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2017/18 semester 1 events in Mathematics

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Mon 11/09/17 15:00
Fulton G20
Mathematics Seminar
Prof. Alexander Gorban (University of Leicester)
Stochastic Separation Theorems
abstract

Abstract

The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples where the system works properly. We demonstrate that in (moderately) high dimension this separation could be achieved with probability close to one by linear discriminants. Based on fundamental properties of measure concentration, we show that for M< aexp(bn) random M-element sets in Rn are linearly separable with probability p, p > 1−ϑ, where 1>ϑ>0 is a given small constant. Exact values of a,b>0 depend on the probability distribution that determines how the random M-element sets are drawn, and on the constant ϑ. Classical measure concentration theorems state that random points are concentrated in a thin layer near a surface (a sphere, an average or median level set of energy or another function, etc.). The stochastic separation theorems describe thin structure of these thin layers: the random points are not only concentrated in a thin layer but are all linearly separable from the rest of the set even for exponentially large random sets. These stochastic separation theorems provide a new instrument for the development, analysis, and assessment of machine learning methods and algorithms in high dimension. Theoretical statements are illustrated with numerical examples. e-print: https://arxiv.org/abs/1703.01203

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Mon 18/09/17 15:00
Fulton G20
Mathematics Seminar
Dr. Theodosios Papathanasiou (Brunel University London)
Finite Elements for the Hydroelastic Response of Very Large Floating Structures under Long Wave Excitation
abstract

Abstract

The study of wave action on large, elastic floating bodies has received considerable attention, finding applications in both geophysics and marine engineering problems. Pontoon-type Very Large Floating Structures (VLFSs) share the same hydroelastic characteristics with ice floes and as a result the methodologies developed for their study bear great resemblance. Among these methodologies, efficient Finite Element schemes for the simulation of hydroelastic interactions in shallow water environments have been recently proposed. These numerical procedures, based on the vertical method of lines, will be presented and analysed. The talk will cover issues regarding (i) weak forms, properties and stability characteristics of the governing equations, (ii) Finite Element approximation and suitable time integration techniques, (iii) Energy norm error estimates and (iv) numerical examples of hydroelastic interactions in variable bathymetry regions.

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Mon 02/10/17 15:00
Fulton G20
Mathematics Seminar
Dr. Sasa Radomirovic (University of Dundee)
The Shared One-Time Pad problem
abstract

Abstract

A number of agents want to broadcast messages to each other in an unconditionally secure way. Every agent has a copy of a secret sequence of random bits and may need to communicate a message to the other agents at any time. How can they use this sequence of bits as a one-time pad to exchange as much information as possible over an asynchronous communication network? Variants of this problem have been considered by Di Crescenzo and Kiayias, by Fitzi, Nielsen and Wolf, and recently by van de Graaf. The problem as stated above has a trivial solution for two agents and an interesting solution for three agents. No deterministic, optimal solution is known for more than three agents.

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Mon 09/10/17 15:00
Fulton G20
Mathematics Seminar
Dr. Radu Cimpeanu (University of Oxford)
Electrohydrodynamic control of multi-fluid systems at small scales
abstract

Abstract

With sizes not exceeding a few centimetres and usage ranging from blood sample analysis to cooling devices and integrated circuit components (and far beyond), liquid systems at small scales play an indispensable part in our lives. Their production and maintenance however is a highly sensitive and expensive process. This body of research focuses on eliminating these drawbacks by creating efficient mechanisms that do not use any moving parts and do not require the presence of an oncoming flow. We analyse electrostatic control procedures in simple geometrical configurations, such as channels containing layers of immiscible fluids. Stability theory guides the imposition of voltage distributions that efficiently manipulate the fluid-fluid interface and harness the classical instabilities arising in such systems. We then formulate nonlinear asymptotic models that enable the study of the interfacial motion far beyond the level of small perturbations. Finally, we implement state-of-the-art computational tools based on the volume-of-fluid method in both two- and three-dimensional contexts to steer these arguments towards practical applications involving microfluidic mixing, pumping and directed polymer assembly.

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Mon 16/10/17 15:00
Fulton G20
Mathematics Seminar
Dr. Carolina Kuepper-Tetzel (University of Dundee)
Using Cognitive Science to Teach Mathematics in Higher Education
abstract

Abstract

Cognitive Science has revealed a number of learning and memory phenomena that have direct, practical implications for learning and teaching. I will highlight the most promising strategies alongside their original research findings and will elaborate on ways on how to implement these in large lectures and small seminars in mathematics. The talk will conclude with an opportunity discuss the feasibility of implementation and ways to overcome potential obstacles.

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Mon 23/10/17 15:00
Fulton G20
Mathematics Seminar
Prof. Christian Engwer (University of Münster)
Bridging the gap between numerical analysis and brain research
abstract

Abstract

Brain source analysis is an important tool in brain research. It is used for example during operation planing for epilepsy patients. Given EEG (electroencephalography) and MEG (magnetoencephalography) measurements the goal is to reconstruct the brain activity, i.e. the electric potential in the brain. This poses an inverse problem. It was observed in experiments, the accuracy of the inverse problem strongly depends on the quality of the forward simulation, in particular the head model. We discuss how modern numerical method like discontinuous Galerkin (dG) methods and cut-cell techniques can increase robustness of the forward problem and simplify the overall work-flow. Hardware oriented design of numerical methods allow to improve speed of the inverse simulation by making use of modern hardware resources. In order to compute the forward problem efficiently we propose an algebraic multigrid solver for cut-cell dG methods. We introduce into the challenges of EEG/MEG inverse modelling and discuss how different parts of the problem can be improved using modern numerical methods.

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Mon 30/10/17 15:00
Fulton G20
Mathematics Seminar
Dr. Tadahiro Oh (University of Edinburgh)
On singular stochastic dispersive PDEs
abstract

Abstract

In this talk, I will go over some of the recent developments on nonlinear dispersive PDEs, such as the nonlinear Schrödinger equations (NLS) and nonlinear wave equations (NLW), with rough and random data and/or forcing. In particular, I will discuss the case of the two-dimensional NLW and how a renormalization appears in the well-posedness theory. Then, taking the stochastic NLS and the stochastic nonlinear heat equations as examples, I will describe the difference between the dispersive and parabolic problems.

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Mon 06/11/17 15:00
Fulton G20
Mathematics Seminar
Dr. Mark Wilkinson (Heriot-Watt University)
Existence of Weak Solutions of the Free Surface Semi-Geostrophic Equations
abstract

Abstract

The semi-geostrophic equations (SG for short) constitute a set of semi-linear transport equations which can be derived as a formal asymptotic limit of the compressible Navier-Stokes equations. SG is deemed by meteorologists to be a model of atmospheric fluids which describes the creation and subsequent dynamics of temperature fronts in the troposphere. In this talk, we present results on the existence of distributional solutions of the semi-geostrophic equations in a finite-depth domain with a free upper boundary. The proof of our existence theorem makes use of techniques from the theory of Optimal Transport, and also the theory of Hamiltonian flows in Wasserstein spaces of measures due to Ambrosio and Gangbo. This work is joint with Mike Cullen (UK Met Office), Tobias Kuna (Reading University) and Beatrice Pelloni (Heriot-Watt University).

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Mon 13/11/17 14:15
Fulton G20
Mathematics Seminar
Prof. Pietro-Luciano Buono (University of Ontario Institute of Technology)
Symmetry in Motion - trotting, swarming and choreographies
abstract

Abstract

Symmetry has been recognized for a long time as an important principle in exploring scientific fields such as physics and chemistry. Symmetry in mathematical models is encoded using group theory and recently, the use of group-theoretic methods has become more prevalent in biology. In this talk, I will start by presenting some highlights of the use of group theoretic methods in nonlinear dynamics. Then, I will review some results from a mathematical model of quadruped locomotion and a model of animal aggregation. I will conclude by going back to physics and discuss choreographies in the N-body problem.

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Thu 16/11/17 15:00
Fulton G20
Mathematics Seminar
Dr. Björn Stinner (University of Warwick)
On a diffuse interface approach to PDEs on surfaces and networks
abstract

Abstract

Diffuse interface models based on the phase field methodology have been developed for various free boundary problems. They involve representing the interfaces by thin layers. Some applications feature phenomena on the interfaces described by PDEs for interface resident fields. We will address the questions of how to model such phenomena in the diffuse interface setting and how to numerically approximate the solutions, which may require special consideration due to degeneracies. The approach can be generalised to networks and bubble clusters. One key challenge then is to correctly recover the conditions in the triple junctions formed by three interfaces. The research is motivated by surface active agents (surfactants) in multi-phase flow which can be effectively modelled by Cahn-Hilliard-Navier-Stokes systems. Some preliminary results for a novel numerical scheme will be presented, as well as some simulations to support the theoretical findings.

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Mon 20/11/17 15:00
Fulton G20
Mathematics Seminar
Dr. Kris van der Zee (University of Nottingham)
Discretisation of PDEs in Banach-space settings: Eliminating Gibbs phenomena and resolving non-Hilbert solutions
abstract

Abstract

Is it possible to obtain near-best approximations to solutions of partial differential equations (PDEs) in a general Banach-space setting? Can this be done with guaranteed stability? I will address these questions by introducing the nonstandard, nonlinear Petrov-Galerkin (NPG) discretisation. The NPG discretisation is imperative for PDEs with rough data or nonsmooth solutions having discontinuities. Its theory generalises and extends Babuska’s theory for the classical Petrov-Galerkin method, as well as recent theories for residual-minimisation methods, such as the discontinuous Petrov-Galerkin method (due to Demkowicz and Gopalakrishnan) and residual minimisation in L^p (due to Guermond). Crucial in the formulation of the NPG method is the nonlinear duality map, which is the natural extension of the Riesz map. I will show the stability of the NPG method and prove its quasi-optimality by extending a classical projection identity which goes back to Kato. To illustrate the significance of the new discretisation framework, I will consider its application to the advection-reaction PDE and the Laplacian. Two of the main benefits of moving to Banach-space settings will be highlighted: 1. The ability to eliminate the notorious Gibbs phenomena of numerical overshoots when the solution contains discontinuities. 2. The ability to approximate rough non-Hilbert solutions, which can not be handled by standard methods in Hilbert spaces. This is joint work with Ignacio Muga from Pontificia Universidad Catolica de Valparaiso.

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