# 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:** 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 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: ** 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.)

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**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: ** 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: ** 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.

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