University of Durham --- Department of Mathematical Sciences

Yorkshire and Durham Geometry Days

A Yorkshire and Durham Geometry Day will be held on Friday March 21, 2014 in the Department of Mathematical Sciences at Durham University. The timetable is as follows:

10:30 Coffee Mathematics Department Common Room CM211

11:15 Brendan Guilfoyle (IT Tralee) in CM221

"A global version of a classical result of Joachimsthal and the slice problem for knots"
Abstract : The slice problem - determining whether a knot in the boundary of a 4-manifold bounds a properly embedded disc in its interior - is of significance both to knot theory and to 4-manifold topology. In this talk we propose a geometric obstruction to sliceness which arises from an ambient neutral Kaehler structure (namely the 4-dimensional space of oriented lines of euclidean 3- space). In euclidean space, it leads to a global version of a Theorem of Joachimsthal on pairs of surfaces intersecting with constant angle. In the 4-manifold, it nails down the total complex index (generalizing the total umbilic index in the classical picture) of properly immersed discs in terms of geometric boundary data. This represents the first step in exerting geometric control of the topological invariants that arise in the slice problem. The result will be discussed in the context of (neutral) Casson handles and its implications for 4-manifold topology. The talk is based on joint work with W. Klingenberg.

12:15 Lunch break

1.50 Leave from CM211

2.00 Alix Deruelle (Warwick University) in CG218 (note the change of venue in an adjacent building)

"Stability of non compact expanding gradient Ricci solitons "
Abstract: The Ricci flow, introduced by Hamilton, can be interpreted as a dynamical system on the set of Riemannian metrics of a fixed manifold modulo the action of diffeomorphisms and homotheties. The fixed points of such dynamical system are called Ricci solitons. There are three types according to their lifetime, in particular, expanding Ricci solitons are immortal solutions that can be used to smooth out metric cones. We investigate the stability of such non negatively curved Ricci expanders.

3:00 Tea break in CM211

3:30 Werner Ballmann (Max Planck Institute and Hausdorff Center for Mathematics, Bonn) in CG218

"Small eigenvalues of Laplacians on surfaces"

Abstract : Eigenvalues of the Laplacian on closed hyperbolic surfaces are called small if they lie below $1/4$, the bottom of the spectrum of the Laplacian on the hyperbolic plane. Buser showed that, for any $\varepsilon>0$, the surface $S$ of genus $g\ge2$ carries a hyperbolic metric such that $\lambda_{2g-3}<\varepsilon$, where the eigenvalues are counted according to their magnitude. He also showed that $\lambda_{2g-2}\ge c>0$, where $c$ is independent of genus and hyperbolic metric. Buser and Schmutz conjectured that $c=1/4$ is the best constant. I will discuss this conjecture, its recent solution by J.-P. Otal and E. Rosas, and some further developments.

4:40 Sasha Veselov (Loughborough University) in CG218

"Configuration spaces, Gaudin subalgebras and separation of variables"

Abstract: Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno--Drinfeld Lie algebra $t_n$, associated with configuration space of $n$ distinct points on a plane. It turns out that Gaudin subalgebras form a smooth algebraic variety isomorphic to the Deligne-Mumford-Knudsen moduli space $\bar M_{0,n+1}$ of stable genus zero curves with $n+1$ marked points. I will explain how the real version of this result allows to describe the moduli space of all orthogonal separation coordinates on the unit sphere. The talk is based on joint works with L. Aguirre and G. Felder and with K. Schoebel.

6:00 Leave for dinner at the The Capital from CM211

Travel: Durham is easy to get to by car and train, and so is the Department of Mathematics, located on the Science Site. Click here for relevant information.

If you intend to drive to Durham, please inform one of the Durham organisers, who will arrange for a car parking permit for you.

Yorkshire and Durham Geometry Days are jointly organised by the Universities of Durham, Leeds and York, and occur at a frequency of three meetings per academic year. Financial support is provided by the London Mathematical Society through a Scheme 3 grant, currently administered by the University of York. A record of previous meetings may be found here.

The local organizers are:

John Bolton & Wilhelm Klingenberg, University of Durham

Martin Speight & John Wood, University of Leeds

Ian McIntosh & Chris Wood, University of York