arXiv subjects

Henry Talbott

Publications and source records attributed to Henry Talbott.

4 recordsLinked to original sources

arXiv2025

Weyl-invariant subspaces are (usually) not generic

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\theta$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_\theta(G) \subseteq V$ of semistable points. We give necessary, sufficient, and in some cases equivalent conditions for the existence of $\theta$ such that a maximal torus $T$ of $G$ acts on $V^{ss}_\theta(T)$ with finite stabilizers. In such cases, the stack quotient $[V^{ss}_\theta(G)/G]$ is is known to be Deligne-Mumford. Our proof uses the combinatorial structure of the weights of irreducible representations of semisimple groups. As an application we generalize the Grassmannian flop example of Donovan-Segal.

math.RT↗

arXiv2025

Dynamical zeta functions and resonance chains for infinite-area hyperbolic surfaces with large funnel widths

We quantitatively relate the resonance sets of topologically finite infinite-area hyperbolic surfaces with no cusps to the resonance sets of certain metric graphs via the spine graph construction. In particular, we prove the existence of approximate resonance chains in resonance sets of these surfaces in the long-boundary-length regime. Our results are similar in spirit to those obtained in recent independent work by Li-Matheus-Pan-Tao, although our perspective, hypotheses, and methods are somewhat different. Our results also generalize older results obtained for three-funneled spheres by Weich. We primarily make use of transfer operators for holomorphic iterated function schemes, along with certain geometric bounds.

math.DS↗

arXiv2025

Critical Exponents on Hyperbolic Surfaces with Long Boundaries and the Asymptotic Weil-Petersson Form

We study the critical exponent random variable $\delta_X$ on moduli spaces of hyperbolic surfaces with boundary, using the normalized Weil-Petersson measures $d\mu_{WP}$ as probability measures. We use the spine graph construction of Bowditch and Epstein to compare this random variable to the corresponding critical exponent random variable $\delta_\Gamma$ on moduli spaces of metric ribbon graphs with the normalized Kontsevich measures $d\mu_K$, proving an asymptotic convergence-in-mean result in the long boundary length regime. In particular, we show that $d\mu_K$ approximately pulls back to $d\mu_{WP}$ with quantitative uniform estimates.

math.GT↗

arXiv2021

Dynamical moduli spaces and polynomial endomorphisms of configurations

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.

math.AG↗