# Spring Term 2015

Alexander Kuznetsov (Steklov Mathematical Institute). Geometry of FGM varieties and birational transformations between them. Friday Jan 16th, Huxley 340, 1.30-2.30pm.

Abstract: An FGM (Fano-Gushel-Mukai) variety is a Fano variety with Picard number 1 coindex 2 and degree 10 (with respect to the generator of the Picard group). These varieties were classified in works of Gushel and Mukai who showed that each of those can be obtained either as an intersection of Gr(2,5) with a quadric, or a double cover of Gr(2,5) ramified in a quadric, or a linear section of one of these. I will discuss a completely new proof of this result based on the use of the excess normal bundle. I will also describe the geometry of FGM varieties, their relation to EPW (Eisenbud-Popescu-Walter) sextics, the period map, and the birational transformations between these varieties.
This is a joint work in progress with Olivier Debarre.

Mark Haskins (Imperial College). New $G_2$ holonomy cones and exotic nearly Kahler structures on the 6-sphere and on the product of two 3-spheres.  Friday Jan 23rd, Huxley 340, 1.30-2.30pm.

Abstract: A long-standing problem in almost complex geometry has been the question of existence of (complete) inhomogeneous nearly Kahler 6-manifolds. One of the main motivations for this question comes from $G_2$ geometry: the Riemannian cone over a nearly Kahler 6-manifold is a singular space with holonomy $G_2$. Viewing Euclidean 7-space as the cone over the round 6-sphere, the induced nearly Kahler structure is the standard $G_2$-invariant almost complex structure on the 6-sphere induced by octonionic multiplication. We resolve this problem by proving the existence of exotic (inhomogeneous) nearly Kahler metrics on the 6-sphere and also on the product of two 3-spheres.
This is joint work with Lorenzo Foscolo, Stony Brook.

Simon Donaldson (Imperial College and Stony Brook). Tangent cones of limits of Kahler manifolds. Friday Jan 30th, Huxley 340, 1.30-2.30pm.

Abstract: The talk is based on joint work with Song Sun. Cheeger and Colding showed that non-collapsed limits of sequences of Riemannian manifolds with bounded Ricci curvature have metric “tangent cones”. When the manifolds are complex projective, and the metrics are Kahler, the limit is a normal algebraic variety and one can ask for an algebro-geometric description of the tangent cones. We will discuss some results in this direction.

Yoshihiko Matsumoto (Tokyo Institute of Technology). Proper harmonic maps between asymptotically hyperbolic manifolds. Friday Feb 6th, Huxley 340, 1.30-2.30pm.

Abstract: In 1990s, P. Li and L.-F. Tam studied the asymptotic Dirichlet problem on proper harmonic maps between the hyperbolic spaces, and showed an existence and uniqueness result under the \$C^1\$ boundary regularity. We generalize it to asymptotically hyperbolic manifolds. Analogously to Eells–Sampson’s theorem for closed manifolds and Hamilton’s theorem for compact manifolds-with-boundary, the unique existence in each relative homotopy class is shown under some assumption. This talk is based on a joint work with K. Akutagawa.

Paolo Stellari (Università di Milano ). Bridgeland stability conditions on abelian and Calabi-Yau threefolds. Friday Feb 13th, Huxley 340, 1.30-2.30pm.

Abstract: We produce examples of stability conditions on the bounded derived category of coherent sheaves of any abelian threefold. This extends (with a completely different proof) results of Maciocia-Piyaratne for abelian threefolds. As a consequence we provide the first examples of stability conditions for smooth projective Calabi-Yau manifold answering a long standing question. This is a joint work with A. Bayer and E. Macri’.

There will be no seminar on Friday Feb 20th.

Alessio Corti (Imperial College). A factorization theorem for volume-preserving birational maps of Mori fibred Calabi-Yau pairs. Friday Feb 27th, Huxley 340, 1.30-2.30pm.

Abstract: Despite the technical-looking title this is a result arising naturally from the theory of cluster algebras. I generalise to higher dimension a theorem first shown by Usnich for surfaces. This is joint work with A.-S. Kaloghiros

Davoud Cheraghi (Imperial College). Topological branched coverings and invariant complex structures . Friday Mar 6th, Huxley 340, 1.30-2.30pm.

Abstract: Let f be an orientation preserving branched covering of the two dimensional sphere. Is f realized (up to homotopy) by a rational map of the sphere? If yes, is the corresponding rational map unique up to the Mobius transformations (the rigidity)? These questions amount to the existence and uniqueness of a complex structure that is invariant under the action of (the homotopy class of) f.
The geometric and topological structure of the “post-critical set of f”, defined as the closure of the orbits of the branched points of f, play a key role in these problems. When the post-critical set of f has finite cardinality, a classical result of Thurston provides a topological characterization of the branched coverings that are realized by rational maps (and the uniqueness). On the other hand, when the post-critical set of f forms a more complicated set of points, say a Cantor set, the questions have been extensively studied over the last three decades. In this talk we survey the main results of these studies, and describe a recent advance made on the uniqueness part using a renormalization technique.

Felix Shulze (UCL). A local regularity theorem for mean curvature flow with triple edges. Friday Mar 13th, Huxley 340, 1.30-2.30pm.

Abstract: We consider the evolution by mean curvature flow of surface clusters, where along triple edges three surfaces are allowed to meet under an equal angle condition. We show that any such smooth flow, which is weakly close to the static flow consisting of three half-planes meeting along the common boundary, is smoothly close with estimates. Furthermore, we show how this can be used to prove a smooth short-time existence result. This is joint work with B. White.

Ed Segal (Imperial College). The Quintic 3-fold and a Fano 11-fold. Friday Mar 20th, Huxley 340, 1.30-2.30pm.

Abstract: I’ll describe some new examples where the derived category of one variety can be embedded inside the derived category of a second variety. These examples are all given by a single construction, and they include the quintic 3-fold sitting in a certain Fano 11-fold, and a new derived equivalence between two Calabi-Yau 5-folds. We also recover the `Pfaffian-Grassmannian’ derived equivalence between two non-birational Calabi-Yau 3-folds.
The proof is inspired by string theory, and uses a non-abelian gauged linear sigma model to relate both derived categories to some categories of (global) matrix factorizations. This is joint work with Richard Thomas.

Ken Baker (University of Miami). New surgeries between the Poincare Sphere and lens spaces. Friday Mar 27th, Huxley 340, 1.30-2.30pm.

Abstract: We exhibit an infinite family of hyperbolic knots in the Poincare Homology Sphere with tunnel number 2 and a lens space surgery and discuss the implications. Notably, this is in contrast to the previously known examples due to Hedden and Tange which are all doubly primitive.