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Aideen Fay

Publications and source records attributed to Aideen Fay.

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Hasse Obstructions to Rationality for Special Fourfolds

For every nonempty Hassett divisor $\mathcal C_d$ parameterizing special cubic fourfolds, we consider the quaternion class $\beta_d=(d/2,-3)$ in the two-torsion Brauer group. We prove that if a Hodge-general member of ${C}_d$ is rational then $\beta_d=0$ -- or, equivalently, that Huybrechts' twisted-K3 condition $(**')$ holds. Consequently, a very general member of $\mathcal C_d$ is irrational if $\beta_d\ne0$. Thus, a very general cubic fourfold containing a smooth cubic scroll or a Veronese surface is irrational, and hence so is a very general K\"uchle fourfold of type $(\mathrm{c7})$. We also obtain an analogous obstruction for Hodge--special Gushel--Mukai fourfolds: a very general member in the discriminant--$d$ locus can be rational only if $d$ is a sum of two squares. Consequently, a very general Gushel--Mukai fourfold containing a cubic scroll is irrational.

math.AG

An atomic criterion for irrationality without quantum computations

The birational invariants introduced by Katzarkov-Kontsevich-Pantev-Yu allows one to obtain irrationality results for varieties whose quantum cohomology is well-behaved. We observe that under certain cohomological conditions, we can deduce irrationality of a very general member from the theory of atoms without actually computing them, using only monodromy equivariance of quantum multiplication and irreducibility of the monodromy representation. Our criterion applies to the very general cubic and Gushel-Mukai fourfolds, whose irrationalities were already known, but also to the very general K{\"u}chle fourfold of type (c5), which is a Fano manifold of index one.

math.AG

Equivariant irrationality of very general symmetric Verra fourfolds

We prove that a very general complex symmetric Verra fourfold is not \(\ZZ/2\)-birational to \(\PP^4\), using the \emph{theory of atoms} introduced by Katzarkov--Kontsevich--Pantev--Yu~\citep{Katzarkov2025BirationalIF} in the equivariant setting as in Cavenaghi--Katzarkov--Kontsevich~\citep{cavenaghi2026atomsmeetsymbols}.

math.AG

The Very General Verra Fourfold is Irrational

We show that the very general Verra fourfold is irrational, using the Hodge atom framework of Katzarkov--Kontsevich--Pantev--Yu. Two novel points are: a refined analysis of Hodge atoms, based on the involution on the cohomology of $X$, and a derivation of the quantum multiplication matrix from the quantum differential operator. This gives the first successful application of the method of Hodge atoms to a space with Picard rank greater than one.

math.AG

LLMail-Inject: A Dataset from a Realistic Adaptive Prompt Injection Challenge

Indirect Prompt Injection attacks exploit the inherent limitation of Large Language Models (LLMs) to distinguish between instructions and data in their inputs. Despite numerous defense proposals, the systematic evaluation against adaptive adversaries remains limited, even when successful attacks can have wide security and privacy implications, and many real-world LLM-based applications remain vulnerable. We present the results of LLMail-Inject, a public challenge simulating a realistic scenario in which participants adaptively attempted to inject malicious instructions into emails in order to trigger unauthorized tool calls in an LLM-based email assistant. The challenge spanned multiple defense strategies, LLM architectures, and retrieval configurations, resulting in a dataset of 208,095 unique attack submissions from 839 participants. We release the challenge code, the full dataset of submissions, and our analysis demonstrating how this data can provide new insights into the instruction-data separation problem. We hope this will serve as a foundation for future research towards practical structural solutions to prompt injection.

cs.CR

The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology

Existing interpretability methods for Large Language Models (LLMs) predominantly capture linear directions or isolated features. This overlooks the high-dimensional, relational, and nonlinear geometry of model representations. We apply persistent homology (PH) to characterize how adversarial inputs reshape the geometry and topology of internal representation spaces of LLMs. This phenomenon, especially when considered across operationally different attack modes, remains poorly understood. We analyze six models (3.8B to 70B parameters) under two distinct attacks, indirect prompt injection and backdoor fine--tuning, and show that a consistent topological signature persists throughout. Adversarial inputs induce topological compression, where the latent space becomes structurally simpler, collapsing the latent space from varied, compact, small-scale features into fewer, dominant, large-scale ones. This signature is architecture-agnostic, emerges early in the network, and is highly discriminative across layers. By quantifying the shape of activation point clouds and neuron-level information flow, our framework reveals geometric invariants of representational change that complement existing linear interpretability methods.

cs.LG

Get my drift? Catching LLM Task Drift with Activation Deltas

LLMs are commonly used in retrieval-augmented applications to execute user instructions based on data from external sources. For example, modern search engines use LLMs to answer queries based on relevant search results; email plugins summarize emails by processing their content through an LLM. However, the potentially untrusted provenance of these data sources can lead to prompt injection attacks, where the LLM is manipulated by natural language instructions embedded in the external data, causing it to deviate from the user's original instruction(s). We define this deviation as task drift. Task drift is a significant concern as it allows attackers to exfiltrate data or influence the LLM's output for other users. We study LLM activations as a solution to detect task drift, showing that activation deltas - the difference in activations before and after processing external data - are strongly correlated with this phenomenon. Through two probing methods, we demonstrate that a simple linear classifier can detect drift with near-perfect ROC AUC on an out-of-distribution test set. We evaluate these methods by making minimal assumptions about how users' tasks, system prompts, and attacks can be phrased. We observe that this approach generalizes surprisingly well to unseen task domains, such as prompt injections, jailbreaks, and malicious instructions, without being trained on any of these attacks. Interestingly, the fact that this solution does not require any modifications to the LLM (e.g., fine-tuning), as well as its compatibility with existing meta-prompting solutions, makes it cost-efficient and easy to deploy. To encourage further research on activation-based task inspection, decoding, and interpretability, we release our large-scale TaskTracker toolkit, featuring a dataset of over 500K instances, representations from six SoTA language models, and a suite of inspection tools.

cs.CR