arXiv subjects

Samuel Trautwein

Publications and source records attributed to Samuel Trautwein.

6 recordsLinked to original sources

arXiv2025

Universal Hitchin moduli spaces

We study metric aspects of the universal moduli space of solutions to Hitchin's equations as the complex structure $J$ varies over the Teichm\"uller space $\mathcal{T}$ of a closed surface $\Sigma$. Our approach is gauge theoretical and builds on the theory of K\"ahler fibrations and the moment map interpretation of constant scalar curvature K\"ahler metrics. Our first main result establishes that, over the moduli space of cscK metrics, the universal moduli space of solutions to Hitchin's equations carries a natural complex structure together with a family of pseudo-K\"ahler metrics forming a K\"ahler fibration with a K\"ahler Ehresmann connection. We then investigate a second universal moduli space, constructed from the space of flat $G$-connections over $\mathcal{T}$, which admits a nontrivial $J$-dependent K\"ahler fibration structure discovered by Hitchin. Using symplectic reduction, we build universal moduli spaces of solutions to the harmonicity equations depending on a coupling constant $\alpha$, obtaining natural complex and pseudo-K\"ahler structures and an explicit K\"ahler potential. The main novelty here is that this moduli space is defined by a system coupling the scalar curvature with a cubic term in the Higgs field. Finally, we propose a conjectural relationship between the two resulting families of moduli spaces in the weak-coupling limit $\alpha\to 0$, inspired by the twistor geometry of Hitchin's hyperk\"ahler moduli space.

math.DG↗

arXiv2025

Gandalf the Red: Adaptive Security for LLMs

Current evaluations of defenses against prompt attacks in large language model (LLM) applications often overlook two critical factors: the dynamic nature of adversarial behavior and the usability penalties imposed on legitimate users by restrictive defenses. We propose D-SEC (Dynamic Security Utility Threat Model), which explicitly separates attackers from legitimate users, models multi-step interactions, and expresses the security-utility in an optimizable form. We further address the shortcomings in existing evaluations by introducing Gandalf, a crowd-sourced, gamified red-teaming platform designed to generate realistic, adaptive attack. Using Gandalf, we collect and release a dataset of 279k prompt attacks. Complemented by benign user data, our analysis reveals the interplay between security and utility, showing that defenses integrated in the LLM (e.g., system prompts) can degrade usability even without blocking requests. We demonstrate that restricted application domains, defense-in-depth, and adaptive defenses are effective strategies for building secure and useful LLM applications.

cs.LG↗

arXiv2021

Complex structures, moment maps, and the Ricci form (Extended Version)

The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmuller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kahler cone.

math.SG↗

arXiv2019

The hyperkähler metric on the almost-Fuchsian moduli space

Donaldon constructed a hyperkähler moduli space $\mathcal{M}$ associated to a closed oriented surface $Σ$ with $\textrm{genus}(Σ) \geq 2$. This embeds naturally into the cotangent bundle $T^*\mathcal{T}(Σ)$ of Teichmüller space or can be identified with the almost-Fuchsian moduli space associated to $Σ$. The later is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in $(-1,1)$. Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bundle of Teichmüller space, the hyperkähler structure on $\mathcal{M}$ can be viewed as the Feix--Kaledin hyperkähler extension of the Weil--Petersson metric. The almost-Fuchsian moduli space embeds into the $\textrm{SL}(2,\mathbb{C})$-representation variety of $Σ$ and the hyperkähler structure on $\mathcal{M}$ extends the Goldman holomorphic symplectic structure. Here the natural complex structure corresponds to the second complex structure in the first picture. Moreover, the area of the minimal surface in an almost-Fuchsian manifold provides a Kähler potential for the hyperkähler metric. The various identifications are obtained using the work of Uhlenbeck on germs of hyperbolic $3$-manifolds, an explicit map from $\mathcal{M}$ to $\mathcal{T}(Σ)\times \bar{\mathcal{T}(Σ)}$ found by Hodge, the simultaneous uniformization theorem of Bers, and the theory of Higgs bundles introduced by Hitchin.

math.DG↗

arXiv2018

Convergence of the Yang-Mills-Higgs flow on gauged holomorphic maps and applications

The symplectic vortex equations admit a variational description as global minimum of the Yang-Mills-Higgs functional. We study its negative gradient flow on holomorphic pairs $(A,u)$ where $A$ is a connection on a principal $G$-bundle $P$ over a closed Riemann surface $Σ$ and $u: P \rightarrow X$ is an equivariant map into a Kähler Hamiltonian $G$-manifold. The connection $A$ induces a holomorphic structure on the Kähler fibration $P\times_G X$ and we require that $u$ descends to a holomorphic section of this fibration. We prove a Lojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the $W^{1,2}\times W^{2,2}$-topology when $X$ is equivariantly convex at infinity with proper moment map, $X$ is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang-Mills-Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet's Kobayashi-Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment-weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

math.DG↗

arXiv2015

A survey of the GIT picture for the Yang-Mills equation over Riemann surfaces

The purpose of this paper is to give a self-contained exposition of the Atiyah-Bott picture for the Yang-Mills equation over Riemann surfaces with an emphasis on the analogy to finite dimensional geometric invariant theory. The main motivation is to provide a careful study of the semistable and unstable orbits: This includes the analogue of the Ness uniqueness theorem for Yang-Mills connections, the Kempf-Ness theorem, the Hilbert-Mumford criterion and a new proof of the moment-weight inequality following an approach outlined by Donaldson. A central ingredient in our discussion is the Yang-Mills flow for which we assume longtime existence and convergence.

math.DG↗