## 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 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 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 06/10/11: Quantum Hall Effect

Quantum Hall Effect

Brian Dolan

NUI Maynooth

Thursday 6 October 2011

1pm, Room KA3-011, DIT Kevin Street

## 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

## Seminar 29/09/11: Teaching the weak student

Teaching the weak student

David McCarthy

Dublin Institute of Technology

Thursday 29 September 2011

1pm, Room KEB-008, DIT Kevin Street