arXiv subjects

Eloise Hamilton

Publications and source records attributed to Eloise Hamilton.

7 recordsLinked to original sources

arXiv2026

Relative Geometric Invariant Theory: Reductive and Non-reductive

We construct good quotients for equivariant actions of group homomorphisms on morphisms of schemes. Using Geometric Invariant Theory, we obtain explicit open semistable loci in the source with Hilbert-Mumford descriptions admitting good quotients relative to a given good quotient of the target. In particular, we obtain quotients for reductive groups acting on projective-over-affine morphism. In the non-reductive case, we consider equivariant actions on affine morphisms which are 'graded' by a multiplicative group and satisfy certain unipotent stabiliser assumptions. This recovers known results in projective non-reductive GIT as a special case, which also proves the Hilbert-Mumford criterion in that setting. As applications, we consider moduli of unstable objects, representations of quivers with multiplicities and jets.

math.AG↗

arXiv2026

The science and practice of proportionality in AI risk evaluations

A global challenge in artificial intelligence (AI) regulation lies in achieving effective risk management without compromising innovation and technical progress. The European Union (EU) Artificial Intelligence Act represents the first regulatory attempt worldwide to navigate this tension in the form of a binding, risk-based framework. In August 2025, obligations for providers of general-purpose AI (GPAI) models under the EU AI Act entered into application. They require providers of the most advanced GPAI models to evaluate possible systemic risks stemming from their models. This raises the regulatory challenge of ensuring that the evaluations provide meaningful risk information without imposing excessive burden on providers. The principle of proportionality, a binding requirement under EU law, requires the regulator to calibrate its actions to their intended objectives. The application of proportionality to model evaluations for AI risk opens opportunities to develop scientific methods that operationalize such calibration within concrete evaluation practices.

cs.CY↗

arXiv2024

Affine Non-Reductive GIT and moduli of representations of quivers with multiplicities

We give an explicit approach to quotienting affine varieties by linear actions of linear algebraic groups with graded unipotent radical, using results from projective Non-Reductive GIT. Our quotients come with explicit projective completions, whose boundaries we interpret in terms of the original action. As an application we construct moduli spaces of semistable representations of quivers with multiplicities subject to certain conditions, which always hold in the toric case for a generic stability condition.

math.AG↗

arXiv2023

Complete collineations for maximum likelihood estimation

We import the algebro-geometric notion of a complete collineation into the study of maximum likelihood estimation in directed Gaussian graphical models. A complete collineation produces a perturbation of sample data, which we call a stabilisation of the sample. While a maximum likelihood estimate (MLE) may not exist or be unique given sample data, it is always unique given a stabilisation. We relate the MLE given a stabilisation to the MLE given original sample data, when one exists, providing necessary and sufficient conditions for the MLE given a stabilisation to be one given the original sample. For linear regression models, we show that the MLE given any stabilisation is the minimal norm choice among the MLEs given an original sample. We show that the MLE has a well-defined limit as the stabilisation of a sample tends to the original sample, and that the limit is an MLE given the original sample, when one exists. Finally, we study which MLEs given a sample can arise as such limits. We reduce this to a question regarding the non-emptiness of certain algebraic varieties.

math.ST↗

arXiv2021

Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients

We establish a method for calculating the Poincaré series of moduli spaces constructed as quotients of smooth varieties by suitable non-reductive group actions; examples of such moduli spaces include moduli spaces of unstable vector or Higgs bundles on a smooth projective curve, with a Harder-Narasimhan type of length two. To do so, we first prove a result concerning the smoothness of fixed point sets for suitable non-reductive group actions on smooth varieties. This enables us to prove that quotients of smooth varieties by such non-reductive group actions, which can be constructed using Non-Reductive GIT via a sequence of blow-ups, have at worst finite quotient singularities. We conclude the paper by providing explicit formulae for the Poincaré series of these non-reductive GIT quotients.

math.AG↗

arXiv2019

Three instability stratifications of the stack of Higgs bundles on a smooth projective curve

We study three instability stratifications of the stack of twisted Higgs bundle of a fixed rank and degree on a smooth complex projective curve. The first is the Harder-Narasimhan (HN) stratification, defined by the instability type of the Higgs bundle. The second is the bundle Harder-Narasimhan (bHN) stratification, defined by the instability type of the underlying bundle. While an unstable HN stratum fibres over the stack parametrising Higgs bundles which are isomorphic to their graded, this is not true for Higgs bundles of unstable bHN type. Obtaining such a fibration requires refining the bHN stratification; this is the third instability stratification. After introducing these three stratifications, we establish comparison results. In particular we obtain explicit criteria for determining semistability of a Higgs bundle of low rank with unstable underlying bundle. Then we show how the HN and bHN stratifications can be used to filter the stack of Higgs bundles by global quotient stacks in two different ways. Finally we use these filtrations to relate the HN and bHN stratifications to GIT instability stratifications, and the refined bHN stratification to a Bialynicki-Birula stratification.

math.AG↗

arXiv2019

Classifying complete $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$

We address the problem of classifying complete $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$. A discrete invariant for this classification problem is the semigroup of orders of the elements in a given $\mathbb{C}$-subalgebra. Hence we can define the space $\mathcal{R}_Γ$ of all $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$ with semigroup $Γ$. After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that $\mathcal{R}_Γ$ is an affine variety by describing its defining equations in an ambient affine space in terms of an explicit algorithm. Moreover, we identify certain types of semigroups $Γ$ for which $\mathcal{R}_Γ$ is always an affine space, and for general $Γ$ we describe the stratification of $\mathcal{R}_Γ$ by embedding dimension. We also describe the natural map from $\mathcal{R}_Γ$ to the Zariski moduli space in some special cases. Explicit examples are provided throughout.

math.AG↗