# Mathematics Seminars 2018 - 2019

## Mathematics Seminars 2018 - 2019

# Seminar Titles and Abstracts 2018-2019

**UCC Mathematics Seminar**

**Time and location: **4-5pm, Tuesday 9 April 2019, WGB G04

**Speaker: ** Padraig Ó Catháin (Worcester Polytechnic Institute)

**Title: **Morphisms of Complex Hadamard matrices

**Abstract:**

Let $M$ be a matrix with complex entries of unit norm. A well-known theorem of Hadamard bounds the magnitude of the determinant of $M$ as a function of its dimension, and $M$ is a complex Hadamard matrix if $M$ meets Hadamard's bound with equality. In this talk I will survey some known results on existence of special types of complex Hadamard matrices, in particular matrices with entries in the $k^{\textrm{th}}$ roots of unity. I will report on recent joint work with Ronan Egan and Eric Swartz on the existence of tensor-product-like maps which reduce the number of entries in a complex Hadamard matrix at the cost of increasing the dimension. This work generalises previous constructions of Turyn and Compton-Craigen-de Launey of real Hadamard matrices from certain complex Hadamard matrices with entries in the fourth and sixth roots of unity respectively.

**UCC Mathematics Seminar**

**Time and location: **4-5pm, Tuesday 2 April 2019, WGB G04

**Speaker: ** Vladimir Dotsenko (Trinity College Dublin)

**Title: **Homotopy type of the moduli space of stable rational curves

**Abstract:**

The moduli space of stable rational curves, also known as the Deligne-Mumford compactification \bar{M}_{0,n} of the moduli space of rational curves with marked points, has been studied in many different areas of mathematics for decades, but some questions about it have remained open until now. An instance of such questions is the rational homotopy type of this space. I shall show that the rational cohomology of this space is a Koszul algebra (answering a question of Yu. I. Manin, D. Petersen and V. Reiner), and explain how this allows one to compute the rational homotopy invariants of this space in a very explicit way.

**UCC Mathematics Seminar**

**Time and location: **4-5pm, Tuesday 19 March 2019, WGB G04

**Speaker: ** Oliver Mason (Maynooth University)

**Title: **The Joint Spectral Radius and Extremal Norms in Nonnegative and Max Algebra

**Abstract:**

The joint spectral radius (JSR) helps to characterise the asymptotic behaviour of non-homogeneous matrix products, and plays an important role in the stability analysis of difference and differential inclusions. In this talk, I will outline some classical results concerning the JSR before going on to discuss recent work on the existence of extremal and Barabanov norms for semigroups of nonnegative matrices and matrices over the max algebra. Time-permitting, I will also describe some applications to persistence theory for switched models in epidemiology.

**UCC Mathematics Seminar**

**Time and location: **4-5pm, Tuesday 5 March 2019, WGB G04

**Speaker: ** Eduardo Mota Sánchez (University College Cork)

**Title: **Constant Mean Curvature Surfaces and Heun's Differential Equations

**Abstract:**

The generalised Weierstrass representation for surfaces with constant mean curvature allows to describe any conformal constant mean curvature immersion in R3, H3 or S3 with four ingredients: a Riemann surface, a base point, a meromorphic loop Lie algebra valued 1-form and finally the initial condition for a linear system of ODE's.

Associating to the linear system a second order differential equation from the class of Heun's Differential Equations, we prescribe certain singularities in the linear system that appear in the resulting surface. Regular singularities produce asymptotically Delaunay ends and irregular singularities produce irregular ends. We discuss global issues such as period problems and asymptotic behavior involved in the construction of this kind of surfaces. Finally, using the generalised Weierstrass representation, we construct new parametric families of constant mean curvature surfaces in R3 with genus zero that possess at least one irregular end.

**UCC Mathematics Seminar**

**Time and location: **4-5pm, Tuesday 19 February 2019, WGB G04

**Speaker: ** Alexandru Nica (University of Waterloo)

**Title: **A free probabilistic approach to meandric systems

**Abstract:**

I will consider a family of diagrammatic objects (well-known to mathematical physicists and to combinatorialists) which go under the name of ``meandric systems''. These objects offer some very appealing, yet difficult problems -- in particular, denoting by $E_n$ the expected number of components of a random meandric system of order $n$, there are no precise results concerning the asymptotic behaviour of $E_n$ for large values of $n$. Numerical experiments suggest the conjecture that the limit of $E_n / n$ should exist, with a value around 0.23. In this talk I will present a result obtained in joint work with Ian Goulden and Doron Puder, giving some evidence in favour of the above conjecture. Quite interestingly, our result is intimately related to the combinatorial side of an area of research called free probability, in particular to a very basic notion of ``free additive convolution'' which is used in free probability (and which will be reviewed as part of the talk)

**UCC Mathematics Seminar**

**Time and location: **4-5pm, Tuesday 12 February 2019, WGB G17

**Speaker: ** Jonathan Hickman (University of St. Andrew’s)

**Title: **On convergence of Fourier integrals

**Abstract:**

In the first half of the 20th century great advances were made in understanding convergence of Fourier series and integrals in one dimension. Many natural convergence problems in higher dimensions are still poorly understood, however, despite great attention by many prominent mathematicians over the last five decades. In this talk I will introduce the basic questions, describe their rich underlying geometry, and explain some recent developments in joint works with L. Guth (MIT) and M. Iliopoulou (UC Berkeley) and K. Rogers (ICMAT) which have applied tools from incidence and algebraic geometry to these problems.

**Time and location: **4-5pm, Tuesday 29 January 2019, WGB G17

**Speaker: ** Clifford Gilmore (University College Cork)

**Title: **The Dynamics of Linear Operators

**Abstract:**

Linear dynamics has been a rapidly evolving area since the early 1990s. It lies at the intersection of operator theory and topological dynamics, and its central property is hypercyclicity. In this talk I will give a general introduction to hypercyclicity and the stronger property of frequent hypercyclicity, whence I will demonstrate that many natural continuous linear maps turn out to possess these properties.

I will finish by outlining some recent joint work with Eero Saksman and Hans-Olav Tylli (University of Helsinki) on the growth rates of harmonic functions that are frequently hypercyclic with respect to the partial differentiation operator.

**Time and location: **4-5pm, Tuesday 22 January 2019, WGB 402

**Speaker: ** Arundhathi Krishnan (University College Cork)

**Title:** On the continuity of the pseudospectrum

**Abstract:** The pseudospectrum of an element of a unital Banach algebra is a particular subset of the complex plane which contains properly the spectrum, and is determined by the norm of the resolvent function. Under certain conditions, the pseudospectrum has the important property of being stable under perturbations.

This property is not shared by the spectrum. We discuss some of these continuity properties.

**Speaker:** Donnacha Oisin Kidney (UCC)

**Time and location: **Tuesday 27 November 4-5pm WGB G02

**Title:** TBC

**Speaker: Michiel van den Berg (University of Bristol)**

**Time and location: **4-5pm, Tuesday 6 November 2018, WGB G02

**Title:** Sign changing solutions of Poisson's equation

Abstract: Michiel van den Berg

**Claus Köstler (University College Cork, 11 September)**

Title: Markovian fun with F

Abstract: Markovianity is a stochastic phenomenon which does not care about the past - the presence 'dictates' the future. Unexpectedly this phenomenon is closely linked to representations of the Thompson group F. I will playfully introduce you to this new connection between randomness and symmetry. Toy examples are given by moving marbles on a two-dimensional grid. I will explain why this entails the general result that every stationary Markov chain induces a representation of the Thompson group F. Furthermore I will briefly address that, conversely, a large class of representations of F yields stationary Markov chains. Finally I will introduce 'partial spreadability' as a new distributional symmetry, aiming at a de Finetti type characterization of Markovianity. The presented results are based on ongoing research with Rajarama Bhat, Gwion Evans, Rolf Gohm, Arundhathi Krishnan, Vijaya Kumar, and Stephen Wills. My talk should be accessible to a general mathematical audience.

**Fran Burstall (University of Bath, 4 September)**

Title: Conformal submanifold geometry for beginners

Abstract: I shall describe those aspects of the geometry of surfaces in R^3 which are invariant under angle preserving transformations. I shall eschew the technical machinery of the subject so as to (try to) make the talk accessible to all.