# Analysis Seminars

### Analysis Seminars for this Semester

**Tuesday 25th September, 2018 **
(place: UCD
Science North 125 )
3.30pm

*Speaker:* H. Render

*Title:* The Green function for the exterior cylinder

**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.

**Tuesday 9th October, 2018 **
(place: UCD
Science North 125 )
4.00pm

*Speaker:* R. Levene

*Title:* State conversion using von Neumann algebras

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

**Tuesday 16th October, 2018 **
(place: UCD
Science North 125 )
4.00pm

*Speaker:* M. Golitsyna

*Title:* Overconvergence Properties of Dirichlet series

**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.

**Tuesday 30th October, 2018 **
(place: UCD
Science North 125 )
3.00pm

*Speaker:* M. Ghergu

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

**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)

**Tuesday 6th November, 2018 **
(place: UCD
Science North 125 )
4.00pm

*Speaker:* C. Boyd

*Title:* Localisation and Positivity of Orthogonally Additive Polynomials

**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).

**Tuesday 13th November, 2018 **
(place: UCD
Science North 125 )
4.00pm

*Speaker:* M. Manolaki

*Title:* Optimal polynomial approximants

**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.)

**Tuesday 20th November, 2018 **
(place: UCD
Science North 125 )
3.00pm

*Speaker:* R. Ryan (NUIG)

*Title:* The Diameter Norm

**Abstract**
The *diameter* of a continuous function $f$ on a compact Hausdorff
space $K$ is
$\sup\lbrace |f(s)-f(t)|, s, t \mbox{ belonging to } K\rbrace$.
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)

**Tuesday 27th November, 2018 **
(place: UCD
Science North 125 )
4.00pm

*Speaker:* S. Gardiner

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

**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.)

**Tuesday 4th December, 2018 **
(place: UCD
Science North 125 )
4.00pm

*Speaker:* R. Smith

*Title:* User-friendly polyhedrality

**Abstract**
The concepts of upper and lower $p$-estimates, where $1 < p < \infty$,
play an important role when studying the geometry of Banach spaces. We
introduce an analogue of upper $p$-estimate, in the case $p = \infty$,
and obtain some results that make the task of finding equivalent
polyhedral norms on certain Banach spaces, having either a Schauder
basis or an unconditional basis, easier and more transparent.

C. Boyd .

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