We discuss questions of isospectrality for hyperbolic orbisurfaces, examining the relationship between the geometry of an orbisurface and its Laplace spectrum. We show that certain hyperbolic orbisurfaces cannot be isospectral, where the obstructions involve the number of singular points and genera of our orbisurfaces.…
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We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.
Study spectral invariants for polygons and orbisurfaces.
problem Compute spectral invariants for polygons and orbisurfaces.
method Study the heat kernel and asymptotic expansion of the heat trace.
result Explicit formulas for heat invariants of polygons and orbisurfaces.
The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsionmath.SP
New formula connects surface singularity zeta function to Reidemeister-Turaev torsion.
problem Calculating Reidemeister-Turaev torsion for non-unitary representations.
method Ruelle zeta function and Reidemeister-Turaev torsion for compact hyperbolic orbisurfaces.
result Value of Ruelle zeta function at 0 equals Reidemeister-Turaev torsion.
New solutions found for complex structures on specific manifolds.
problem Constructing smooth solutions to the Hull-Strominger system.
method Using fibrations over K3 orbisurfaces.
result Proved existence of solutions for certain manifolds.
New solutions found for system using K3 orbifolds.
problem Finding smooth solutions to the Hull-Strominger system.
method Torus fibrations over K3 orbifolds, adapted Serre construction for singular settings.
result Constructed new smooth solutions to the Hull-Strominger system.
Study calculates heat coefficients from geodesic polygon corners.
problem Heat coefficients from geodesic polygon corners.
method Formulas for cone points contributions to heat trace.
result Determines Laplacian of Gauss curvature at corners.
We consider how the geometry and topology of a compact -dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in t…