# Analysis Seminars

### Analysis Seminars for this Semester

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

Speaker: R. Smith
Title: Approximation of norms in Banach spaces

Abstract This talk follows on from one I gave in May 2017. Let $X$ be a Banach space and let $\mathbf P$ be a property of norms. We say that a norm $\|\cdot\|$ on $X$ (equivalent to the original norm) can be approximated by norms having $\mathbf P$ if, given $\varepsilon>0$, there exists another norm $|||\cdot|||$ on $X$ with $\mathbf P$, such that $\|x\| \leq |||x||| \leq (1+\varepsilon)\|x\|$ for all $x \in X$. There are a number of papers in the literature that consider the question of whether or not all (equivalent) norms on a given space can be approximated in this way.
For a number of classes of Banach spaces $X$, including $c_0(\Gamma)$ (where $\Gamma$ is an arbitrary set), certain Orlicz spaces and Lorentz predual spaces, and a class of $C(K)$ spaces (where $K$ comes from a class of compact spaces having unbounded scattered height), we show that all equivalent norms on $X$ can be approximated by $C^\infty$-smooth norms or polyhedral norms.
This is joint work with Stanimir Troyanski, University of Murcia, Spain, and Institute of Mathematics, Bulgarian Academy of Sciences

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

Speaker: A. M\"uller-Hermes (Copenhagen)
Title: Positive maps in quantum information theory

Abstract Many problems in quantum information theory are connected to properties of positive linear maps between matrix algebras. After a brief introduction to some basic concepts of quantum information theory I want to focus on the problem of entanglement distillation. I will explain how this problem is connected to the existence problem of positive linear maps that stay positive under taking tensor powers and that are neither completely positive nor completely co-positive. If time permits I will outline some constructions of interesting positive maps and how non-decomposability arises naturally from tensorisation.

Tuesday 20th February, 2018 (place: UCD Science North 125 ) 4.00pm

Speaker: R. Harte
Title: 1973 and all that

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

Title: On the least doubling constant of a metric space

Abstract We will explore the question of determining the least doubling constant among all doubling measures defined on a metric space. We will show that in many natural instances this constant is at least 2. This is based on work in progress with J. Soria (Barcelona).

Tuesday 6th March, 2018 (place: UCD Science North 125 ) 3.00pm

Speaker: J. Giacomoni (Pau, France)
Title: Diaz Saa Inequality for $p(x)$-laplacian and applications

Abstract In this talk, I will present a recent work with P. Takac. It concerns a new extension of the well-known inequality by Diaz-Saa which in our case, involves an anisotropic operator, such as $p(x)$-Laplacian. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions, comparison principles and stabilization for some anisotropic quasilinear elliptic and parabolic equations.

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

Speaker: S. Gardiner
Title: Universal Fourier and Taylor series

Abstract It has long been known that there exist trigonometric series, the partial sums of which possess universal approximation properties on the unit circle. It turns out that, for most smooth functions on the unit circle, the partial sums of the associated Fourier series, when extended to the plane, have universal approximation properties off the circle. There are also related results for pairs of Taylor and Laurent series arising from functions that are holomorphic off a Jordan curve. (This is joint work with Vassili Nestoridis and Christos Papadimitropoulos.)

Tuesday 20th March, 2018 (place: UCD Science North 125 ) 4.00pm

Speaker: P. Mellon
Title: Holomorphic dynamics on bounded symmetric domains

Abstract The open unit ball, $B$, of a Banach space is homogeneous if given any two points $z,w$ in $B$, there is a biholomorphic map sending $z$ to $w$. Such balls classify the bounded symmetric domains, include many classical spaces and ensure a Jordan structure on the underlying space. Let $f:B\mapsto B$ be a holomorphic fixed-point free map. The behaviour of the sequence of iterates, $f^n=f\circ f^{n-1}$, of $f$ is the subject of much study since the Wolff Denjoy results for the complex disc $\Delta$ in 1926. Generally, in infinite dimensions, $(f^n)$ does not converge, even in the Hilbert space case. Our work therefore seeks to establish the 'location' of accumulations points of $(f^n)$, with respect to the topology of local uniform convergence on $B$.
This seminar will present results in this direction, using a recently proved Wolff type theorem for infinite dimensional bounded symmetric domains.

Tuesday 27th March (place: UCD Science North 125 ) 3:00pm

Speaker: M. Ghergu
Title: Isolated singularities for semilinear elliptic systems with power-law nonlinearity

Abstract We discuss the behaviour around isolated singularity for nonnegative solutions of semilinear elliptic systems. Unlike the standard methods available in the literature which rely on moving plane methods, we apply several tools that pertain to free boundary problems in order to pursue our investigation.
This talk is based on a joint work with Sunghan Kim (National University Seoul, South Korea) and Henrik Shahgholian (KTH Stockholm).

Tuesday 27th March (place: UCD Science North 125 ) 4:15pm

Speaker: C. Boyd
Title: Integral and Nuclear Polynomials on tree spaces

Abstract We examine the Radon-Nikodym Property and Asplundness, in particular their connection to integral and nuclear mappings and polynomials. We show that the structure given to us by tree spaces provides the ideal setting to uncover the intricacies of the relationship between integral and nuclear polynomials. (This is joint work with C. Poulios and M. Venkova.)

Tuesday 10th April, 2018 (place: UCD Science North 125 ) 4.00pm

Speaker: H. Render
Title: Generalized Bernstein operators

Abstract In this talk we discuss the existence and properties of generalized Bernstein operators in the context of extended Chebyshev spaces. Special analysis is given to Bernstein operators in the polynomial setting which fix the constant function 1 and the function $x^3$ with respect to an interval $[a,b]$ containing 0.
This talk is based on a joint work with J.M. Aldaz (Universidad Autonoma de Madrid)

Tuesday 24th April, 2018 (place: UCD Science North 125 ) 4.00pm

Speaker: M. Golitsyna
Title: Overconvergent properties of Dirichlet series

Abstract In this talk I will describe overconvergent properties of Dirichlet series with regard to the behavior of their partial sums near infinity. The results are inspired by corresponding phenomena for Taylor series, which were recently discovered using the notion of thinness.