Spring Term 2016

Saul Schleimer (University of Warwick). Veering Dehn Surgery. Friday 22nd Jan, 1.30-2.30pm. Huxley 139.

Abstract: (Joint with Henry Segerman.) It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced “ideal triangulations” for studying manifolds with torus boundary; Lackenby introduced “taut ideal triangulations” for studying the Thurston norm ball; Agol introduced “veering triangulations” for studying punctured surface bundles over the circle. Veering triangulations are very rigid; one current conjecture is that any fixed three-manifold admits only finitely many veering triangulations.
After giving an overview of these ideas, we will introduce “veering Dehn surgery”. We use this to give the first infinite families of veering triangulations with various interesting properties.

David Holmes (University of Leiden). A Néron model of the universal jacobian. Friday 29th Jan, 1.30-2.30pm. Huxley 139.

Abstract: Every non-singular algebraic curve C has a jacobian J, which is a smooth projective group variety. Given a family of non-singular curves one can construct a family of jacobians. We are interested in what happens to the family of jacobians when the family of non-singular curves degenerates to a singular curve. In the case where the base-space of the family has dimension 1, this is completely understood due to work of André Néron in the 1960s. However, when the base space has higher dimension things become more difficult. We describe a seemingly-new combinatorial invariant which controls these degenerations. In the case of the jacobian of the universal stable curve, we will use this to construct a `minimal’ base-change after which a Néron model exists.

Nick Shepherd-Barron (King’s College). From exceptional groups to del Pezzo surfaces via principal bundles over elliptic curves. Friday 5th Feb, 1.30-2.30pm. Huxley 139.

Abstract: In this talk we extend the construction by Brieskorn, Grothendieck and Springer (BGS) of simultaneous resolutions of a du Val singularity inside the corresponding simple algebraic group. to give a group-theoretical construction of simultaneous log resolutions of simply elliptic singularities. In particular, we give a path from exceptional groups to del Pezzo surfaces that is geometrical rather than merely combinatorial; this path leads through stacks of principal bundles, and their reductions, over elliptic curves. We will also comment on how things are different over supersingular curves.
This is joint work with Ian Grojnowski.

Chenyang Xu (Peking University). Dual complex of a singular pair. Friday 12th Feb, 1.30-2.30pm. Huxley 139.

Abstract: Given a pair (X,D) which is a variety X with a divisor D, the dual complex is the combinatorial datum which characterizes how the components of D intersect with each other. We will discuss the recent progress on using the minimal model program to study it. Especially, we will explain the finiteness of the fundamental group of the dual complex associated to a log Calabi-Yau pair (X,D).

Dan Ketover (Imperial College). Minimal surfaces in 3-manifold topology. Friday 19th Feb, 1.30-2.30pm. Huxley 139.

Abstract: I will explain some recent work using minimal surfaces to address problems in 3-manifold topology. Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface. If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting. I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

Yevgeniy Liokumovich (Imperial College). Problems in quantitative topology and geometric calculus of variations. Friday 26th Feb, 1.30-2.30pm. Huxley 139.

Abstract: A theorem of Lyusternik and Shnirelman states that on every Riemannian 2-sphere there are 3 simple closed geodesics. We will show that their lengths are bounded in terms of the diameter of the sphere (joint with A. Nabutovky and R. Rotman). We will discuss other results about lengths and volumes of stationary objects (such as minimal surfaces, closed geodesics or geodesic nets) in Riemannian manifolds. The idea is to find “nice” ways of constructing non-trivial homology/homotopy classes in the space of cycles, so that their mass can be bounded from above in a nearly optimal way. We will consider: the space of 1-cycles on Riemannian surfaces; the space of (n-1)-cycles on manifolds with Ricci curvature bounded from below (joint with P. Glynn-Adey); the space of 1-cycles on manifolds with Ric>0 (joint with X. Zhou).

Marco Marengon (Imperial College). Concordance maps in knot Floer homology. Friday 4th Mar, 1.30-2.30pm. Huxley 139.

Abstract: Knot Floer homology (HFK) is a bi-graded vector space, which is an invariant of a knot in S^3. Given a (decorated) knot concordance between two knots K and L (that is, an embedded annulus in S^3 x [0,1] that K and L co-bound), Juhász defined a map between their knot Floer homologies. We prove that this map preserves the bigrading of HFK and is always non-zero. This has some interesting applications, in particular the existence of a non-zero element in HFK(K) associated to each properly embedded disc in B^4 whose boundary is the knot K in S^3.
This is joint work with András Juhász.

Yvette Kosmann-Schwarzbach (Ecole polytechnique, Palaiseau). On the generalized geometry of Lie groupoids. Friday 11th Mar, 1.30-2.30pm. Huxley 139.

Abstract: The generalized geometry of smooth manifolds was introduced by Nigel Hitchin in 2003. I aim to report on results concerning the case where the base manifold is a Lie groupoid. I shall first review the concept of Nijenhuis torsion for an endomorphism of the generalized tangent bundle of a manifold when the bundle is equipped with either the Courant or the Dorfman bracket. The “multiplicative generalized complex structures” on Lie groupoids are defined by imposing a compatibility condition between the generalized complex structure and the groupoid multiplication. The integration theorem of Jotz, Stiénon and Xu (2016) relates their infinitesimal counterparts on Lie algebroids to the structures on the Lie groupoid, assumed to have simple-connected source fibers. This theorem yields as corollaries the known theorem on the symplectic realization of Poisson manifolds and a theorem on the integration of holomorphic Lie bialgebroids into holomorphic Poisson groupoids.