Autumn term: Room M314

Spring term: Room M100

(some) Tuesdays 13.00-14.00

Organiser:

Dr Nikos Katzourakis

November 14 2017 13.00-14.00, M314

COLLOQUIUM IN ANALYSIS

Title: How operators read factorial languages

Abstract: Factorial languages arise in the context of Automata Theory. Essentially, they give the sequences of allowable operations an automaton can perform. The interplay with Operator Theory goes back to the work of Cuntz-Krieger and Matsumoto. A factorial language can be quantized in Hilbertian operators by using a Fock space construction, similar to what is done in Quantum Mechanics. In this talk I will present two algebras of operators that can be considered, and the level of rigidity they offer. Our study is carried in the intersection of C*-correspondences, subproduct systems, dynamical systems and subshifts. I will give the basic steps of our results with some comments on their proofs. The talk is based on joint works with Shalit and Barrett.

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October 31 2017 13.00-14.00, M314

Speaker: Marco Squassina (Brescia, Italy)

Title: Approximation results for magnetic Sobolev norms

Abstract:  We discuss recent results on the approximation of classical magnetic Sobolev norms with

some physically relevant nonlocal magnetic energies.

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[PLEASE NOTE THE ONE-OFF CHANGE OF DAY AND ROOM]

FRIDAY October 27 2017 13.00-14.00, M100

Speaker: Agissilaos Athanasoulis (Dundee, UK)

Title: Rogue Waves and Penrose-modulation stability analysis for waves with continuous spectra

Abstract: Rogue Waves can be thought of as extreme events in quasi-homogeneous noisy wavefields. They are known to appear in the ocean, as well as other noisy nonlinear settings, including optic fibres, Bose-Einstein condensates and plasmas, and they are several orders of magnitude more common than straightforward statistical analysis of "typical waves" would indicate. Recently there has been a flurry of studies on various aspects of Rogue Waves, however, there has been no satisfactory understanding of the phenomenon. This can seen in the lack of likelihood estimation for their emergence and in the the lack of ways to control and manipulate their emergence and qualitative features. In this work we analyse the problem at the level of the autocorrelation function. A linear stability analysis provides a Penrose-type condition determining whether the background wavefield is stable or unstable. In case of instability, the time and length scales for the emergence of localised instabilities are recovered. For the special case of plane waves we recover the modulation instability, but a key difference is that our analysis can be applied to any autocorrelation function.  When this approach is applied to realistic oceanographic data the result is a plausible description of Rogue Waves, and a first-ever quantitative explanation of certain empirically known facts. Includes joint work with G. Athanassoulis (NTUA), T. Sapsis (MIT) and M. Ptashnyk (Dundee).

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October 3 2017 13.00-14.00, M314

Speaker: Andrew Morris (Birmingham, UK)

Title: HuygensÕ Principle for Hyperbolic Equations with L° coefficients via First-Order Systems

Abstract: We prove that strongly continuous groups generated by first- order systems on Riemannian manifolds have finite propagation speed. Our procedure provides a new direct proof for self-adjoint systems that has an extension to operators on metric measure spaces. As an application, we present a new approach to the weak HuygensÕ principle for second-order hyperbolic equations with L° coefficients. This is joint work with Alan McIntosh.

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Event: 7th meeting of the Reading-Bath-Cardiff network on generalised solutions for nonlinear PDE

Venue: Department of Mathematics and Statistics, University of Reading, UK

Room: M113

Date: 13th of June 2017

Programme:

10.30-11.00 Coffee and discussions

11.00-11.40 Melanie Rupflin (Oxford, UK)

11.50-12.40 Michael Ruzhansky (Imperial, UK)

12.50-14.00 Lunch and coffee

14.00-14.40 Ali Taheri (Sussex, UK)

14.50-15.40 Abderrahim El Moataz Billah (Caen, France)

15.50-16.40 Lucia Scardia (Bath, UK)

16.50-17.30 Discussions

17.30-18.30 Supper

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April 4 2017 13.00-14.00, M212

Speaker: Jonas Azzam (Edinburgh, UK)

Title: Harmonic measure, absolute continuity, and rectifiability

Abstract: For reasonable domains $\Omega\subseteq\mathbb{R}^{d+1}$, and given some boundary data $f\in C(\partial\Omega)$, we can solve the Dirichlet problem and find a harmonic function $u_{f}$ that agrees with $f$ on $\partial\Omega$. For $x_{0}\in \Omega$, the association $f\rightarrow u_{f}(x_{0})$. is a linear functional, so the Riesz Representation gives us a measure $\omega_{\Omega}^{x_{0}}$ on $\partial\Omega$ called the harmonic measure with pole at $x_{0}$. One can also think of the harmonic measure of a set $E\subseteq \partial\Omega$ as the probability that a Brownian motion of starting at $x_{0}$ will first hit the boundary in $E$. In this talk, we will survey some very recent results about the relationship between the measure theoretic behavior of harmonic measure and the geometry of the boundary of its domain. In particular, we will study how absolute continuity of harmonic measure with respect to $d$-dimensional Hausdorff measure implies rectifiability of the boundary and vice versa.

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31 January 2017 13.00-14.00, M212

Title: Algebraic aspects of Integrable equations - the Lax-Darboux scheme

Abstract: The first part of the talk will be an introduction to the modern theory of integrable equations. When possible I will mention some historical elements. More precisely, I will present a classical result, due to Darboux and Crum, regarding the covariance properties of the Sturm-Liouville eigenvalue problem. The ramifications of the Darboux-Crum theorem to the modern theory of integrable equations via the so called Lax representation, of an integrable equation, will be reviewed for the case of the Korteweg-de Vries equation. Finally, I will briefly mention some recent developments of the field.

In the second part of the talk, I will present some recent results that I obtained with my colleagues on the Darboux transformations of  the vector nonlinear Schrodinger equation (vNLS-Manakov system) and the representation of its n-soliton solution in terms of ratio of determinants. The Backlund transformation of the vNLS will also be presented. Finally, it will be shown that the general Darboux transformation is invariant under the action of an involutive automorphism of the underlying Lie algebra and that it is parametrised in terms of the Grassmannian  Gr(k,N).

The second part of the talk contains results of a recent joint work with A. Doikou and P.M. Adamopoulou from the Heriot-Watt University.

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24 January 2017 13.00-14.00, M113

Title: A glimpse of Calculus of Variations in L° through the non-expertÕs keyhole

Abstract: A basic problem in Riemannian Geometry (which is an extension of the Yamabe problem) asks about the existence of metrics that minimise the maximum of the scalar curvature in a class of conformal changes. A relevant problem in applied mathematics (Data Assimilation, PDE-constrained optimisation, etc) asks about the existence of minimisers of the maximum error of a measurable quantity depending on the particular model.

In this talk I will describe a general framework of vectorial Calculus of Variations in L° which is the appropriate setup in which such problems can be studied effectively and in a unified fashion. I will also discuss a general existence-uniqueness-structural result I obtained recently with R. Moser from Bath which demystifies the structure of the so-called  °-Biharmonic functions.

No knowledge of differential geometry or of applied mathematics is required to attend this talk, only a knowledge of basic Analysis.

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17 January 2017 13.00-14.00, M212

Speaker: John Meyers (Birmingham, UK)

Title: Localised and front solutions to a Cauchy problem for a Semi-linear parabolic PDE with trivial initial data

Abstract: In this presentation we consider whether or not there exist spatially inhomogeneous classical solutions to a class of Cauchy problems for reaction-diffusion equations with spatially homogeneous initial data. In addition we consider the sharpness of a functional derivative estimate of Schauder type for solutions to associated Cauchy problems.

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13 December 2016 13.00-14.00, M212

Speaker: Enea Parini (Aix-Marseille, France)

Title: The eigenvalue problem for the fractional p-Laplacian

Abstract: In this talk I will present some results on the eigenvalue problem for the fractional p-laplacian, a nonlinear, nonlocal differential operator. In particular, the minimization of the first and the second eigenvalues will be discussed, as well as the stability of the variational spectrum when the operator tends to its local counterpart, the p-laplacian. The results were obtained in collaboration with Lorenzo Brasco (Ferrara), Erik Lindgren (Stockholm), Marco Squassina (Brescia).

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22 November 2016 13.00-14.00, M212

Title: A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in L°

Abstract: After introducing the main objects of vectorial Calculus of Variations in L° in an accessible way, I will describe a new result which establishes that generalised solutions to the relevant PDE system describing critical points can be characterised via local affine variations of the energy functional. This is talk is based on recent joint work with N. Katzourakis.

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8 November 2016 13.00-14.00, M212

Speaker: Andrew Comech (Texas A&M University, USA & IITP, Moscow)

Title: On stability of solitary waves in the nonlinear Dirac equation

Abstract: We consider the point spectrum of non-selfadjoint Dirac operators which arise as linearizations at solitary wave solutions to the nonlinear Dirac equation. We show that in the model with the Soler-type nonlinearity, in the nonrelativistic limit (small-amplitude solitary waves with frequency near m), the spectral stability and linear instability results essentially parallel the case of the nonlinear Schrodinger equation. Besides analytic results, we present numerical computations of the spectrum in dimensions up to three. Results are partially based on the preprint "On spectral stability of the nonlinear Dirac equation" (with Nabile Boussaid), http://arxiv.org/abs/1211.3336  to appear in Journal of Functional Analysis.

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1 November 2016 13.00-14.00, M212

Speaker: Alessia Kogoj (Salermo, Italy)

Title: Liouville theorems for  Hypoelliptic Partial Differential Operators on Lie Groups

Abstract: We present several Liouville-type theorems for caloric and subcaloric functions on Lie groups in R^{N}. Our results apply in particular to the heat operator on Carnot groups, to linearized Kolmogorov operators and to operators of Fokker-Planck-type like the Mumford operator. An application to the uniqueness for the Cauchy problem is also showed.

The results presented are obtained in collaboration with A. Bonfiglioli, E. Lanconelli, Y.Pinchover and S. Polidoro.

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18 October 2016 13.00-14.00, M212

Title: Sobolev spaces on non-Lipschitz sets with application to boundary integral equations on fractal screens

Abstract: The scattering of a time-harmonic acoustic wave by a planar screen with Lipschitz boundary is classically modelled by boundary integral equations (BIEs). If the screen is not Lipschitz, e.g. has fractal boundary, the correct Sobolev space setting to pose the problem is not obvious, because many of the relations between the standard definitions of Sobolev spaces on subsets of Euclidean space (e.g. restriction, completion of spaces of smooth functions, interpolation...) that hold in the Lipschitz case, fail to hold in general.

To extend the BIE framework to general screens, we study properties of the classical fractional Sobolev spaces (or Bessel potential spaces) on general non-Lipschitz subsets of Rn. In particular, we extend results about duality, s-nullity (whether a

set with empty interior can support distributions with given Sobolev regularity), and about the equivalence or not between alternative space definitions, providing several examples.

An interesting application is the approximation of variational (integral or differential) problems posed on fractal sets by problems

posed on prefractal approximations.

This is a joint work with S.N. Chandler-Wilde (Reading) and D.P. Hewett (UCL).

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11 October 2016 13.00-14.00, M212

Speaker: Igor Velcic (Zageb, Croatia)

Title: Homogenization of thin structures in nonlinear elasticity- periodic and non-periodic

Abstract: We will give the results on the models of thin plates and rods in nonlinear elasticity by doing simultaneous homogenization and dimensional reduction. In the case of bending plate we are able to obtain the models only under periodicity assumption and assuming some special relation between the periodicity of the material and thickness of the body. In the von K\'arm\'an regime of rods and plates and in the bending regime of rods we are able to obtain the models in the general non-periodic setting. In this talk we will focus on the derivation of the rod model in the bending regime without any assumption on periodicity.

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