Geometry and Analysis Seminar

I will discuss an ascending filtration on the Chow group of zero-cycles on a smooth projective variety obtained roughly by considering the successive kernels of the iterates of some modified diagonal embedding of the variety. This filtration is particularly relevant in the case of abelian varieties and of hyper-Kähler varieties, where it is expected to be opposite to the conjectural Bloch-Beilinson filtration. In the case of abelian varieties, it can in fact be described explicitly in terms of the Beauville decomposition, while in the case of hyper-Kähler varieties, I conjecture (and prove in some cases) that it coincides with a filtration introduced earlier by Claire Voisin. As a by-product we obtain in joint work with Olivier Martin a criterion involving second Chern classes for two effective zero-cycles on a moduli space of stable objects on a K3 surface to be rationally equivalent, generalising a result of Marian-Zhao.

A 2-connected, rational homotopy 7-sphere is classified up to diffeomorphism by three invariants: its (finite) 4th cohomology group, its q-invariant and its Eells-Kuiper invariant.  The q-invariant is a quadratic refinement of the linking form and determines the homeomorphism type, while the Eells-Kuiper invariant then pins down the diffeomorphism type.  In this talk, I will discuss the diffeomorphism classification of a certain family of non-negatively curved, 2-connected, rational homotopy 7-spheres, discovered by Sebastian Goette, Krishnan Shankar and myself, which contains, in particular, all $S^3$-bundles over $S^4$ and all exotic 7-spheres.

Symplectic implosion is an abelianisation construction in symplectic geometry. The implosion of the cotangent bundle of a group K plays a universal role in the implosion of manifolds with a K-action.  This universal implosion, which is usually a singular variety, can also be viewed as the non-reductive Geometric Invariant Theory quotient of the complexification G of K by its maximal unipotent subgroup. 

In this talk, we describe joint work with Johan Martens and Nick Proudfoot which uses point-counting techniques to calculate the intersection cohomology of the universal implosion.

I will discuss the Brill-Noether theory of a general elliptic $K3$ surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general $k$-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

The well-known Milnor-Wood inequality gives a bound on the Toledo invariant of a representation of the fundamental group of a compact surface in a non-compact Lie group of Hermitian type. While a lot is known regarding the counting of maximal Toledo components, and their role in higher Teichmueller theory, the non-maximal case remains elusive. In this talk, I will present a strategy to count the number of such non-maximal Toledo connected components. This is joint work in progress with Brian Collier and Jochen Heinloth, building on previous work with Olivier Biquard, Brian Collier and Domingo Toledo.

A Riemannian metric is said to be  Einstein if it has constant Ricci curvature. In dimensions 2 or 3, this is actually equivalent to requiring the metric to have constant sectional curvature. However,  in dimensions 4 and higher, the Einstein condition becomes significantly weaker than constant sectional curvature, and this has rather dramatic consequences. In particular, it turns out that there are  high-dimensional smooth closed manifolds that admit pairs of Einstein metrics with Ricci curvatures of opposite signs. After explaining how one constructs such examples, I will then discuss some recent results exploring the coexistence of Einstein metrics with zero and positive Ricci curvatures.

Contact topology and hyperbolic geometry are two well-established, yet so far largely unrelated subfields of 3-manifold topology. We will discuss a recent result relating phenomena in these two fields. Specifically, we will demonstrate that tightness of certain contact structures on hyperbolic manifolds is detected by the behaviour of geodesics in the underlying hyperbolic geometry. A key geometric tool we will discuss is the deformation theory for hyperbolic manifolds. 

Complex dynamics is the study of the behaviour, under iteration, of complex polynomials and rational functions. This talk is about an application of combinatorial algebraic geometry to complex dynamics. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point. Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. Two long-standing open questions are: (1) Is G_n is irreducible over Q? (2) Is Per_n connected? I will sketch an argument showing that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a special degeneration of degree-2 rational maps that tells us that Per_n has smooth point with Q-coordinates "at infinity”.