# Analysis Seminars 2018/19

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**Title:** The Green function for the exterior cylinder**Speaker:** Hermann Render (UCD)**Date:** Tuesday, 25th September 2018**Time:** 3.30 pm **Location:** Room 1.25, O’Brien Centre for Science (North)

**Abstract:**

In this talk we present a formula for the Green function of the exterior cylinder and we discuss basic properties and estimates of the Green function. Related to this are suitable estimates of the cross product of scaled Bessel functions.

**Title: ** State conversion using von Neumann algebras

**Speaker: **Rupert Levene (UCD)

**Date: **Tuesday, 9th October 2018

**Time: **4pm

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

**Nielsen's theorem gives a simple characterisation of the pairs of pure finite-dimensional quantum states for which one may be converted to the other using "local operations and classical communication". We will give a mathematical introduction to this important result from quantum information theory, along with a new generalisation to an infinite-dimensional von Neumann algebraic context.**

Abstract:

Abstract:

This is based on joint work with Jason Crann, David Kribs and Ivan Todorov.

**Title: ** Overconvergence Properties of Dirichlet series

**Speaker: ** Mayya Golitsyna (UCD)

**Date: ** Tuesday, 16th October 2018

**Time: ** 4pm

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

In this talk we discuss the properties of the subsequences of the partial sums of general Dirichlet series. It is known that a Dirichlet series of the form $\sum_{j=0}^\infty a_je^{-\lambda_js}$ either diverges, converges on some half-plane $\{\mathrm{Re}(s)>c\}$ to a holomorphic function $f$ or converges on the whole complex plane. In case where the series converges on a half-plane it is possible that the function $f$ has a holomorphic extension to a larger domain that strictly contains the half-plane. We will give sufficient conditions for a subsequence of partial sums of the series to converge at every regular point of $f.$

We apply potential theoretic techniques to prove the results.

**Title:** Isolated Singularities for a semi-linear elliptic system

**Speaker: ** Marius Ghergu (UCD)

**Date: ** Tuesday, 30th October 2018

**Time:** 3pm

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

We are concerned with the study of a semi-linear elliptic system featuring power type non linearities in the critical case. We classify all non negative solutions around their isolated singularity by using moving spheres method and invariant quantities.

This is a joint work with H. Shagholian (Stockholm) and S. Kim (Seoul)

** **

**Title: **Localisation and Positivity of Orthogonally Additive Polynomials

**Speaker**: Christopher Boyd (UCD)

**Date**: Tuesday, 6^{th} November 2018

**Time: **4pm

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

** **

**Abstract:**

We show how the localisation technique allows to characterise orthogonally additive polynomials which are the power of a linear functional and to bound the norm of the absolute value of an orthogonally additive polynomial, $P$, by the norm of $P$.

These results are joint work with R. Ryan and N. Snigireva (NUI Galway).

**Title: ** Optimal polynomial approximants**Speaker: ** Myrto Manolaki (UCD)**Date: ** Tuesday, 13th November 2018**Time: ** 4pm **Location: ** Room 1.25, O’Brien Centre for Science (North)**Abstract:**

Given a Hilbert space $H$ of analytic functions on the unit disc and a function $f$ in $H$, a polynomial $p_n$ is called an optimal polynomial approximant of degree $n$ of $1/f$ if $p_n$ minimizes $\|pf - 1\|$ over all polynomials $p$ of degree at most $n$. This notion was introduced to investigate the phenomenon of cyclicity in certain function spaces, including the classical Hardy, Bergman and Dirichlet spaces. In this talk, we will discuss the behaviour of the sequence of optimal polynomial approximants on subsets of the unit circle. Our main theorem uses a new result on simultaneous zero-free approximation, which is of independent interest.

(Joint work with Catherine B\'en\'eteau, Oleg Ivrii and Daniel Seco.)

**Title:** The Diameter Norm**Speaker:** Ray Ryan (NUI Galway)**Date:** Tuesday, 20th November 2018**Time:** 3pm **Location:** Room 1.25, O’Brien Centre for Science (North)

**Abstract:**

The diameter of a continuous function f on a compact Hausdorff space K is sup{ |f(s)-f(t)|, s, t belonging to K}. We look at some recent results about diameter-preserving operators between spaces of continuous functions and we give some applications to the geometry of spaces of orthogonally additive polynomials on C(K) spaces.

(Joint work with C. Boyd and N. Snigireva)

**Title:** An analogue of Rado's theorem for subharmonic functions

**Speaker: ** Stephen Gardiner (UCD)

**Date: ** Tuesday, 27th November 2018

**Time:** 4pm

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

This talk will verify a conjecture of Kral (1985), that a continuously differentiable function, which is subharmonic outside its critical set, is subharmonic everywhere.

(This is joint work with Tomas Sjodin.)

**Title: **The approximation property and holomorphic mappings

**Speaker: ** Pilar Rueda (Universitat de Valencia)

**Date: ** Tuesday, 29th January 2019

**Time: ** 4pm

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

Abstract:

In 1973 Per Enflo obtained the first example of a separable Banach space that lacks the Approximation property. This example provoked a renewed interest in investigating which spaces fulfill the approximation property and how this property can be extended beyond linear operators. After recalling some classical essentials regarding the approximation property, and inspired by joint work with C. Boyd and S. Dineen, and with E. \c Cal{\i}\c{s}kan, we will make a brief tour visiting some of the results that relate the approximation property to spaces of holomorphic functions.

**Title: **Dynamics of maps from the exponential family

**Speaker: ** Neil Dobbs (UCD)

**Date: ** Tuesday, 5th February 2019

**Time: ** 4pm

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

In this introductory talk, we will investigate what happens when one iterates maps of the form $f_\lambda : z \mapsto c e^z$, where c is a non-zero complex number. In some interesting cases, the set of periodic points is dense, a generic point will have a dense orbit, and a Lebesgue-typical orbit may have different properties again.

**Title: ** Optimal polynomial approximants II

**Speaker: ** Myrto Manolaki (UCD)

**Date: ** Tuesday, 12th February 2019

**Time: ** 4pm

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

Given a Hilbert space H of analytic functions on the unit disc and a function f in H, a polynomial p_n is called an optimal polynomial approximant of degree n of 1/f if p_n minimizes ||pf - 1|| over all polynomials p of degree at most n. This notion was introduced to investigate the phenomenon of cyclicity in certain function spaces, including the classical Hardy, Bergman and Dirichlet spaces. In this talk, we will discuss the behaviour of the sequence of optimal polynomial approximants on subsets of the unit circle. Our main theorem uses a new result on simultaneous zero-free approximation, which is of independent interest. (Joint work with Catherine Bénéteau, Oleg Ivrii and Daniel Seco.)

**Title: ** Symbolic dynamics for billiards

**Speaker: ** Yuri Lima (Universidade Federal do Ceará)

**Date: ** Friday 15th February 2019

**Time: ** 12pm (please note different day and time)

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

One of the classical examples in dynamical systems are billiards: introduced more than a century ago, they are a simplification of Boltzmann's model in statistical mechanics.

However, due to the presence of singularities, most of the usual theory in dynamics does not apply to them. In this talk, I will present an approach, using symbolic dynamics,

that allows to better understand billiards. The main result, apparently expected since the 70s (when symbolic dynamics was extensively applied to hyperbolic diffeos/flows), was only recently obtained. Joint work with Carlos Matheus.

**Title:** Global and blow-up solutions for quasilinear elliptic systems

**Speaker: ** Marius Ghergu (UCD

**Date: ** Tuesday 19th February 2019

**Time: ** 3pm (please note different time)

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

We study positive radial solutions for elliptic systems driven by the p-Laplace operator. First, we classify all positive solutions in a ball according to their behaviour at the boundary. Then, we discuss the system in the whole space. In such a setting, we describe the precise behaviour at infinity for global radial solutions by using properties of three component cooperative and irreducible dynamical systems. This talk is based on a joint work with G. Singh (Trinity College Dublin) and J. Giacomoni (Univ Pau, France)

**Title: ** Spectral disjointness and partially hyperinvariant subspaces**Speaker:** Robin Harte (TCD)**Date: ** Tuesday 26th February 2019**Time: ** 4pm**Location:** Room 1.25, O’Brien Centre for Science (North)**Abstract:**

Disjointness of their spectrums confirms a sort of linear independence on pairs of linear operators

**Title: ** Multifractal analysis for self-affine iterated function systems

**Speaker:** Thomas Jordan (University of Bristol)

**Date: ** Tuesday 5th March 2019

**Time: ** 4pm

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

I will start by describing a simple example (due to Besicovitch) of multifractal analysis which consider the frequency of digits in binary expansions. I’ll then show how this can be put in the framework of self-similar sets. I’ll then describe how this problem can be generalised to self-affine sets. In this setting the talk will describe how the problem relates to the study of random matrix products and describe some progress towards solving the problem which is joint work with Ba\’lazs B\’ar\’any, Antti K\”aenm\”aki and Micha\l Rams.

**Title: ** Simultaneous Polynomial Approximation and Universality Properties of Taylor Series

**Speaker:** Juergen Mueller (University of Trier, Germany)

**Date: ** Tuesday 26th March 2019

**Time: ** 4pm

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

Let $H(\D)$ denote the space of functions holomorphic in the unit disc $\D$ endowed with the topology of locally uniform convergence. It is known that for a residual set of functions in $H(\D)$ the partial sums of the Taylor series behave extremely erratic outside $\D$. According to classical results, for functions in Banach spaces of holomorphic functions in $\D$ as for example the Hardy spaces or the Dirichlet space the situation is quite different. We show that, however, to a certain extend erratic behaviour still occurs. As a basic tool, results on simultaneous approximation by polynomials play a role.

**Title: ** A characterization of annular domains by quadrature identities I

**Speaker:** Stephen Gardiner (UCD)

**Date: ** Tuesday 2nd April 2019

**Time: ** 4pm

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

We will show that annular domains may be characterized as quadrature domains for harmonic functions with respect to a uniformly distributed measure on a sphere. This verifies an old conjecture of Armitage and Goldstein.

(Joint work with Tomas Sjödin.)

**Title: **About Spin Factors

**Speaker**: Michael Mackey (UCD)

**Date**: Tuesday 16^{th} April 2019

**Time: **4pm

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

** **

**Abstract:**

Spin factors, also known as spaces of Spinors, or as Type IV Cartan factors, form a somewhat obscure class of Banach spaces. We will briefly review their algebraic structure paying particular attention to their holomorphic automorphisms with a view to identifying the existence of fixed points.

**Title: **Magnums: Counting Sets with Surreals

**Speaker**: Peter Lynch

**Date**: Tuesday 23^{rd} April 2019

**Time: **3pm

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

** **

**Abs****tract:** How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This jars with our intuition. Our objective is to define a measure of the magnitude of subsets of the natural numbers that corresponds to our intuition. The class of surreal numbers is the largest possible ordered field. Using the surreals, we define the "magnum" for elements of the power set of the natural numbers. The magnum of a proper subset of a set is strictly less than the magnum of the set itself. There are difficulties evaluating limits over the surreals. We will review some recent progress in developing surreal analysis. For the real numbers, 0.999... = 1. For the surreals, this is not the case; 0.999... = 1 - 10^{-\omega} < 1. Many more similar examples can be given.

**Title: **A topological characterization of dual strict convexity in Asplund spaces I

**Speaker**: Richard Smith

**Date**: Tuesday 23^{rd} April 2019

**Time: **4.20pm

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

** **

**Abs****tract:** In 1975 Lindenstrauss asked whether it is possible to characterize Banach spaces $X$ that admit an equivalent strictly convex norm, in terms of other linear or topological properties of $X$. We provide a partial answer to this problem in the case of dual spaces, by showing that if $X$ is an Asplund space, then it admits an equivalent norm having a strictly convex dual norm if and only if the dual unit sphere $S_{X^*}$ (equivalently $X^*$), endowed with the $w^*$-topology, possesses a certain topological property. It follows that this ostensibly geometric property of the space can in fact be characterised in purely non-linear, topological terms.

In the first of two talks we will cover the background to the problem, the basic concepts involved, and give an introduction to the main technique of the proof, which is a type of set-theoretic derivation or 'eating' process. Such processes are usually indexed by natural numbers or ordinal numbers, but this one is indexed instead by elements of a certain tree.

**Title: **A topological characterization of dual strict convexity in Asplund spaces II

**Speaker**: Richard Smith

**Date**: Tuesday 30^{th} April 2019

**Time: **4.00pm

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

** **

**Abs****tract:** In 1975 Lindenstrauss asked whether it is possible to characterize Banach spaces $X$ that admit an equivalent strictly convex norm, in terms of other linear or topological properties of $X$. We provide a partial answer to this problem in the case of dual spaces, by showing that if $X$ is an Asplund space, then it admits an equivalent norm having a strictly convex dual norm if and only if the dual unit sphere $S_{X^*}$ (equivalently $X^*$), endowed with the $w^*$-topology, possesses a certain topological property. It follows that this ostensibly geometric property of the space can in fact be characterised in purely non-linear, topological terms. In the second of two talks we will sketch the proof of this characterization.

**Title: **Non-commutative graph parameters and zero-error capacity bounds

**Speaker**: Ivan Todorov

**Date**: Tuesday 7^{th} May 2019

**Time: **4.00pm

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

** **

**Abs****tract:** The problem of information transmission with zero error was first studied by Shannon, who defined the zero-error capacity of a classical information channel in terms of the independence number of a graph, canonically associated with the channel. This gave rise to fruitful interactions between Graph Theory and Information Theory and led to the introduction of informationally inspired graph parameters such as the Lovasz number. Analogous developments have been taking place in Quantum Information Theory in the past few years. The current fruitful approach in the quantum case involves methods from Operator Algebra Theory, and in particular, the notion of an operator system. In this talk, I will discuss these developments and some open questions faced by the area at present.

**Title: ** Operator Structures and Hybrid Classical and Quantum Information

**Speaker: ** David Kribs (University of Guelph)

**Date: ** Tuesday, 28th May 2019

**Time: ** 4.00pm

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

**Abstract:**I will discuss recent works with collaborators on topics in quantum information that involve aspects of operator algebras, operator systems, and operator theory. Specifically I'll touch on a hybrid error correction framework used for the simultaneous transmission of classical and quantum information, and work to characterise the convertibility and distinguishability of entangled states under quantum local operation and classical communication schemes.

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