Autumn Term 2013


Hans-Joachim Hein (Imperial and Nantes). Asymptotically cylindrical Calabi-Yau manifolds. Friday October 11th. Huxley Room 130, 1.30-2.30pm.

I will discuss complete Ricci-flat Kahler manifolds that are asymptotic to cylinders at infinity. The main result roughly states that – after some natural modifications – every such space can be written as a compact projective algebraic variety minus an anticanonical divisor with trivial normal bundle. The converse statement holds as well. Joint work with Mark Haskins and Johannes Nordstrom.

Ionut Ciocan-Fontanine (Minnesota). Stable quasimaps, wall-crossings, and Mirror Symmetry. Friday October 18th. Huxley Room 130, 1.30-2.30pm.

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas and will explain why they can be viewed as generalizations  (in several directions) of Givental’s toric mirror theorems. The talk is based on joint work with Bumsig Kim, and partly also with Davesh Maulik.

Friday October 25th. No seminar this week due to the Groups and Geometry in the Southeast meeting at UCL that day.

Ali Craw (Bath). Quiver embeddings for Mori Dream Spaces. Friday November 1st. Huxley Room 130, 1.30-2.30pm.

A globally generated vector bundle on a projective variety determines a morphism to a Grassmannian. More generally, a collection of globally generated vector bundles determines a morphism to an iterative Grassmann-bundle called a framed quiver variety.  I’ll describe the case where the projective variety is a Mori Dream Space and the bundles have rank one, in which case the defining ideal of the image can be described explicitly.

Lukas Lewark (Durham). The Khovanov-Rozansky concordance invariants. Friday November 8th. Huxley Room 130, 1.30-2.30pm.

The Khovanov-Rozansky homologies induce a family of knot concordance invariants (among them the Rasmussen invariant) which give strong lower bounds to the slice genus. We will see why some of those concordance invariants are distinct from the rest, using amongst others various spectral sequences that relate the different Khovanov-Rozansky homologies.

November 15th : No seminar due to LMS Annual General Meeting with Simon Donaldson and Graeme Segal speaking.

Dario Martelli (King’s College London). Supersymmetric Gauge Theories on Curved Manifolds and their Gravity Duals. Friday November 22nd. Huxley Room 130, 1.30-2.30pm.

I will discuss recent results concerning supersymmetric field theories on curved manifolds preserving (rigid) supersymmetry, both in three and for four dimensions. Moreover, I will present gravity dual solutions to such gauge theories, providing new non-trivial tests of the gauge/gravity duality.

Vlad Markovic (Cambridge and Caltech). Homology of curves and surfaces in closed hyperbolic 3-manifolds. Friday November 29th. Huxley Room 130, 1.30-2.30pm.

We prove the following two topological statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by a connected closed quasi-Fuchsian subsurface. Second, every rationally nullhomologous closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds a quasi-Fuchsian subsurface. This talk reports on joint work with Yi Liu.

Matt Rathbun (Imperial College) Band surgeries and crossing changes between fibered links. Friday December 6th. Huxley Room 130, 1.30-2.30pm.

Fibered links are links whose complements fiber over the circle, with each fiber a copy of a Seifert surface for the link. With motivations coming from a certain biological system of protein-DNA interactions, we investigate an operation called band surgery on links in the case that the original link and the resulting link are both fibered. We give a complete characterization of such band surgeries, in terms of the action of the monodromy map of the fibration on arcs in fiber surfaces. We further give a complete characterization of when a generalized crossing change on a fibered link results in a new fibered link in terms of the monodromy action. This talk reports on joint work with Dorothy Buck, Kai Ishihara, and Koya Shimokawa.

Alexei Kovalev (Cambridge). Asymptotically cylindrical manifolds with holonomy Spin(7). Friday December 13th. Huxley Room 130, 1.30-2.30pm.

We construct examples of asymptotically cylindrical Riemannian 8-manifolds with holonomy group Spin(7). To our knowledge, these are the first such examples. The construction uses an extension to asymptotically cylindrical setting of Joyce’s existence result for torsion-free Spin(7)-structures. One source of examples arises from `Fano-type’ Kaehler 4-orbifolds with smooth anticanonical Calabi-Yau 3-fold divisors and with compatible antiholomorphic involution. We give examples using weighted projective spaces and calculate basic topological invariants of the resulting Spin(7)-manifolds.

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