Special Lectures

Monday, April 6 2009

16:30 - 17:30    Kirwan Lecture Theatre

History of Mathematics Lecture

Rod Gow (UCD):
From George Salmon and 27 lines to octonions, E_6 and beyond

The best known geometrical achievement of the Irish mathematician George Salmon is his enumeration in 1849 of the 27 lines on a non-singular cubic surface. While the study of this unusual configuration was pursued by many mathematicians, including C. Jordan, Cayley, Sch\"afli and Steiner, Salmon himself made no further contributions to the subject. In 1901, L. E. Dickson showed how the 27 lines could be used to construct a cubic form in 27 dimensional space, and he investigated the group preserving this form. The cubic form had previously arisen in the 1894 thesis of E. Cartan, in connection with a realization of the exceptional 78-dimensional Lie algebra E_6. In the 1950's, work of Freudenthal showed that the cubic form just described was a so-called determinant of an exceptional simple Jordan algebra of dimension 27.
  The automorphism group of the algebra is a group of type F_4 and the group preserving the cubic form is of type E_6. Interestingly enough, the exceptional Jordan algebra can be described using 3\!\times\! 3 Hermitian matrices with entries in the division algebra of octonions. Here we make another point of contact with Irish mathematics, since the octonions had been discovered in 1843 by the amateur mathematician John Graves, a friend and contemporary of William Rowan Hamilton, about two months after Hamilton's discovery of quaternions.
  We intend in this talk to enlarge upon the circle of ideas and interconnections briefly alluded to above.

18:00 - 19:00    Kirwan Lecture Theatre

Public Lecture

Thomas Körner (University of Cambridge):
Mathematics and Smallpox

Smallpox was and is a rightly dreaded disease. In the 18th century a new technique of innoculation was introduced from Turkey. Since, in effect, the innoculator deliberately gave an uninfected person what was usually (but, unfortunately, not always) a milder form of the disease the idea was controversial.
  Bernoulli tried to use early statistical ideas to see whether the case for innoculation was valid. Although the lecturer will not seek present day analogies, the audience may well feel that some of the ideas discussed have relevance today.

Tuesday, April 7 2009

17:30 - 18:30    Kirwan Lecture Theatre

Afternoon Lecture

Tom Laffey (UCD):
The Nonnegative Inverse Eigenvalue Problem

The nonnegative inverse eigenvalue problem (NIEP) asks for necessary and sufficient conditions on a list \sigma:=(\lambda_{1},\ldots,\lambda_{n}) of complex numbers, in order that \sigma be the list of eigenvalues of an (entry-wise) nonnegative real matrix. If \sigma is the spectrum of a nonnegative matrix A, we say that \sigma is {\it realizable} and refer to A as a {\it realizing matrix}.
  The Perron-Frobenius theorem yields the necessary condition that \rho:=\max\{\mid\lambda_{j}\mid~\mid~j=1,\ldots,n\} be in \sigma, if \sigma is realizable. We call \rho the Perron root of \sigma. We assume that \lambda_{1}=\rho. While the NIEP was formulated by Suleimanova sixty years ago, and despite the work of many authors, the general problem remains unsolved. We will present an account of the current state of knowledge on it.
  Boyle and Handelman (Ann.Math 133 (1991) 249-316) solved the corresponding problem where one is allowed to augment the list \sigma by adding an arbitrary number of zeroes. Much recent research has been aimed at obtaining a constructive proof of their theorem, with an effective bound on the minimum number of zeroes which must be appended for realizability. The solution of this problem in the case where all the (non-Perron) \lambda_{j} (j>1) have negative real parts, and also when \sigma consists of two positive real numbers and n-2 negative ones, obtained by H. \v{S}migoc and the author (LAA 416 (2006) 148-159, ELA 17 (2008) 333-342) will be discussed. In both cases, realizing matrices of a certain block companion type are found. Work of Kim, Ormes and Roush (JAMS 13 (2004) 773-806) on realizable spectra for which the corresponding characteristic polynomials have integer coefficients, can also be shown to lead to matrices of this type, and there is some optimism that all realizable spectra can be realized in this way. Some recent results on realizable spectra for which a symmetric realizing matrix may be found, will also be presented.