Spring Term 2014

Jorgen Rennemo (Imperial). Homology of Hilbert schemes of points on a locally planar curve. Friday Jan 17th, Huxley 130. 1.30-2.30pm.

We consider Hilbert schemes of points on a singular curve. These spaces have in recent years become important through a connection to invariants which count curves in a Calabi-Yau 3-fold, and through the related discovery that the topology of the Hilbert schemes gives interesting information about the singularities of the curve.

Recent work of Maulik-Yun and Migliorini-Shende gives a formula for the homology of these spaces in the case where the curve has planar singularities. I will describe a new proof of this formula, which works by constructing an action of a certain Heisenberg algebra on the homologies of the Hilbert schemes. I will also tell you how this sheds some light on a question about CY3 curve counting invariants.

Yanki Lekili (King’s College London). Equivariant Lagrangian branes and Representations. Friday Jan 24th, Huxley 130. 1.30-2.30pm.

Classical Bott-Borel-Weil theory constructs irreducible representations of semisimple Lie algebras on the section spaces of homogeneous vector bundles on homogeneous spaces. In this talk, we decribe a construction in symplectic geometry which is meant to serve as the mirror dual to Bott-Borel-Weil construction. Building on Seidel-Solomon’s fundamental work, we define the notion of an “equivariant Lagrangian brane” in an exact symplectic manifold M (which is meant to be the mirror dual of a homogenous variety G/P), and construct representations of the Lie algebra g= Lie G, on Floer cohomology of equivariant Lagrangian branes. We will make our construction completely explicit in the case of sl2 and comment on generalizations to arbitrary semisimple Lie algebras. This is a joint work with James Pascaleff.

Friday Jan 31st, Huxley 130. No seminar due to the London School of Geometry and Number Theory (EPSRC CDT in Geometry and Number Theory at the interface) open afternoon at 3pm at UCL.

Tom Coates (Imperial). Mirror symmetry without localisation.  Friday Feb 7th, Huxley 130. 1.30-2.30pm.
Mirror Symmetry predicts a surprising relationship between the virtual numbers of degree-d rational curves in a target space X and variations of Hodge structure on a different space X’, called the mirror to X.  Concretely, it predicts that one can compute genus-zero Gromov–Witten invariants (which are the virtual numbers of rational curves) in terms of hypergeometric functions (which are the solutions to a certain differential equation coming from the variation of Hodge structure).  Existing proofs of this rely on beautiful but fearsomely complicated localization calculations in equivariant cohomology.  I will describe a new proof of the Mirror Theorem, for a broad range of target spaces X, which is much simpler and more conceptual. This is joint work with Cristina Manolache.

Yohsuke Imagi (Kyoto). Uniqueness Theorems for Smoothing Special Lagrangians. Friday Feb 14th, Huxley 130. 1.30-2.30pm.

Special Lagranigian submanifolds are area-minimizing Lagrangian submanifolds of Calabi–Yau manifolds. One can define the moduli space of compact special Lagrangian submanifolds of a (fixed) Calabi–Yau manifold. Mclean proved it has the structure of a manifold (of dimension finite). It isn’t compact in general, but one can compactify it by using geometric measure theory. Kontsevich conjectured a mirror symmetry, and special Lagrangians should be “mirror” to holomorphic vector bundles. By using algebraic geometry one can compactify the moduli space of holomorphic vector bundles. By “counting” holomorphic vector bundles in Calabi-Yau 3-folds Richard Thomas defined holomorphic Casson invariants (Donaldson-Thomas invariants). It is an open question (probably very difficult) whether one can “count” special Lagrangians, or define a nice structure on the (compactified) moduli space of special Lagrangians. To do it one has to study singularities of special Lagrangians.

One can smooth singularities in suitable situations: given a singular special Lagrangian, one can construct smooth special Lagrangians tending to it (by the gluing technique). I’ve proved a uniqueness theorem in a “symmetric” situation: given a symmetric singularity, there’s only one way to smooth it (the point of the proof is that the symmetry reduces the problem to an ordinary differential equation).

More recently I’ve studied a non-symmetric situation together with Dominic Joyce and Joana Oliveira dos Santos Amorim. Our method is based on Lagrangian Floer theory, and is effective at least for pairs of two (special) Lagrangian planes intersecting transversely. I’ll give the details in the talk.

Helene Esnault (Berlin). Étale fundamental groups and local systems. Friday Feb 21st, Huxley 130, 1.30-2.30pm.

Over the field of complex numbers, one understands the relation between the étale fundamental group, which is the profinite completion of the topological one, with the local systems, or, equivalently, with regular singular algebraic flat connections. In short, the étale fundamental group controls the local systems. In characteristic $p>0$, we don’t understand yet the corresponding statement fully, except when $X$ is projective. We will discuss natural problems and conjectures.

Andre Neves (Imperial). Applications of Almgren-Pitts Morse Theory in Geometry. Friday Feb 28th, Huxley 130, 1.30-2.30pm.

I will talk about how Almgren-Pitts Morse Theory can be used to solve open problems in geometry. This is joint work with Fernando Marques.

No seminar on Friday Mar 7th.

Ulrike Tillmann (Oxford). Homology stability and (symmetric) diffeomorphism groups. Friday March 14th, Huxley 130, 1.30-2.30pm.

Homology stability for families of discrete groups such as the symmetric groups, linear groups, braid groups and mapping class groups are well-known. Extensions to diffeomorphism groups of manifolds more generally have only been proved recently in a few special cases. We will revisit some classical results on configuration spaces, extend them to the equivariant setting, and prove homology stability for so-called symmetric diffeomorphism groups for arbitrary manifolds.

Mu-Tao Wang (Columbia). Surface geometry and general relativity. Friday March 21st, Huxley 130, 1.30-2.30pm.

Many classical results regarding surfaces in 3-dimensional Euclidean space such as Weyl’s isometric embedding problem and the Minkowski inequality have their counterparts for surfaces in spacetime. These generalization are not only of mathematical interest, but also of physically relevant importance. They are closely related to fundamental concepts such as gravitational energy and Cosmic censorship. In my talk, I shall discuss some recent developments in these directions.

No seminar on Friday March 28th.