Shearing deformations in Hitchin representations are computed for a symplectic form.
problem Computing symplectic form pairings for Hitchin representations.
method Shearing deformations of Hitchin representations.
result Pairings of shearing deformations computed for the Atiyah-Bott-Goldman symplectic form.
Infinite volume found in the thick part of PSLn(R)-Hitchin-Riemann moduli space.
problem Proving infinite volume in the thick part of PSLn(R)-Hitchin-Riemann moduli space. method Employing Goldman flows and internal sequences to find an infinite series of subsets of identical volume.
result Infinite total Atiyah--Bott--Goldman volume for n>2. Study shows infinite volumes of moduli spaces for certain groups.
problem Infinite volumes of Hitchin-Riemann moduli spaces for specific groups.
method Employed Goldman flows to find infinite disjoint subsets of identical volume.
result Proved infinite Atiyah-Bott-Goldman covolume for mapping class group actions.
The paper defines and calculates Reidemeister torsion for a specific class of representations.
problem Defining and calculating Reidemeister torsion for G-Anosov representations.
method Symplectic chain complex method to establish a novel formula for R-torsion.
result Reidemeister torsion is well-defined and calculated for G-Anosov representations.
We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functio…
For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an h…
Explicit computation of symplectic form for PGLn(R)-Hitchin component.
problem Symplectic structure of PGLn(R)-Hitchin component. method Atiyah-Bott-Goldman symplectic form and global coordinates.
result Coefficients of the symplectic form are constant.
Symplectic coordinates found on a Hitchin component for a hyperbolic surface.
problem Parametrizing the PSL3(R)-Hitchin component with canonical coordinates. method Proved global Darboux coordinates with half canonical Goldman coordinates.
result Global Darboux coordinates exist for the PSL3(R)-Hitchin component. Random representations of surface groups approach asymptotic freeness in large n limit.
problem Asymptotic freeness of Haar unitary matrices for surface groups.
method Interplay between Dehn's work and classical invariant theory.
result Expected value of trace of a fixed non-identity element is bounded as no∞. Symplectic coordinates found on projective structures on orbifolds.
problem Symplectic structure on deformation spaces of convex projective structures.
method Global Darboux coordinates system construction and symplectic space decomposition.
result Symplectic form on deformation space of convex projective structures.
The {\em rank n swapping algebra} is a Poisson algebra defined on the set of ordered pairs of points of the circle using linking numbers, whose geometric model is given by a certain subspace of (Kn×Kn∗)r/GL(n,K). For any ideal triangulation of Dk---a disk wit…
The paper derives formulas for symplectic volume forms on surface representation varieties.
problem Calculating symplectic volume forms on surface representation varieties.
method Multiplicative gluing formulas and Heusener-Porti results.
result Symplectic volume form on Σg,0 is a product of forms on Σ2,1 and Σ2,2. Paper constructs moduli spaces of Higgs bundles and connects them to Teichmüller space structures.
problem Constructing moduli spaces of Higgs bundles on varying Riemann surfaces.
method Gauge theoretic construction, Teichmüller space, isomonodromic foliation, Atiyah-Bott-Goldman symplectic structure.
result Surprising relationships between Higgs bundles, isomonodromic foliation, and Teichmüller space structures.
We study a particular class of representations from the fundamental groups of punctured spheres Σ0,n to the group PSL(2,R) (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple clos…
This paper introduces complex Chern-Simons bundles in families setting and proves their crystalline nature.
problem Characterizing projective structures of Riemann surfaces and establishing holomorphic torsion formulas.
method Develops a formalism for direct images of characteristic classes, uses deformation theory of harmonic maps, and relies on non-abelian Hodge theory.
result Establishes the crystalline nature of the relative complex Chern-Simons bundle and its holomorphic extension.