Analysis Seminars 2017-18
Tuesday 3rd October, 2017 (place: UCD Science North 125 ) 4.00pm
Speaker: R. Smith
Title: The continuity of betweenness
Abstract Given a set $X$, we can use a suitable ternary relation $[\cdot,\cdot,\cdot] \subseteq X^3$ to express the notion of `betweenness' on $X$: $x$ is between $a$ and $b$ if and only if $[a,x,b]$ holds. We assume that this relation is "basic": $[a,a,b]$ and $[a,b,b]$ always hold, $[a,x,b]$ implies $[b,x,a]$, and $[a,x,a]$ implies $x=a$. Many natural examples of betweenness arise when $X$ is endowed with some additional order-theoretic or topological structure. Given $a,b \in X$, we can define the "interval" $[a,b] = \lbrace x \in X\,:\,[a,x,b]\rbrace\;(= [b,a])$. If $X$ has additional topological structure, it is reasonable to ask whether the assignment $\lbrace a,b\rbrace \mapsto [a,b]$ has good continuity properties, given a suitable hyperspace topology. We examine this question in the context of "Menger betweenness" on metric spaces $(X,d)$ ($[a,x,b]$ holds if and only if $d(a,b)=d(a,x)+d(x,b)$), and the "K-interpretation of betweenness" on topological continua ($[a,x,b]$ holds if and only if $x$ is an element of every subcontinuum that includes $a$ and $b$). This is joint work with Paul Bankston (Marquette University, WI) and Aisling McCluskey (NUI Galway).
Tuesday 10th October (place: UCD Science North 125 ) 3:00pm
Speaker: R. Levene
Title: Non-commutative graph parameters and quantum channel capacities
Abstract We generalise some graph parameters to non-commutative graphs (a.k.a. operator systems of matrices) and quantum channels. In particular, we introduce the quantum complexity of a non-commutative graph, generalising the minimum semidefinite rank. These parameters give upper bounds on the Shannon zero-error capacity of a quantum channel which can beat the best general upper bound in the literature, namely the quantum Lovász theta number. This is joint work with Vern Paulsen (Waterloo) and Ivan Todorov (Belfast).
Tuesday 10th October (place: UCD Science North 125 ) 4:15pm
Speaker: M. Whittaker (Glasgow)
Title: Fractal substitution tilings and applications to noncommutative
geometry
Abstract Starting with a substitution tiling, such as the Penrose tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles typically have fractal boundary. As an application of fractal tilings, we construct an odd spectral triple on a C*-algebra associated with an aperiodic substitution tiling. Even though spectral triples on substitution tilings have been extremely well studied in the last 25 years, our construction produces the first truly noncommutative spectral triple associated with a tiling. My work on fractal substitution tilings is joint with Natalie Frank and Sam Webster, and my work on spectral triples is joint with Michael Mampusti.
Tuesday 17th October, 2017 (place: UCD Science North 125 ) 4.00pm
Speaker: N. Dobbs
Title: Nearby Birkhoff averages
Abstract Birkhoff averages (of an observable along orbits) are objects of interest when investigating statistical behaviour of a dynamical system. If there is a unique physical measure, the Birkhoff averages will converge, for almost every orbit, to the space average (i.e. the integral) of the observable, so the physical measure captures important statistical properties of the dynamical system. However, in the quadratic family, for example, physical measures don't always exist, and even when they do, they don't necessarily depend continuously on the parameter. In joint work with Alexey Korepanov, we examine what happens for finite time Birkhoff averages for nearby parameters.
Tuesday 24th October, 2017 (place: UCD Science North 125 ) 4.00pm
Speaker: R. Timoney
Title: TROs and Morita equivalence
Abstract It is possible to recast the theory of Morita equivalence in terms of the elementary theory of Ternary Rings of Operators (TROs). In particular the Morita correspondence between primitive ideals follows by extending irreducible representations from the right C*-algebra to the linking C*-algebra. The celebrated Brown-Green-Reiffel theorem characterising Morita equivalence as stable isomorphism in the separable case follows by using a Lemma of Brown to show that separable stable TROs are TRO isomorphic to C*-algebras.
Tuesday 7th November, 2017 (place: UCD Science North 125 ) 4.00pm
Speaker: C. Boyd
Title: Real Extreme points of Spaces of Complex Polynomials
Abstract Given a Banach space $E$ and a positive integer $n$ we let $\mathcal P_I(^nE)$ denote the space of all $n$-homogeneous integral polynomials on $E$. This space generalise the trace class operators and plays an important role in the duality theory of spaces of homogeneous polynomials. When $E$ is a real Banach space and $n\ge 2$ it is known that the set of extreme points of the unit ball of $\mathcal P_I(^nE)$ is equal to the set $\lbrace\pm\varphi^n:\|\varphi\|=1\rbrace$. When $E$ is a complex Banach space a characterisation of the set of extreme points of the unit ball of $\mathcal P_I (^nE)$ is not so easy to establish. In this talk, I will look at what can be said for low values of $n$ and small linear combinations of extreme points. This is joint work with Anthony Brown.
Tuesday 14th November (place: UCD Science North 125 ) 3:00pm
Speaker: S. Gardiner
Title: Isoperimetric inequalities for Bergman analytic content I
Abstract The analytic content of a plane domain measures the distance between $\bar z$ and a given space of holomorphic functions on the domain. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This talk will review known results about analytic content, and then focus on the Bergman space of $L^p$ integrable holomorphic functions. It will describe isoperimetric-type inequalities for Bergman p-analytic content in terms of the St Venant functional for torsional rigidity, and address the cases of equality with the upper and lower bounds. (This is joint work with Marius Ghergu and Tomas Sjödin.)
Tuesday 14th November (place: UCD Science North 125 ) 4:15pm
Speaker: H. Render
Title: The differential equation of second order for the cross product of Bessel functions
Abstract Bessel functions play an important role for problems with cylindrical symmetry. The cross product of Bessel functions is used for solving boundary value problems of an annular cylinder. In this talk we shall present the construction of a second order differential equation for the cross product. The method applies in a more general setting and various examples will be given. For the case of half-integers the potential of the cross product can be explicitly computed and examples show that the potential seems to have a special form, having a unique maximum at one point $x_0$ and it is increasing for $x < x_0$ and decreasing for $x > x_0$.
Tuesday 21st November (place: UCD Science North 125 ) 3:00pm
Speaker: B. Lemmens (Kent)
Title: The Denjoy-Wolff theorem for Hilbert geometries
Abstract The classical Denjoy-Wolff theorem asserts that all orbits of a fixed point free holomorphic self-mapping of the open unit disc in the complex plane, converge to a unique point in the boundary of the disc. Since the inception of the theorem by Denjoy and Wolff in the nineteen-twenties a variety of extensions have been obtained. In this talk I will discuss some extensions of the Denjoy-Wolff theorem to certain real metric spaces, namely Hilbert geometries. Hilbert geometries are a natural generalisation of Klein's model of the real hyperbolic space, and play in important role in the analysis of linear, and nonlinear, operators on cones.
Tuesday 21st November (place: UCD Science North 125 ) 4:15pm
Speaker: S. Gardiner
Title: Isoperimetric inequalities for Bergman analytic content II
Tuesday 28th November (place: UCD Science North 125 ) 3:00pm
Speaker: G. Singh
Title: Nonlocal Pertubations of Fractional Choquard
Equation
Abstract We study the Nonlocal Fractional Choquard equation. These kind of problems (involving fractional Laplace and nonlocal operators) arise in various applications, such as continuum mechanics, phase transitions, population dynamics, optimization, finance and many others. First, the existence of a groundstate solutions using minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.
Tuesday 28th November (place: UCD Science North 125 ) 4:15pm
Speaker: M. Ghergu
Title: Exact behaviour of positive solutions near isolated singularity in a
logarithmic setting
Abstract We study the exact behaviour around the isolated singularity at the origin for $C^2$-positive solutions of $-\Delta u=u^\alpha |\log u|^\beta$ in a punctured neighbourhood of the origin of $R^n$, $n\ge 3$. This talk discusses first the power case $\beta =0$ then underlines the difficulties in dealing with logarithmic terms and presents further directions of investigation. This is based on a joint work with Henrik Shahgholian (KTH, Stockholm) and Sunghan Kim (National University of Seoul, Korea).
Tuesday 5th December (place: UCD Science North 125 ) 3:00pm
Speaker: M. Golitsyna
Title: Overconvergence properties of harmonic homogeneous polynomial expansions with gaps
Abstract The series expansions of holomorphic functions with gaps have been studied since the beginning of the twentieth century and found applications in the recent study of universal Taylor series. In this talk I will discuss analogous theory for harmonic functions and its application to show non-existence of universal harmonic homogeneneous expansions on certain type of domains in $R^N$.
Tuesday 5th December (place: UCD Science North 125 ) 4:15pm
Speaker: M. Mackey
Title: Inner iteration of holomorphic functions on hyperbolic domains
Abstract If $(f_n)$ is a sequence of holomorphic self-maps of a domain then the associated inner iterated function system is $(F_n)$ where $F_n=f_1\circ\cdots\circ f_n$. We survey results of Gill, Lorentzen, Beardon et al., Keen and Lakic, and Bracci concerning the convergence of such systems, focusing (in the sprit of the Denjoy-Wolff theorem) on conditions which guarantee that limit points are constant.
Tuesday 6th February, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: R. Smith
Title: Approximation of norms in Banach spaces
Abstract
This talk follows on from one I gave in May 2017. Let $X$ be a Banach space
and let $\mathbf P$ be a property of norms. We say that a norm $\|\cdot\|$
on $X$ (equivalent to the original norm) can be
approximated by norms having $\mathbf P$ if,
given $\varepsilon>0$, there exists another norm
$|||\cdot|||$ on $X$ with $\mathbf P$, such that $\|x\| \leq |||x||| \leq
(1+\varepsilon)\|x\|$ for all $x \in X$. There are a number of papers in
the literature that consider the question of whether or not all
(equivalent) norms on a given space can be approximated in this way.
For a number of classes of Banach spaces $X$, including $c_0(\Gamma)$
(where $\Gamma$ is an arbitrary set), certain Orlicz spaces and Lorentz
predual spaces, and a class of $C(K)$ spaces (where $K$ comes from a class
of compact spaces having unbounded scattered height), we show that all
equivalent norms on $X$ can be approximated by $C^\infty$-smooth norms or
polyhedral norms.
This is joint work with Stanimir Troyanski, University of Murcia, Spain,
and Institute of Mathematics, Bulgarian Academy of Sciences
Tuesday 13th February, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: A. M\"uller-Hermes (Copenhagen)
Title: Positive maps in quantum information theory
Abstract Many problems in quantum information theory are connected to properties of positive linear maps between matrix algebras. After a brief introduction to some basic concepts of quantum information theory I want to focus on the problem of entanglement distillation. I will explain how this problem is connected to the existence problem of positive linear maps that stay positive under taking tensor powers and that are neither completely positive nor completely co-positive. If time permits I will outline some constructions of interesting positive maps and how non-decomposability arises naturally from tensorisation.
Tuesday 20th February, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: R. Harte
Title: 1973 and all that
Tuesday 27th February, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: P. Tradacete (Madrid Carlos III)
Title: On the least doubling constant of a metric space
Abstract We will explore the question of determining the least doubling constant among all doubling measures defined on a metric space. We will show that in many natural instances this constant is at least 2. This is based on work in progress with J. Soria (Barcelona).
Tuesday 6th March, 2018 (place: UCD Science North 125 ) 3.00pm
Speaker: J. Giacomoni (Pau, France)
Title: Diaz Saa Inequality for $p(x)$-laplacian and applications
Abstract In this talk, I will present a recent work with P. Takac. It concerns a new extension of the well-known inequality by Diaz-Saa which in our case, involves an anisotropic operator, such as $p(x)$-Laplacian. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions, comparison principles and stabilization for some anisotropic quasilinear elliptic and parabolic equations.
Tuesday 13th March, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: S. Gardiner
Title: Universal Fourier and Taylor series
Abstract It has long been known that there exist trigonometric series, the partial sums of which possess universal approximation properties on the unit circle. It turns out that, for most smooth functions on the unit circle, the partial sums of the associated Fourier series, when extended to the plane, have universal approximation properties off the circle. There are also related results for pairs of Taylor and Laurent series arising from functions that are holomorphic off a Jordan curve. (This is joint work with Vassili Nestoridis and Christos Papadimitropoulos.)
Tuesday 20th March, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: P. Mellon
Title: Holomorphic dynamics on bounded symmetric domains
Abstract
The open unit ball, $B$, of a Banach space is homogeneous if given
any two points $z,w$ in $B$, there is a biholomorphic map sending
$z$ to $w$. Such balls classify the bounded symmetric domains,
include many classical spaces and ensure a Jordan structure on the
underlying space. Let $f:B\mapsto B$ be a holomorphic fixed-point
free map. The behaviour of the sequence of iterates, $f^n=f\circ
f^{n-1}$, of $f$ is the subject of much study since the Wolff Denjoy
results for the complex disc $\Delta $ in 1926. Generally, in
infinite dimensions, $(f^n)$ does not converge, even in the Hilbert
space case. Our work therefore seeks to establish the 'location'
of accumulations points of $(f^n)$, with respect to the topology
of local uniform convergence on $B$.
This seminar will present results in this direction, using a recently
proved Wolff type theorem for infinite dimensional bounded symmetric
domains.
Tuesday 27th March (place: UCD Science North 125 ) 3:00pm
Speaker: M. Ghergu
Title: Isolated singularities for semilinear elliptic systems with power-law nonlinearity
Abstract
We discuss the behaviour around isolated singularity for
nonnegative solutions of semilinear elliptic systems. Unlike the standard
methods available in the literature which rely on moving plane methods, we
apply several tools that pertain to free boundary problems in order to
pursue our investigation.
This talk is based on a joint work with Sunghan Kim (National University
Seoul, South Korea) and Henrik Shahgholian (KTH Stockholm).
Tuesday 27th March (place: UCD Science North 125 ) 4:15pm
Speaker: C. Boyd
Title: Integral and Nuclear Polynomials on tree spaces
Abstract We examine the Radon-Nikodym Property and Asplundness, in particular their connection to integral and nuclear mappings and polynomials. We show that the structure given to us by tree spaces provides the ideal setting to uncover the intricacies of the relationship between integral and nuclear polynomials. (This is joint work with C. Poulios and M. Venkova.)
Tuesday 10th April, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: H. Render
Title: Generalized Bernstein operators
Abstract
In this talk we discuss the existence and properties of generalized
Bernstein operators in the context of extended Chebyshev spaces.
Special analysis is given to Bernstein operators in the polynomial
setting which fix the constant function 1 and the function $x^3$
with respect to an interval $[a,b]$ containing 0.
This talk is based on a joint work with
J.M. Aldaz (Universidad Autonoma de Madrid)
Tuesday 24th April, 2018 (place: UCD Science North 125 ) 4.00pm
Speaker: M. Golitsyna
Title: Overconvergent properties of Dirichlet series
Abstract In this talk I will describe overconvergent properties of Dirichlet series with regard to the behavior of their partial sums near infinity. The results are inspired by corresponding phenomena for Taylor series, which were recently discovered using the notion of thinness.