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William Fisher

Publications and source records attributed to William Fisher.

5 recordsLinked to original sources
arXiv2026

Exchange theorems and coherent duality in six functors

We define the notion of an exchange theorem and show that any two functors satisfying an exchange theorem are canonically related via twisted norm maps. This is done by identifying the universal category receiving a pair of functors satisfying an exchange theorem. Additionally, we show that the twists occurring are $K$-theoretic in nature, parametrized by a categorified analogue of virtual vector bundles. As an application, we show that every 3-functor formalism has a canonical extension which encodes Poincar\'e duality and Thom twists internal to the formalism. This gives a 1-categorical realization of the "coherent six operations" outlined by Hoyois. In the process of proving universality, techniques for computing categories associated to bi- and $n$-simplicial spaces are developed. Many of the results in this direction may be viewed as model-independent rederivations of work of Liu--Zheng.

math.CT↗
arXiv2025

Incorporating Legal Structure in Retrieval-Augmented Generation: A Case Study on Copyright Fair Use

This paper presents a domain-specific implementation of Retrieval-Augmented Generation (RAG) tailored to the Fair Use Doctrine in U.S. copyright law. Motivated by the increasing prevalence of DMCA takedowns and the lack of accessible legal support for content creators, we propose a structured approach that combines semantic search with legal knowledge graphs and court citation networks to improve retrieval quality and reasoning reliability. Our prototype models legal precedents at the statutory factor level (e.g., purpose, nature, amount, market effect) and incorporates citation-weighted graph representations to prioritize doctrinally authoritative sources. We use Chain-of-Thought reasoning and interleaved retrieval steps to better emulate legal reasoning. Preliminary testing suggests this method improves doctrinal relevance in the retrieval process, laying groundwork for future evaluation and deployment of LLM-based legal assistance tools.

cs.CL↗
arXiv2024

Collaborative Design of Controlled Experiments in the Presence of Subject Covariates

We consider the optimal experimental design problem of allocating subjects to treatment or control when subjects participate in multiple, separate controlled experiments within a short time-frame and subject covariate information is available. Here, in addition to subject covariates, we consider the dependence among the responses coming from the subject's random effect across experiments. In this setting, the goal of the allocation is to provide precise estimates of treatment effects for each experiment. Deriving the precision matrix of the treatment effects and using D-optimality as our allocation criterion, we demonstrate the advantage of collaboratively designing and analyzing multiple experiments over traditional independent design and analysis, and propose two randomized algorithms to provide solutions to the D-optimality problem for collaborative design. The first algorithm decomposes the D-optimality problem into a sequence of subproblems, where each subproblem is a quadratic binary program that can be solved through a semi-definite relaxation based randomized algorithm with performance guarantees. The second algorithm involves solving a single semi-definite program, and randomly generating allocations for each experiment from the solution of this program. We showcase the performance of these algorithms through a simulation study, finding that our algorithms outperform covariate-agnostic methods when there are a large number of covariates.

stat.ME↗
arXiv2019

Ionic-to-electronic current amplification in hybrid perovskite solar cells: ionically gated transistor-interface circuit model explains hysteresis and impedance of mixed conducting devices

Mobile ions in hybrid perovskite semiconductors introduce a new degree of freedom to electronic devices suggesting applications beyond photovoltaics. An intuitive device model describing the interplay between ionic and electronic charge transfer is needed to unlock the full potential of the technology. We describe the perovskite-contact interfaces as transistors which couple ionic charge redistribution to energetic barriers controlling electronic injection and recombination. This reveals an amplification factor between the out of phase electronic current and the ionic current. Our findings suggest a strategy to design thin film electronic components with large, tuneable, capacitor-like and inductor-like characteristics. The resulting simple equivalent circuit model, which we verified with time-dependent drift-diffusion simulations of measured impedance spectra, allows a general description and interpretation of perovskite solar cell behaviour.

physics.app-ph↗