Winter 2023

Title: J-holomorphic curves in the pseudo 6-sphere and surface group representations

Abstract: The space of norm 1 vectors in $R^{3,4}$ is a pseudo-Riemannian analogue of the 6-sphere. Like the compact 6-sphere it has a nonintegrable almost complex structure which arises from octonionic multiplication, and the group of  isometries which preserve this almost complex structure is known as the split real form of G_2. In this talk we will describe a special class of J-holomorphic curves in the pseudo 6-sphere that are equivariant with respect to the action the fundamental group of a closed surface. Such curves have previously been studied for Hitchin representations, and we show they exist and parameterize their moduli in general. This in particular identifies particularly interesting families of surface group representations into the split real form of G_2 which  are not connected components of the character variety. This is based on joint work with Jeremy Toulisse.


Title: The space of metric structures on hyperbolic groups

Abstract: Teichmuller space is a classical construction that, for a given closed hyperbolic surface, parameterizes the geometric actions of its fundamental group on the hyperbolic plane. I will talk about a generalization of this space, where for an arbitrary hyperbolic group we consider a moduli space of its geometric actions on Gromov hyperbolic spaces. Even in the surface group case, this space turns out to be much larger than Teichmuller space, and its points can be defined from negatively curved Riemannian metrics, Anosov representations, random walks, geometric actions on CAT(0) cube complexes, etc. Equipped with a natural Lipschitz metric, this space is contractible and geodesic, and when the group is torsion-free, it has a “thick” part that is cocompact for the isometric action of the outer automorphism group. This is joint work with Stephen Cantrell.


Title:  Finiteness properties of algebraic fibers of group extensions.

Abstract: I will discuss some results about the existence and properties of algebra fibrations of group extensions, with particular regards for those thatarise as fundamental groups of Kaehler manifolds, such as (iterated) Kodaira fibrations.


Title: Superexponential Dehn functions inside CAT(0) groups

Abstract: We construct 4–dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are exp(n)(xm) for integers n, m ≥ 1 and 6–dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are exp(n)(xα) for integers n ≥ 1 and α dense in [1, ∞). This significantly expands the known geometric behavior of subgroups of CAT(0) groups. This is a joint project with Noel Brady.



Title: Two notions of duality for geodesic currents

Abstract: Geodesic currents are a suitable closure of the space of curves on a hyperbolic surface introduced by Bonahon in 1986. Notions such as the geometric intersection number of curves extend to geodesic currents. I will discuss two equivalent viewpoints on geodesic currents: as dual curve functionals and as dual spaces. On the one hand, a geodesic current induces a functional on the space of curves on the surface via intersection number with the current. We say that such a curve functional is ``dual to the current''. In joint work with Dylan Thurston, we give sufficient and necessary conditions for curve functionals to be dual to geodesic currents. On the other hand, a geodesic current together with a choice ofmhyperbolic metric induces a Gromov hyperbolic space, that we call a ``dual space of the current''. In joint work with Luca De Rosa, we describe the metric structure of such spaces.


Title :  Homological Mirror Symmetry for Theta Divisors

Abstract: Symplectic geometry is a relatively new branch of geometry. However, a string theory-inspired duality known as “mirror symmetry” reveals more about symplectic geometry from its mirror counterparts in complex geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS results were then proved for symplectic mirrors to Calabi-Yau and Fano manifolds. Those mirror to general type manifolds have been studied in more recent years, including my research. In this talk, we will introduce HMS through the example of the 2-torus T^2. We will then outline how it relates to HMS for a hypersurface of a 4-torus T^4, in joint work with Haniya Azam, Heather Lee, and Chiu-Chu Melissa Liu. From there, we generalize to hypersurfaces of higher dimensional tori, otherwise known as “theta divisors.” This is also joint with Azam, Lee, and Liu.


Title: On the hyperbolic Bloch transform

Abstract: Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform of Fuchsian groups, that we call the hyperbolic Bloch transform. I will prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" and I will introduce a modified, geometric, Bloch transform, that transforms wave functions to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan.