)

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Mon 27/02/17 14:00
Fulton G20
Mathematics Seminar
Prof. Chris Sangwin (University of Edinburgh)
Automatic assessment of reasoning by equivalence
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Abstract

Deductive reasoning and proof is one of the hallmarks of mathematics, and is an important factor in distinguishing mathematics from empirical sciences. Fluency in calculation, including symbolic manipulation in algebra and calculus, sit alongside deduction, reasoning and problem solving. "Core pure mathematics" is that essential amalgam which is universally studied by all mathematics, science and engineering students. It starts with traditional algebra, trigonometry and calculus, culminating with De Moivre's theorem and its consequences while stopping short of real analysis. Presentations of core pure mathematics often contain little explicit "proof" beyond formulaic proof by induction, but it is where proof starts for pure mathematicians. Core pure mathematics contains a key activity "reasoning by equivalence". This is reasoning and is key in many of the deductions at this level, but it is very close to a calculation. Indeed, in many situations it can be treated formally as a calculation. This talk will look at the interplay between calculation and reasoning, with a focus on automatic assessment. To what extent can we automate the assessment of reasoning now, and where are the limits of automatic assessment in the future?

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Wed 01/03/17 14:00
Fulton H2
Mathematics Seminar
Dr. Raimondo Penta (Universidad Politecnica de Madrid, Spain)
Investigation of multiphase composites via asymptotic homogenization and its application to the bone hierarchical structure
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Abstract

A multiscale approach is developed for three dimensional multiphase elastic composites via asymptotic homogenization. Each phase is assumed to behave as a linear, possibly anisotropic and inhomogeneous, elastic solid. Discontinuities of the elastic constants across the interface between the host medium (matrix) and any subphase interface are allowed. The classical stress balance equations are stated in each phase, where volume forces and inertia are neglected. Coupling among phases is enforced via continuity of stresses and displacements across every interface. Asymptotic expansion of the displacements is carried out to exploit the sharp length scale separation between the spatially periodic structure (fine scale) and the whole material (coarse scale). The coarse scale mechanics is described by a standard anisotropic elastic model, where the role of the fine scale geometry is encoded in the effective elasticity tensor, which is to be computed solving elastic-­type problems on the appropriate periodic cell. The model is general with respect to the number of subphases and periodic cell shapes. The cell problems are equipped with stress discontinuities, which are proportional to the jumps of the elastic constants across interfaces. The effective elasticity tensor and the auxiliary strains which arise from the cell problems computation are characterized by specific properties and representations which lead to a consistent effective elasticity tensor definition, in terms of symmetries and energetic bounds [1]. A novel three dimensional numerical study is performed assuming an isotropic and homogeneous linear elastic rheology for each phase. The periodic cell geometrical setup is chosen to properly compare the model response to that provided by Eshelby based techniques and point out analogies and differences between the two approaches. The model is benchmarked by comparing our method to well established semi-­analytical schemes [2]. An example of application to the hierarchical structure of the bone (see, e.g., [3]), where the host medium is identified with the collagen matrix and the subphases to the mineral inclusions is provided. We account for the formation of a continuous mineral foam [5], which is represented extending the mineral inclusions up to the periodic cell boundary. Such a physiological condition (which characterizes aged bone tissue) cannot be captured by simple average field techniques, which are widely exploited in the bone literature (see [4]). REFERENCES: [1] Penta, Raimondo, and Alf Gerisch. "The asymptotic homogenization elasticity tensor properties for composites with material discontinuities." Continuum Mechanics and Thermodynamics 29.1 (2017): 187-206. [2] Penta, Raimondo, and Alf Gerisch. "Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study." Computing and Visualization in Science 17.4 (2015): 185-201. [3] S. Weiner and H. D. Wagner. The material bone: Structure-­mechanical function relations. Annual Reviews of Materials Science, 28:271–298, 1998. [4] Penta, R., et al. "Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues." Bioinspiration & biomimetics 11.3 (2016): 035004. [5] Sara Tiburtius, Susanne Schrof, Ferenc Molnar, Peter Varga, Franc¸oise Peyrin, Quentin ´ Grimal, Kay Raum, and Alf Gerisch. On the elastic properties of mineralized turkey leg tendon tissue: multiscale model and experiment. Biomechanics and modeling in mechanobiology, 13:1003–1023, 2014.

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Mon 06/03/17 15:00
Fulton G20
Mathematics Seminar
Prof. Frédéric Hecht (Université Pierre et Marie Curie, Paris)
Multiphysics and HPC with FreeFem++
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Abstract

FreeFem++ is a powerful and flexible software to solve numerically partial differential equations (PDE) in IR2) and in IR3) with finite elements methods. The FreeFem++ language allows for a quick specification of linear PDE’s, with the variational formulation of a linear steady state problem and the user can write they own script to solve non linear problem and time depend problem. You can solve coupled problem or problem with moving domain or eigenvalue problem, do mesh adaptation , compute error indicator, etc ... This talk will give an overview of the main characteristics of FreeFem++ and demonstrate its application to a range of academic examples.

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Mon 13/03/17 14:00
Fulton G20
Mathematics Seminar
Prof. Andy Wright (University of St Andrews)
The theoretical foundation of 3D Alfven resonances
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Abstract

The resonant coupling of magnetohydrodynamic fast and Alfven waves has received much attention in 1D and 2D where it is well-understood. The process has received very little attention in 3D to date. We present some numerical solutions for 3D resonant Alfven waves whose interpretation leads us to propose a theoretical framework that can be used to understand them. It turns out that the 3D case exhibits a number of properties do not exist in 1D or 2D.

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Mon 20/03/17 14:00
Fulton G20
Mathematics Seminar
Dr. Miguel Bernabeu (University of Edinburgh)
Bringing together experimental and computational methods for the study of vascular patterning
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Abstract

The mechanisms underpinning vascular development remain incompletely understood. It is therefore crucial to gain further insight into how complex vascular networks form and which are the key players in this process. The clinical translation of these results holds the key to the improvement of therapies aimed at modulating vascular patterning for the treatment of retinopathies or cancer. In this talk, I will present work undertaken in collaboration with experimental colleagues in order to develop computational blood flow models capable of recapitulating the haemodynamic environment encountered in developing vasculature. These models have already contributed to further our understanding of how blood vessels remodel during development [1-3]. Ongoing work is also focused on the study of vascular remodelling in the context of diabetic retinopathy for the identification of flow-based biomarkers of disease progression. [1] Bernabeu MO, Franco CA, Jones ML, Nielsen JH, Krüger T, Nash RW, Groen D, Hetherington J, Gerhardt H, and Coveney PV “Computer simulations reveal complex distribution of haemodynamic forces in a mouse retina model of angiogenesis” Journal of the Royal Society Interface 11(99):20140543, 2014. [2] Franco CA, Jones ML, Bernabeu MO, Geudens I, Mathivet T., Rosa A, Lopes FM, Lima AP, Ragab A, Collins RT, Phng LK, Coveney PV, and Gerhardt H. “Dynamic endothelial rearrangements drive developmental vessel regression” PLoS Biology 13(4):e1002125, 2015. [3] Franco CA, Jones ML, Bernabeu MO, Vion A-C, Fan J, Mathivet T, Ragab A, Yamaguchi TP, Coveney PV, Lang RA, and Gerhardt H. “Non-canonical Wnt signalling modulates the endothelial shear stress flow sensor in vascular remodelling” eLife 5:e07727, 2016.

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Mon 08/05/17 14:00
Fulton J20
Mathematics Seminar
Prof. Yves Capdeboscq (University of Oxford)
Small inclusions for the Time Harmonic Maxwell Equations
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Abstract

I will review recent result on the topic of small perturbative inclusions for the Time Harmonic Maxwell Equation : we show that the general theory developed over a decade ago for conductivity equation can be extended, in similar generality, for the Maxwell Equation, by revisiting elliptic regularity theory in this context.

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Mon 15/05/17 14:30
Fulton J20
Mathematics Seminar
Dr. Silvia Dalla (University of Central Lancashire)
3D Modelling of Solar Energetic Particle Propagation
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Abstract

Solar Energetic Particles (SEPs) are ions and electrons accelerated during flares and coronal mass ejections at the Sun. They can escape from the corona into interplanetary space and may reach near-Earth locations, where they pose a significant radiation risk to humans in space and satellite hardware. This talk will review our understanding of the origin and transport of SEPs, based on a large body of data gathered by spacecraft detectors and on theoretical models. It will focus on recent results of test particle and kinetic simulations, which show that accurate modelling of SEP propagation requires a 3D approach, due to guiding centre drifts and magnetic field line meandering.

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Mon 22/05/17 14:00
Fulton J20
Mathematics Seminar
Dr. Tommaso Lorenzi (University of St Andrews)
A partial differential equation approach to studying evolutionary dynamics in cancer cell populations
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Abstract

A growing body of evidence supports the idea that solid tumours are complex ecosystems populated by cells with heterogeneous phenotypes, whose dynamics can be described in terms of evolutionary and ecological principles. Under this perspective, it has become increasingly recognised that mathematical modelling can complement experimental cancer research by offering alternative means of interpreting experimental data and by enabling extrapolation beyond empirical observation. This talk deals with mathematical models formulated in terms of partial differential equations which can be used to study evolutionary dynamics in cancer cell populations. In particular, I will present a number of results which illustrate how qualitative analysis and numerical simulation of these equations can help to uncover fresh insights into the critical mechanisms underpinning cancer progression and the emergence of resistance to cytotoxic therapy.

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Fri 26/05/17 16:00
Dalhousie LT2
Mathematics Seminar
Prof. Robert Ghrist (University of Pennsylvania)
Topological Inference from Data
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Abstract

In this talk, I'll argue that the recent advances in applied algebraic topology and topological data analysis point to sheaf theory as a good source of mathematical structure for modelling data tethered to spaces; and cohomology as an especially useful compression of such data. I'll survey a few cutting-edge applications ranging from neuroscience to pursuit-evasion and tracking, closing with how computational issues arise and are addressed with novel mathematical perspectives. Note: Tea in Dalhousie foyer from 4pm. Talk starts at 4.30pm.

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