# 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.