## Lancaster University 50th Anniversary Lectures

Lancaster University 50th Anniversary Lectures

# 50th Anniversary Special Lectures

In 2014, Lancaster University celebrated the 50th anniversary of its foundation. To honour this occasion, the department organised two special Anniversary Lectures on Friday 26th September.

The first lecture was given by Professor E. Brian Davies, FRS, of King’s College, London. Brian was the President of the London Mathematical Society from 2007 to 2009; he is the author of many papers and seven books.

His talk, based on a joint paper with Michael Levitin, was devoted to an easily stated question: what are the eigenvalues of a specific, elementary, non-self-adjoint n by n matrix (with n large)? The answer, however, is not easily found.

Elementary estimates give some rough bounds, and impressive numerical calculations, shown on attractive slides, lead to conjectures on other bounds. The latter have been verified in some cases, using surprisingly advanced mathematics. Brian illustrated beautifully the elusive nature of this approach by showing a short video indicating how the eigenvalue distribution varies with n.

We are far from a secure theory that explains the numerical phenomena, even in this specific example. But the audience was surely convinced that the conjectures of Davies and Levitin must be true, even if nobody could prove them.

The second lecture was given by Professor W. Hugh Woodin. Hugh now holds a joint position in the Departments of Mathematics and of Philosophy at Harvard University; he was previously Professor at the University of California, Berkeley, from 1989 to 2014. He is surely one of the leaders in our era in the quest to understand the fundamental nature of sets and the real numbers, taking forward the journey of Gödel and Tarski into the far reaches of higher cardinals.

His lecture (almost) also marked the 50th anniversary of Gödel’s Incompleteness Theorem. By 1964, we knew that the Continuum Hypothesis (CH) is independent of the axioms ZFC of set theory; now we know that very many statements, including many in mainstream mathematics, are independent. How satisfactory is this? Could (necessarily detailed) examination of higher cardinals lead to a further compelling axiom that ‘resolves’ CH and other questions? Is ‘ultra-L’ such an axiom? These were very challenging ideas, very clearly articulated.