## 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 23/02/12: What is a real number?

What is a real number?

Sean Dineen

University College Dublin

Thursday 23 February 2012

1pm, Room KA3-011, DIT Kevin Street

Abstract:

We give one answer to this question

## Seminar 06/12/11: An introduction to the ring spur assignment problem

An introduction to the ring spur assignment problem

Paula Carroll

Quinn & Smurfit Schools of Business, UCD

Tuesday 6 December 2011

1pm, Room KA3-011, DIT Kevin Street

Abstract:

The Ring Spur Assignment Problem (RSAP) is motivated by a practical situation: a network operator seeks to identify an economical fault tolerant Next Generation Network (NGN). The aim of the presentation is to introduce the RSAP, set it in context relative to related problems in the literature and outline a cutting plane approach to solving the RSAP. Cutting plane approaches have been successful in solving large combinatorial optimisation problems. Using a cutting plane approach, the problem is modelled as a minimal Integer Programming problem, the relaxed LP solved and additional cuts (violated constraints) are separated and added to speed up the branch and bound search.

## Seminar 25/10/11: On the integrability of the perturbed KdV hierarchy with generic self-consistent sources

On the integrability of the perturbed KdV hierarchy with generic self-consistent sources

Georgi Grahovski

Dublin Institute of Technology

Tuesday 25 October 2011

1pm, Room KA3-011, DIT Kevin Street

Abstract:

First we will review briefly the methods for studying soliton perturbations. Then we will describe a class non-holonomic deformations of the KdV hierarchy by using the expansions over the so-called "squared solutions" (squared eigen-functions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability.

## Seminar 15/11/11: Spaces of matrices without non-zero eigenvalues in their field of definition

Spaces of matrices without non-zero eigenvalues in their field of definition

Rachel Quinlan

NUI Galway

Tuesday 15 November 2011

1pm, Room KA3-011, DIT Kevin Street

Abstract:

It was shown by M. Gerstenhaber in 1958 that for (almost) any field F, the maximum possible dimension of a linear subspace of M_{n}(F) in which every element is nilpotent is n(n-1)/2. In this talk we will use a duality relation involving the trace bilinear form to show that the same upper bound applies to the dimension in the more general situation of a subspace of M_{n}(F) in which no element has a non-zero eigenvalue that belongs to F. For example, the space of skew-symmetric n by n matrices over the field of real numbers possesses this latter property but does not consist of nilpotent matrices. The talk will include a discussion of the duality relation in a wider context involving affine spaces of square and rectangular matrices that have special rank properties and special covering properties. Only basic concepts in linear algebra and the theory of bilinear forms are involved.

## Seminar 18/10/11: Polynomials of degree n

Polynomials of degree n

Anthony Brown

University College Dublin

Tuesday 18 October 2011

1pm, Room KA3-011, DIT Kevin Street