Geometry and Analysis Seminar

The Beilinson-Drinfeld Grassmannian, which classifies a G-bundle trivialised away from a finite set of points on a curve, is one of the basic objects in the geometric Langlands programme. Similar construction in higher dimensions in the algebraic and analytic settings are not very interesting because of Hartogs' theorem. In this talk I will discuss a differentiable version. I will also explain a theory of D-modules on differentiable spaces and use it
to define differentiable chiral and factorisation algebras. By linearising the Grassmannian we get examples of differentiable chiral algebras. This is joint work with Dennis Borisov.

We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety.

 We study codimension one holomorphic distributions on projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is an integrable phenomenon. This part of the  talk  is work joint with  M. Jardim(Unicamp) and O. Calvo-Andrade(Cimat).

We also study foliations by curves via the investigation  of their  singular schemes and  conormal  sheaves and we provide a classification  of foliations of degree at most 3 with  conormal  sheaves locally free.  Foliations of degrees  1 and 2 are aways given by a global intersection of two codimension one distributions. In the classification of degree 3 appear Legendrian foliations, foliations whose  conormal sheaves are instantons and other ” exceptional”
type examples. This part of the  talk   is  work joint with  M. Jardim(Unicamp) and S. Marchesi(Unicamp).

Introduced by Konno, hyperpolygon spaces are examples of Nakajima quiver varieties.  The simplest of these is a noncompact complex surface admitting the structure of a gravitational instanton, and therefore fits nicely into the Kronheimer-Nakajima classification of complete ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2).  For more general hyperpolygon spaces, we can speculate on how
this classification might be extended by studying the geometry of hyperpolygons at "infinity". This is ongoing work with Hartmut Weiss.

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$
holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The
asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical
(ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.  

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In
particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.

In this talk, we consider the question of exact (boundary) controllability of wave equations: whether one can steer their solutions from any initial state to any final state using appropriate boundary data. In particular, we discuss new and fully general results for linear wave equations on time-dependent domains with moving boundaries. We also discuss the novel geometric Carleman estimates that are the main tools for proving these controllability results

A C^infinity scheme is a version of a scheme that uses a maximal spectrum. The category of C^infinity schemes contains the category of Manifolds as a full subcategory, as well as being closed under fibre products. In other words, this category is equipped to handle intersection singularities of smooth spaces.

While originally defined in the set up of Synthetic Differential Geometry, C^infinity schemes have more recently been used to describe derived manifolds, for example, the d-manifolds of Joyce. There are applications of this in Symplectic Geometry, such as the describing the moduli space of J-holomorphic forms.

In this talk, I will describe the category of C^infinity schemes, and how this idea can be extended to manifolds with corners. If time, I will mention the applications of this in derived geometry.

I will explain our recent description of the fundamental degrees of freedom underlying a generalized Kahler structure. For a usual Kahler
structure, it is well-known that the geometry is determined by a complex structure, a Kahler class, and the choice of a positive(1,1)-form in this class, which depends locally on only a single real-valued function: the Kahler potential. Such a description for generalized Kahler geometry has been sought since it was discovered in1984. We show that a generalized Kahler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a
holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on
only a single real-valued function, which we call the generalized Kahler potential. To solve the problem we make use of, and generalize,
two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds;
the second is Donaldson's interpretation of a Kahler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.

A C^k-extension of a smooth and connected Lorentzian manifold (M,g) is an isometric embedding of M into a proper subset of a connected Lorentzian manifold (N,h) of the same dimension, where the Lorentzian metric h is C^k regular. If no such extension exists, then we say that (M,g) is C^k-inextendible. The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity.

The Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a C^2 regular metric. In this talk I will describe how one
proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric.

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold,  whose dimension depends on a parameter $N$ unbounded from above. By taking suitable projections we construct sequences of space-time Brownian motion and stochastic parallel transport whose limit as $N\to \infty$ are the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on Perelman’s manifold and for the horizontal Laplacian on the corresponding orthonormal frame bundle.

As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman's manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and Haslhofer.

This is joint work with Robert Haslhofer.