arXiv subjects

Andy Wilson

Publications and source records attributed to Andy Wilson.

9 recordsLinked to original sources

arXiv2026

Superspace coinvariants and inverse systems for $GL_n(\mathbb{F}_q)$

Let $q$ be a prime power and write $\Omega$ for the bigraded algebra of regular differential forms over $\mathbb{F}_q^n$. The general linear group $GL_n(\mathbb{F}_q)$ acts on $\Omega$; write $SI \subseteq \Omega$ for the ideal generated by $GL_n(\mathbb{F}_q)$-invariants with vanishing constant term. The {\em $GL_n(\mathbb{F}_q)$-superspace coinvariant ring} is the quotient $SR := \Omega/SI$. We calculate the bigraded Hilbert series of $SR$ and give an operator-theoretic characterization of the inverse system $SI^\perp$. Our results extend to subgroups $G$ of $GL_n(\mathbb{F}_q)$ which contain $SL_n(\mathbb{F}_q)$.

math.CO↗

arXiv2026

Considerations for the Integration of Randomized Controlled Trials and Real-World Data

As clinical decision-making increasingly moves toward individualized and context-specific treatment recommendations, reliance on any single evidence source, randomized or observational, may be insufficient. Principled integration of randomized controlled trials and real-world data, grounded in explicit causal frameworks, offers a path toward evidence that is both internally credible and externally relevant. In this article, we describe distinct objectives for the integration of randomized controlled trials and real-world data and discuss how these objectives shape key design and analytic considerations, illustrating the resulting choices through example estimands. We highlight practical issues that commonly arise in applied settings, including data relevance and curation, cross-source comparability, estimand specification, and sensitivity analysis. We aim for this article to help readers evaluate and implement principled approaches to integrating randomized controlled trials and real-world data in ways that can support more reliable treatment recommendations while maintaining regulatory-grade evidentiary standards.

stat.ME↗

arXiv2025

From Measurement to Expertise: Empathetic Expert Adapters for Context-Based Empathy in Conversational AI Agents

Empathy is a critical factor in fostering positive user experiences in conversational AI. While models can display empathy, it is often generic rather than tailored to specific tasks and contexts. In this work, we introduce a novel framework for developing and evaluating context-specific empathetic large language models (LLMs). We first analyze a real-world conversational dataset consisting of 672 multi-turn conversations across 8 tasks, revealing significant differences in terms of expected and experienced empathy before and after the conversations, respectively. To help minimize this gap, we develop a synthetic multi-turn conversational generation pipeline and steer responses toward our defined empathy patterns based on the context that more closely matches users' expectations. We then train empathetic expert adapters for context-specific empathy that specialize in varying empathy levels based on the recognized task. Our empirical results demonstrate a significant gap reduction of 72.66% between perceived and desired empathy with scores increasing by an average factor of 2.43 as measured by our metrics and reward models. Additionally, our trained empathetic expert adapters demonstrate superior effectiveness in preserving empathy patterns throughout conversation turns, outperforming system prompts, which tend to dramatically diminish in impact as conversations lengthen.

cs.HC↗

arXiv2025

A proof of the Fields Conjectures

The {\em superspace ring} of rank $n$ is the algebra $\Omega_n$ of differential forms on affine $n$-space. The algebra $\Omega_n$ is bigraded with respect to polynomial and exterior degree and carries a natural action of the symmetric group $\mathfrak{S}_n$. Modding out by $\mathfrak{S}_n$-invariants with vanishing constant term yields the {\em superspace coinvariant ring} $SR_n$. We prove that, as an ungraded $\mathfrak{S}_n$-module, the space $SR_n$ is isomorphic to the sign-twisted permutation action of $\mathfrak{S}_n$ on ordered set partitions of $\{1,\dots,n\}$. We refine this result by calculating the bigraded $\mathfrak{S}_n$-isomorphism type of $SR_n$. This proves the Fields Conjectures of N. Bergeron, L. Colmenarejo, S.-X. Li, J. Machacek, R. Sulzgruber, and M. Zabrocki as well as a related conjecture of V. Reiner.

math.CO↗

arXiv2024

Tutte polynomials in superspace

We associate a quotient of superspace to any hyperplane arrangement by considering the differential closure of an ideal generated by powers of certain homogeneous linear forms. This quotient is a superspace analogue of the external zonotopal algebra, and it further contains the central zonotopal algebra in the appropriate grading. We show that an evaluation of the bivariate Tutte polynomial is the bigraded Hilbert series of this quotient. We then use this fact to construct an explicit basis for the Macaulay inverse. These results generalize those of Ardila-Postnikov and Holtz-Ron. We also discuss enumerative consequences of our results in the setting of hyperplane arrangements.

math.CO↗

arXiv2023

The Hilbert series of the superspace coinvariant ring

Let $\Omega_n$ be the ring of polynomial-valued holomorphic differential forms on complex $n$-space, referred to in physics as the superspace ring of rank $n$. The symmetric group $\mathfrak{S}_n$ acts diagonally on $\Omega_n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq \Omega_n$ be the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term and study the quotient $SR_n = \Omega_n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an `operator theorem' which characterizes the harmonic space $SH_n \subseteq \Omega_n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results which were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Li, Machacek, Sulzgruber, Swanson, Wallach, and Zabrocki.

math.CO↗

arXiv2022

A symmetric function lift of torus link homology

Suppose $M$ and $N$ are positive integers and let $k = \gcd(M, N)$, $m = M/k$, and $n=N/k$. We define a symmetric function $L_{M,N}$ as a weighted sum over certain tuples of lattice paths. We show that $L_{M,N}$ satisfies a generalization of Mellit and Hogancamp's recursion for the triply-graded Khovanov--Rozansky homology of the $M,N$-torus link. As a corollary, we obtain the triply-graded Khovanov--Rozansky homology of the $M,N$-torus link as a specialization of $L_{M,N}$. We conjecture that $L_{M,N}$ is equal (up to a constant) to the elliptic Hall algebra operator $\mathbf{Q}_{m,n}$ composed $k$ times and applied to 1.

math.CO↗

arXiv2010

First Smart Spaces

This document describes the Gloss software currently implemented. The description of the Gloss demonstrator for multi-surface interaction can be found in D17. The ongoing integration activity for the work described in D17 and D8 constitutes our development of infrastructure for a first smart space. In this report, the focus is on infrastructure to support the implementation of location aware services. A local architecture provides a framework for constructing Gloss applications, termed assemblies, that run on individual physical nodes. A global architecture defines an overlay network for linking individual assemblies. Both local and global architectures are under active development.

cs.DC↗

arXiv2010

Architectural Support for Global Smart Spaces

A GLObal Smart Space (GLOSS) provides support for interaction amongst people, artefacts and places while taking account of both context and movement on a global scale. Crucial to the definition of a GLOSS is the provision of a set of location-aware services that detect, convey, store and exploit location information. We use one of these services, hearsay, to illustrate the implementation dimensions of a GLOSS. The focus of the paper is on both local and global software architecture to support the implementation of such services. The local architecture is based on XML pipe-lines and is used to construct location-aware components. The global architecture is based on a hybrid peer-to-peer routing scheme and provides the local architectures with the means to communicate in the global context.

cs.DC↗