Past Geometry and Analysis Seminar

Milnor has shown that three-dimensional Lie groups with left invariant Riemannian metrics furnish examples of 3-manifolds with principal Ricci curvatures of fixed signature --- except for the signatures (-,+,+), (0,+,-), and (0,+,+).  We examine these three cases on a Riemannian 3-manifold, and prove global obstructions in certain cases.  For example, if the manifold is closed, then the signature (-,+,+) is not globally possible if it is of the form -µ,f,f, with µ a positive constant and f a smooth function that never takes the values 0,-µ (this generalizes a result by Yamato '91).  Similar obstructions for the other cases will also be discussed.  Our methods of proof rely upon frame techniques inspired by the Newman-Penrose formalism.  Thus, we will close by turning our attention to four dimensions and Lorentzian geometry, to uncover a relation between null vector fields and exact symplectic forms, with relations to Weinstein structures. 

From a complex analytic perspective, both Teichmüller spaces and
symmetric spaces can be realised as contractible bounded domains, that
have several features in common but also exhibit many differences. In
this talk we will study isometric maps between these two important
classes of bounded domains equipped with their intrinsic Kobayashi metric.

I will discuss some properties of Hermitian metrics on compact complex manifolds, having constant Chern scalar curvature, focusing on the existence problem in fixed Hermitian conformal classes (the "Chern-Yamabe problem"). This is joint work with Daniele Angella and Simone Calamai.

Let X be a smooth projective curve (of genus greater than or equal to 2) over C and M the moduli space of vector bundles over X, of rank 2 and with fixed determinant of degree 1.Then the Fourier-Mukai functor from the bounded derived category of coherent sheaves on X to that of M, given by the normalised Poincare bundle, is fully faithful, except (possibly) for hyperelliptic curves of genus 3,4,and 5

 This result is proved by establishing precise vanishing theorems for a family of vector bundles on the moduli space M.

 Results on the deformation  and inversion of Picard bundles (already known) follow from the full faithfulness of the F-M functor

We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result

A long-standing problem in almost complex geometry has been the question of existence of (complete) inhomogeneous nearly Kahler 6-manifolds. One of the main motivations for this question comes from $G_2$ geometry: the Riemannian cone over a nearly Kahler 6-manifold is a singular space with holonomy $G_2$.

Viewing Euclidean 7-space as the cone over the round 6-sphere, the induced nearly Kahler structure is the standard $G_2$-invariant almost complex structure on the 6-sphere induced by octonionic multiplication. We resolve this problem by proving the existence of exotic (inhomogeneous) nearly Kahler metrics on the 6-sphere and also on the product of two 3-spheres. This is joint work with Lorenzo Foscolo, Stony Brook.

From a geometric viewpoint the irregular Riemann-Hilbert correspondence can be viewed as a machine that takes as input a simple
`additive' symplectic/Poisson manifold and it outputs a more complicated `multiplicative' symplectic/Poisson manifold. In the
simplest nontrivial example it converts the linear Poisson manifold Lie(G)^* into the dual Poisson Lie group G^* (which is the Poisson
manifold underlying the Drinfeld-Jimbo quantum group). This talk will firstly describe some more recent (and more complicated) examples of
such `nonperturbative symplectic/Poisson manifolds', i.e. symplectic spaces of Stokes/monodromy data or `wild character varieties'. Then
the natural generalisations (`fission algebras') of the deformed multiplicative preprojective algebras that occur will be discussed, some
of which are known to be related to Cherednik algebras.

A meromorphic connection on a complex curve can be interpreted as a representation of a simple Lie algebroid.  By integrating this Lie algebroid to a Lie groupoid, one obtains a complex surface on which the parallel transport of the connection is globally well-defined and holomorphic, despite the apparent singularities of the corresponding differential equations.  I will describe these groupoids and explain how they can be used to illuminate various aspects of the classical theory of singular ODEs, such as the resummation of divergent series solutions.  (This talk is based on joint work with Marco Gualtieri and Songhao Li.)

The lecture will introduce the notion of a folded 4-dimensional hyperkähler manifold, give examples and prove a local existence theorem from boundary data using twistor methods, following an idea of Biquard.  

The moduli space of Higgs bundles on a hyperbolic Riemann surface is a complex analytic variety which has a hyperkahler metric on its smooth locus. As such it has several associated symmetry groups including the group of complex analytic automorphisms and the group of isometries. I will discuss the classification of these and some other related groups.

Kodaira dimension provides a very successful classification scheme for complex manifolds. The notion was extended to symplectic 4-manifolds. In this talk, we will define the Kodaira dimension for 3-manifolds through Thurston’s eight geometries. This is compatible with other Kodaira dimensions in the sense of “additivity”. This idea could be extended to dimension 4. Finally, we will see how it is sitting in a potential classification of 4-manifolds by exploring its relations with various Kodaira dimensions and other invariants like Gromov norm.

Hitchin's existence theorem asserts that a stable Higgs bundle of rank two carries a unitary connection satisfying Hitchin's self-duality equation. In this talk we discuss a new proof, via gluing methods, for
elements in the ends of the Higgs bundle moduli space and identify a dense open subset of the boundary of the compactification of this moduli space.
 

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic K3 fibrations whose mirror images are also elliptic K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.

 I will report on recent work on a tropical/symplectic approach to the Horn inequalities. These describe the possible spectra of Hermitian matrices which may be obtained as the sum of two Hermitian matrices with fixed spectra. This is joint work with Anton Alekseev and Maria Podkopaeva.

I will describe a generalization of toric symplectic geometry to a new class of Poisson manifolds which are
symplectic away from a collection of hypersurfaces forming a normal crossing configuration.  Using a "tropical
moment map",  I will describe the classification of such manifolds in terms of decorated log affine polytopes,
in analogy with the Delzant classification of toric symplectic manifolds. 

 I will discuss joint work with Daniel Greb and Matei Toma in which we introduce a notion of Gieseker-stability that depends on several polarisations.  We use this to study the change in the moduli space of Giesker semistable sheaves on manifolds giving new results in dimensions at least three, and to study the notion of Gieseker-semistability for sheaves taken with respect to an irrational Kahler class.

I will present a formula that relates a Higgs bundle on an algebraic curve and Gromov-Witten invariants. I will start with the simplest example, which is a rank 2 bundle over the projective line with a meromorphic Higgs field. The corresponding quantum curve is the Airy differential equation, and the Gromov-Witten invariants are the intersection numbers on the moduli space of pointed stable curves. The formula connecting them is exactly the path that Airy took, i.e., from wave mechanics to geometric optics, or what we call the WKB method, after the birth of quantum mechanics. In general, we start with a Higgs bundle. Then we apply a generalization of the topological recursion, originally found by physicists Eynard and Orantin in matrix models, to this context. In this way we construct a quantization of the spectral curve of the Higgs bundle. 

By solving the Homogeneous Monge-Ampere equation on the deformation to the normal cone of a complex submanifold of a Kahler manifold, we get a canonical tubular neighbourhood adapted to both the holomorphic and the symplectic structure. If time permits I will describe an application, namely an optimal regularity result for certain naturally defined plurisubharmonic envelopes.

In this talk I will discuss results on the geometry of constant mean curvature (H\neq 0) disks embedded in R^3. Among other

things I will prove radius and curvature estimates for such disks. It then follows from the radius estimate that the only complete, simply connected surface embedded in R^3 with constant mean curvature is the round sphere. This is joint work with Bill Meeks.

The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will recall how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.

Noncommutative projective geometry is the study of quantum versions of projective space and other projective varieties.  Starting with the celebrated work of Artin, Tate and Van den Bergh on noncommutative projective planes, a substantial theory of noncommutative curves and surfaces has been developed, but the classification of noncommutative versions of projective three-space remains unknown.  I will explain how a portion of this classification can be obtained, via deformation quantization, from a corresponding classification of holomorphic foliations due to Cerveau and Lins Neto.  In algebraic terms, the result is an explicit description of the deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.

We produce new families of steady and expanding Ricci solitons
that are not of Kahler type. In the steady case, the asymptotics are
a mixture of the Hamilton cigar and the Bryant soliton paraboloid
asymptotics. We obtain some examples of Ricci solitons on homeomorphic
but non-diffeomorphic spaces. We also find numerical evidence of solitons
with more complicated topology.

We consider Fano threefolds on which SL(2,C) acts with a dense

open orbit. This is a finite list of threefolds whose classification

follows from the classical work of Mukai-Umemura and Nakano. Inside

these threefolds, there sits a Lagrangian space form given as an orbit

of SU(2). We prove this Lagrangian is non-displaceable by Hamiltonian

isotopies via computing its Floer cohomology over a field of non-zero

characteristic. The computation depends on certain counts of holomorphic

disks with boundary on the Lagrangian, which we explicitly identify.

This is joint work in progress with Jonny Evans.

I will define and describe in some details a large class of gauge theories in four dimensions. These theories admit a variational principle with the action a functional of only the gauge field. In particular, no metric appears in the Lagrangian or is used in the construction of the theory. The Euler-Lagrange equations are second order PDE's on the gauge field. When the gauge group is taken to be SO(3), a particular theory from this class can be seen to be (classically) equivalent to Einstein's General Relativity. All other points in the SO(3) theory space can be seen to describe "deformations" of General Relativity. These keep many of GR's properties intact, and may be important for quantum gravity. For larger gauge groups containing SO(3) as a subgroup, these theories can be seen to describe gravity plus Yang-Mills gauge fields, even though the associated geometry is much less understood in this case.

I will explain a new formulation of Einstein’s equations in 4-dimensions using the language of gauge theory. This was also discovered independently, and with advances, by Kirill Krasnov. I will discuss the advantages and disadvantages of this new point of view over the traditional "Einstein-Hilbert" description of Einstein manifolds. In particular, it leads to natural "sphere conjectures" and also suggests ways to find new Einstein 4-manifolds. I will describe some first steps in these directions. Time permitting, I will explain how this set-up can also be seen via 6-dimensional symplectic topology and the additional benefits that brings.