## Research Seminars in the School of Mathematical Sciences

The School of Mathematical Sciences host regular research seminars delivered by internal and external researchers on topics across mathematical sciences. Seminars take place on the Kevin Street campus and are open to all.

## Seminar 21/10/16: Noise and dissipation on coadjoint orbits

Noise and dissipation on coadjoint orbits

Alexis Arnaudon

Imperial College London

Friday 21 October 2016

1pm, Blue Room, DIT Kevin Street

Abstract:

In this talk I will start by introducing a stochastic perturbation of mechanical systems with symmetries that is most compatible with the preservation of geometrical invariants, or conserved quantities. In order for interesting phenomenons to emerge in these systems, I will add a dissipative term that also preserves the geometrical structure of the equations. Having both noise and dissipation in these systems will allow me to make a thermodynamical interpretation of the expected solutions and introduce a notion of temperature. Another feature is the existence of the so-called random attractors, a building block in modern random dynamical systems theory, that I will present in some details, with helps of some numerical simulations. My main examples of such systems will be the free rigid body associated to the group of rotations, and the heavy top when this symmetry is broken and requires the theory of semi-direct products. I will discuss some numerical challenges associated to these stochastic processes, and, if time permits, another application of this methodology to landmark dynamics in image matching.

This is joint work with D. Holm and A. De Castro and is an arXiv paper with number 1601.02249

## Seminar 11/03/16: Dynamics of internal wave-current interaction

Dynamics of internal wave-current interaction

Alan Compelli

Dublin Institute of Technology

Friday 11 March 2016

2.30pm, Blue Room, DIT Kevin Street

Abstract:

A bounded system consisting of two fluids is considered. An internal wave, driven by gravity, acts as a free common interface between the fluids. Various current profiles are examined. A Hamiltonian formulation is taken and the resultant equations of motion show that wave-current interaction is not influenced by the current profile outside of the strip adjacent to the internal wave.

## Seminar 11/12/15: Finite k-nets in projective planes

Finite k-nets in projective planes

Nicola Pace

Dublin Institute of Technology

Friday 11 December 2015

1pm, Blue Room, DIT Kevin Street

Abstract:

This talk deals with k-nets embedded in the projective plane PG(2,K) defined over a field K. They are line configurations in PG(2,K) consisting of k≥3 pairwise disjoint line-sets, called *components*, such that any two lines from distinct families are concurrent with exactly one line from each component.

The case k=3 is particularly interesting because, in some instances, it is possible to realise finite groups. A 3-net is said to *realise* a group (G,·) when the following condition holds. If A,B,C are the components, then there exists a triple of bijective maps from G to (A,B,C), say α∶ G→A, β: G →B, γ: G → C, such that a·b =c if and only if α(a),β(b),γ(c) are three collinear points, for any a,b,c∈G. If K has zero characteristic, 3-nets realising a finite group are classified. If the characteristic of the field is p>|G|, then the same classification holds true apart from three possible exceptions: A_{4},S_{4} and A_{5}.

Key ideas and results needed for the classification are presented.

## Seminar 12/02/16: Ways of Seeing Julia Sets: Visualizing the forces that shape fractal Julia sets

Ways of Seeing Julia Sets: Visualizing the forces that shape fractal Julia sets

Ted Burke

School of Electrical and Electronic Engineering, DIT

Friday 12 February 2016

1pm, Blue Room, DIT Kevin Street

Abstract:

Early in the 20th century, work by mathematicians such as Pierre Fatou and Gaston Julia on complex dynamics led to the definition of so-called Julia sets. When a rational complex polynomial function is applied iteratively to a complex number, z, it produces a sequence of complex values called the orbit of z. Depending on the particular function and on the value of z, that orbit may or may not remain bounded. For a given function, the Julia set forms the boundary between those regions of the complex plane where the orbits remain bounded and those where they do not.

Intriguingly, even for quite simple iterative complex functions, Julia sets often take on very striking fractal shapes. With the aid of a computer, it is easy to visualize the Julia set of a function but, for most people, understanding why it takes on a fractal shape is difficult. In my own ongoing struggle to gain a more intuitive understanding of fractals, I have written many short computer programs to visualize them in different ways. In this presentation, I will explore some ways of visualizing Julia sets which I found helpful in understanding why they are fractal.

## Seminar 20/11/15: Mathematical Models of Visual Processing: Explaining Illusory Motion

Mathematical Models of Visual Processing: Explaining Illusory Motion

Garrett Greene

Dublin Institute of Technology

Friday 20 November 2015

1pm, Blue Room, DIT Kevin Street

Abstract:

I will give a brief overview of mathematical models of visual processing in the brain, both in the retina and in the visual cortex. I will then discuss some recent results demonstrating how certain motion illusions can arise from models of non-linear processing in the retina.

The Ouchi Illusion and the Out-of-focus Illusion are stationary images which can produce a perception of motion or "jittering" under certain conditions. These effects are related to small unconscious eye movements, which are ubiquitous in humans and play an important role in human vision. Since we do not normally perceive motion in the world during these eye movements, there must be a mechanism to compensate for the motion of the visual scene across the retina. I will present a computational model of visual processing which compensates for eye-motion under normal viewing conditions. In this model, the effect of the Ouchi and Out-of-focus illusions are predicted as failure modes of the compensation mechanism.