# Analysis Seminars

### Analysis Seminars for this Semester

**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.

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