# Analysis Seminars 2017/18

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**Speaker:**

**Date: **

**Time: ** 3pm

**Location: ** **2nd Seminar**

**Title:** TBC

**Speaker:** TBC

**Date: **

**Time: ** 4.15pm

**Location:**

**Title: ** The continuity of betweenness**Speaker:** Richard Smith (UCD)**Date: ** Tuesday, 3rd October 2017**Time: ** 4pm**Location: ** UCD, 125 Science North (JK Lab)**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) nd Aisling McCluskey (NUI Galway).

**Please note there will be two talks; one at 3pm and a 2nd at 4.15pm****There will be coffee during the interval in Room G26A****Title: ** Non-commutative graph parameters and quantum chanel capacities

**Speaker: ** Rupert Levene - (UCD)**Date: ** Tuesday, 10th October 2017**Time: ** 3pm**Location:** Room 1.25 Science Centre North**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).

Please note there will be two talks; one at 3pm and a 2nd at 4.15pm

There will be coffee during the interval in Room G26A**Title: ** Fractal substitution tilings and applications to noncommutative Geometry**Speaker:** Michael Whittaker (Glasgow)

**Date: ** Tuesday, 10th October 2017**Time: ** 4.15pm**Location: ** Room 1.25 Science Centre North**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 mywork on spectral triples is joint with Michael Mampusti.

This is joint work with Paul Bankston (Marquette University, WI)

and Aisling McCluskey (NUI Galway).

**Title: **Nearby Birkhoff averages

**Speaker:** Neil Dobbs (UCD) Date: Tuesday, 17th October 2017

**Time:** 4pm

**Location:** Room 1.25, O’Brien Centre for Science

**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.

Coffee will be served in Room G0.26A, Ground floor, Science North at 3.45pm.

**Title: ** TROs and Morita equivalence

**Speaker: ** Richard Timoney (TCD)

**Date: ** Tuesday, 24th October 2017**Time: ** 4pm**Location: ** Room 1.25, O’Brien Centre for Science (North)**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.

Coffee will be served in Room G0.26A, Ground floor, Science North at 3.45pm.

**Title: ** Real Extreme points of Spaces of Complex Polynomials

**Speaker:** Christopher Boyd (UCD)

**Date: ** Tuesday, 7th November 2017

**Time: ** 4pm**Location: ** Room 1.25, O’Brien Centre for Science (North)**Abstract:**

Given a Banach space $E$ and a positive integer $n$ we let $\mathcal P_I(^nE)$ denote the space of all $n$-omogeneous 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 $\{ \pm\varphi^n : \|\varphi\|=1 \}$.

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.

Coffee will be served in Room G0.26A, Ground floor, Science North at 3.45pm.

**Title: ** Isoperimetric inequalities for Bergman analytic content I

**Speaker: ** Stephen Gardiner (UCD)

**Date: ** Tuesday, 14th November 2017

**Time:** 3pm (1st talk)

**Location: ** Room 1.25, O’Brien Centre for Science (North)

**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.)

Coffee will be served in Room G0.26A, Ground floor, Science North at 3.45pm.

**Title: ** The differential equation of second order for the cross product of Bessel functions

Speaker: Herman Render (UCD)

**Date: ** Tuesday, 14th November 2017

**Time: ** 4.15pm (2nd talk)

**Location: ** Room 1.25, O’Brien Centre for Science (North)

**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 $x > x_0$.

**Title: ** The Denjoy-Wolff theorem for Hilbert geometries

**Speaker: ** Bas Lemmens (University of Kent)

**Date: ** Tuesday, 21st November 2017

**Time: ** 3pm (1st talk)

**Location: ** Room 1.25, O’Brien Centre for Science (North)

**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.

**Coffee will be served in Room G0.26A, Ground floor, Science North at 4pm prior to second seminar.**

*Coffee will be served in Room G0.26A, Ground floor, Science North at 4pm.***2nd Seminar****Title: ** Isoperimetric Inequalities for Bergman Analytic Content II

**Speaker: ** Stephen Gardiner (UCD)

**Date: ** Tuesday, 21st November 2017

**Time: ** 4.15pm (2nd talk)

**Location:** Room 1.25, O’Brien Centre for Science (North)

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