*Various speakers. ***Geometry Day III. ** Friday May 4th. King’s College London.

Geometry Day III will be taking place on 4 May 2012 in K0.16, King’s Building, Strand Campus of King’s College London.

Speakers are: Simon Donaldson, Francisco Lopez, Baris Coskunuzer, Carlo Sinestrari and YanYan Li.

If you would like to attend, please contact Emily Balls (emily.balls@kcl.ac.uk) with your name, email address and institution, ideally by 20 April 2012. Please also indicate whether you are interested in attending the dinner for which there will be a nominal fee. Any academic questions should be directed to Dr Giuseppe Tinaglia (giuseppe.tinaglia@kcl.ac.uk).

More information at www.kcl.ac.uk/nms/depts/mathematics/geometry3.aspx

*Marc Lackenby (Oxford). ***Links with splitting number one. ** Friday May 11th. Huxley 139, 1.30-2.30pm.

The unknotting number of a knot is an incredibly difficult invariant to compute. In fact, there are many knots which are conjectured to have unknotting number 2 but for which no proof of this is currently available. It therefore remains an unsolved problem to find an algorithm that determines whether a knot has unknotting number one. In my talk, I will show that an analogous problem for links is soluble. We say that a link has splitting number one if some crossing change turns it into a split link. I will give an algorithm that determines whether a link has splitting number one. (In the case where the link has two components, we must make a hypothesis on their linking number.) The proof that the algorithm works uses sutured manifolds and normal surfaces.

*Gabor Szekelyhidi (University of Notre Dame)*. **Filtrations and test-configurations.** Friday May 18th. Huxley 139, 1.30-2.30pm.

Test-configurations are certain degenerations of projective manifolds, which are used in the definition of K-stability. I will explain how filtrations of the homogeneous coordinate ring of a projective manifold can be thought of as sequences of test-configurations, and that they encode the limiting behavior of these sequences. These filtrations arise naturally when studying the Calabi flow, or when trying to minimize the Calabi functional. I will also discuss how filtrations can be used to give a strengthening of the notion of K-stability, and why this is desirable.

*Jan Christophersen (University of Oslo)*. **Simplicial complexes and projective varieties.** Friday May 25th. Huxley 139, 1.30-2.30pm.

The Stanley-Reisner construction associates to a simplicial complex a projective scheme which “looks like” the complex. For combinatorial manifolds, if the scheme is smoothable, it will smooth to special algebraic varieties. For example a sphere will in this way correspond to a Calabi-Yau manifold and a torus to an abelian variety. It turns out that interesting triangulations of manifolds yield interesting algebraic geometry via deformations of Stanley-Reisner schemes and I will describe this connection with several examples.

*No Seminar. * Friday June 1st.

*Roger Bielawski (Leeds). ***Pluricomplex geometry and quaternionic manifolds.** Friday June 8th. Huxley 139, 1.30-2.30pm.

I will describe a new type of geometric structure on complex manifolds. It can be viewed as a deformation of hypercomplex structure, but it also leads to a special type of hypercomplex and hyperkähler geometry. These structures have both algebro-geometric and differential-geometric descriptions, and there are interesting examples arising from physics. Moreover, a class of pluricomplex manifolds leads to quaternion-Kähler metrics, generalising the SO(3)-invariant self-dual Einstein examples of Hitchin.

*Marion Moore Campisi (University of Texas, Austin). ***Bridge distance and exceptional surgeries.** Friday June 15th. Huxley 340, 1.30-2.30pm.

In this talk I will discuss ongoing work which shows that links that have high distance bridge surfaces do not admit exceptional surgeries. The distance d(T) of a bridge surface T is defined in terms of the arc and curve complex for the bridge surface and has become a standard way of measuring the “complexity” of the link. This is joint work with Ryan Blair, Jesse Johnson, Scott Taylor and Maggy Tomova.

*Andrea Brini (Imperial). ***A crepant resolution conjecture for open strings.** Friday June 22nd. Huxley 139, 1.30-2.30pm.

We propose a Crepant Resolution Conjecture in the context of open Gromov-Witten theory. The content of this talk is based on arxiv:1102.0281 and on joint work in progress with R. Cavalieri (CSU), T. Coates (Imperial) and D. Ross (CSU).

*Anne-Sophie Kaloghiros (Imperial). ***Geography of models and the Sarkisov Program.** Friday June 29th. Huxley 139, 1.30-2.30pm.

Given a normal projective variety X and an effective divisor D on X, an important question is to determine whether X has a “good model” for D. When does there exist a birational map X–>X_D, where X_D has manageable singularities and where f_*D is “positive” in a suitable sense? This question can be answered for two remarkable classes of varieties and divisors on them: the first is that of varieties where the Minimal Model Program can be run, and the other that of Mori Dream Spaces. I will define a class that generalizes these two examples, and show how to understand birational maps between good models for distinct divisors. I will give an application to the study of birational maps between Mori fibre spaces (the Sarkisov Program): I will describe generators (work of Hacon-McKernan) and relations in the Sarkisov Program. Part of this talk is based on joint work with Kuronya and Lazic.

*Vincent Borrelli (Lyon)* and *Etienne Ghys (CNRS, Lyon).* **LMS Hardy Lecture.** Friday June 29th. University College London, 3.30-6.30pm.

Vincent Borrelli will speak on *Flat tori in three-dimensional space* at 3.30, and Etienne Ghys will speak on *Cutting cloth according to Chebyshev* at 5.15. Both talks will be directed at a general mathematical audience. A reception will follow at 6.30 at a cost of £35 per person. See http://www.lms.ac.uk/sites/default/files/files/posters/Hardy%2012%20poster.pdf for more details and for contact information.