**Reading Lunchtime PDE Seminar Series**

**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

Speaker: Eugenios
Kakariadis (Newcastle, UK)

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

Speaker: Giorgos
Papamikos (Reading, UK)

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

Speaker: Nikos Katzourakis (Reading, UK)

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

Speaker: Birzhan
Ayanbayev (Reading, UK)

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

Speaker: Andrea Moiola (Reading, UK)

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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