Mahan Mj

Motions of limit sets

Let $\rho_n : \pi_1(S) \to PSL_2(\mathbb{C})$ be a sequence of discrete, faithful representations converging to a representation
$\rho : \pi_1(S) \to PSL_2(\mathbb{C})$.  We study the question: Does the dynamics of $\rho_n(\pi_1(S))$ on the Riemann sphere converge to that of $\rho(\pi_1(S))$? We shall also focus on the locus of chaotic dynamics, also called the limit set.

It is well known, thanks to celebrated work of Mane-Sad-Sullivan that for a parametrized family of quasifuchsian groups (or equivalently
convex cocompact discrete surface subgroups of PSl(2,C)) the limit set moves holomorphically on the Riemann sphere. We shall discuss what happens when a sequence of such groups converges to a non-quasifuchsian group. A discontinuity phenomenon was discovered in joint work with Caroline Series. In further work with Ken'ichi Ohshika, we characterize when precisely these discontinuities arise.

 

Georg Schumacher

Relative Canonical Bundles for families of Calabi-Yau manifolds, twisted Hodge Bundles, and Positivity

Given a holomorphic, polarized family of Calabi-Yau manifolds, the family of Ricci flat Kaehler metrics on the fiber determines a hermitian structure on the relative canonical bundle. This metric is being studied and the curvature is being explicitly computed. If the total space is Kaehler, we show the existence of a Kahler metric, whose restriction to the fibers is Ricci flat. We give further results about twisted Hodge bundles for families of Kaehler manifolds equipped with a positive line bundle.

 

Jacques Hurtubise

The algebraic geometry of Instantons on the Taub-NUT manifold

The talk will address the geometry behind the construction and classification of instantons on the Taub-NUT manifold, and the associated solutions to Nahm’s equations, the “bow solutions” expounded by Cherkis. These share many features of the study of calorons, but with some very interesting twists. Joint work with Sergey Cherkis.

 

Frederic Campana

Holomorphic tensors, fundamental groups and universal covers of compact Kähler manifolds

TBA

 

Baird, Tom

The moduli space of Higgs bundles over a real curve and the real Abel-Jacobi map

The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane. In this talk, I will explain how to compute the mod 2 Betti numbers of M_R using Morse theory. A key role in this calculation is played by the Abel-Jacobi map from symmetric products of X to the Jacobian of X.

 

Misha Verbitsky

Hyperbolic geometry and the proof of Morrison-Kawamata cone conjecture

TBA

 

Frank Loray

Moduli of connections on curves: some examples

We will describe some examples of moduli spaces of parabolic bundles and logarithmic connections on curves of low genus.

 

 

Ajneet Dhillon

Quotient stacks as root stacks

This is joint work with Ivan Kobyzev. After introducing the main objects of study we will give necessary conditions for a root stack to be a quotient stack. The main application of the result we have in mind is to algebraic K- theory. In this direction, our results generalise a result G. Ellingsgurd and K. Lonsted.

 

Kingshook Biswas

On Moebius and conformal maps between boundaries of CAT(-1) spaces

 Motivated by rigidity problems for negatively curved manifolds, in particular the marked length spectrum rigidity problem, we consider Moebius and conformal maps between boundaries of CAT(-1) spaces. We discuss results on extension of Moebius maps to quasi-isometries and in some cases isometries. We also discuss an analogue of the classical Schwarzian derivative for conformal maps between boundaries of CAT(-1) spaces, which measures the deviation of a conformal map from being Moebius."

 

Gautam Bharali

Iterative holomorphic dynamics on compact hyperbolic Riemann surfaces

The phrase ``iterative holomorphic dynamics'' refers to the iteration of a holomorphic correspondence (all orbits of the iteration of a regular self-map of a hyperbolic Riemann surface are periodic, hence uninteresting). We will introduce the regular leaf-space associated to the latter set-up, which is analogous to a construct of Lyubich--Minsky in the space of all backward orbits of a rational map of the 2-sphere. We shall first see some uses the regular leaf-space can be put to. For instance, the orbits of an iteration starting at z_0 are insensitive to small perturbations if the lift of z_0 to the space of orbits lies in in the regular leaf-space. This suggests that the contribution to the complexity of such an iterative dynamical system comes from a very small part, call it B_{\Gamma}, of the Riemann surface in question (where \Gamma is the given correspondence). At this stage, we shall narrow our focus to \Gamma such that d_{top}(\Gamma) > d_{top}(\Gamma_t), where d_{top} denotes the topological degree and \Gamma_t denotes the transpose of \Gamma. With this constraint, there exists an invariant measure (constructed by Dinh--Sibony) that is analogous to the Brolin measure for polynomial maps. We shall use this measure to investigate a couple of geometric features of orbits that originate in B_{\Gamma}.

 

Ritwik Mukherjee

Counting curves in a Linear System with upto eight singular points

We will consider the following question: let L--->X be a sufficiently ample line bundle over a compact complex manifold X. How many curves are there in this linear system, passing through the appropriate number of generic points that have delta nodes and one degenerate singularity of codimension k? We describe a complete solution to this problem upto codimension 8 (i.e. delta + k <= 8). We will explain how to compute the degenerate contribution to the Euler class using local intersection theory. If time permits, we will also describe some natural generalisations to the above question.

 

 Niels Borne

Nori uniformization of algebraic stacks

This is joint work with Indranil Biswas. Inspired by Noohi's work on the étale fundamental group of algebraic stacks, we study the following question : given an algebraic stack $X$ over a field $k$, does there exist a finite group scheme $G$, and a $G$-torsor $Y\to X$, where $X$ is an algebraic space ? After giving a complete characterization of uniformizable algebraic stacks, I will also report on an application of this result. In this work in progress, we study the existence of tamely ramified covers of a scheme with prescribed ramification.

 

Rukmini Dey

Some aspects of Minimal surfaces, maximal surfaces and solitons

We will first talk about the derivation of the Weierstrass-Enneper representation of a minimal surface using hodographic coordinates. We will mention an interesting link between minimal surfaces and Born-Infeld solitons. Then we will talk about some identities we obtain from certain simple Ramanujan's identities. If time permits, we will introduce maximal surfaces in Lorentzian space and talk about analogous results corresponding to these surfaces.

 

Johannes Huisman

Characteristic classes of Real vector bundles

Kahn and Krasnov constructed independently equivariant Chern classes for Real vector bundles on topological spaces endowed with an involution. I will show in this talk that the characteristic classes I constructed some time ago induce theirs.