Publications

2026

1. mathlib4

   

   feat(CategoryTheory/Abelian/Preradical): introduce and characterize radicals

   Blake Farman

   Mar 2026

   Following Stenström, a preradical ‘Φ‘ is called radical if it coincides with its self colon. We encode this as the existence of an isomorphism ‘Φ.colon Φ ≅ Φ‘. We then prove a basic characterization of radical preradicals in terms of the vanishing of ‘Φ.r‘ on ‘Φ.quotient‘.

   GitHub
2. mathlib4

   

   feat(CategoryTheory/Abelian/Preradical): add Stenström’s colon construction for preradicals

   Blake Farman

   Mar 2026

   Given preradicals ‘Φ‘ and ‘Ψ‘ on an abelian category ‘C‘, this file defines their colon ‘Φ : Ψ‘ in the sense of Stenström. Categorically, ‘Φ : Ψ‘ is constructed objectwise as a pullback of the canonical projection ‘Φ.π X : X ⟶ Φ.quotient.obj X‘ along the inclusion ‘Functor.whiskerLeft Φ.quotient Ψ.ι‘.

   GitHub
3. mathlib4

   

   feat(CategoryTheory/Abelian/Preradical): introduce basic definition of preradicals

   Blake Farman

   Mar 2026

   A preradical on an abelian category ‘C‘ is a monomorphism in the functor category ‘C ⥤ C‘ with codomain ‘𝟭 C‘, i.e. an element of ‘MonoOver (𝟭 C)‘.

   GitHub
4. mathlib4

   

   feat(RingTheory/IdealFilter): add ideal filters and Gabriel filters

   Blake Farman

   Jan 2026

   Introduces ideal filters on rings and Gabriel filters in mathlib4, including the uniformity conditions (T1–T3), Gabriel composition, and a characterization of Gabriel filters (T4). Preparatory infrastructure for relating Gabriel filters to Giraud subcategories; merged into mathlib4 via PR #33021

   GitHub
5. mathlib4

   

   feat(RingTheory/IdealFilter): topologies associated to ideal filters

   Blake Farman

   Feb 2026

   Introduces additive and ring topologies on rings arising from ideal filters in mathlib4, including constructions of additive and ring filter bases, characterizations of uniform ideal filters via the existence of a ring filter basis, neighborhood descriptions of the induced topologies, and a proof that the resulting ring topology is linear; provides the topological counterpart to the algebraic theory of ideal filters; merged into mathlib4 via PR #33852.

   GitHub
6. mathlib4

   

   feat(RingTheory/Ideal): Generalize Submodule.colon to sets

   Blake Farman

   Jan 2026

   Generalize Submodule.colon to accept S : Set M and update the surrounding API; merged into mathlib4 via PR #33390.

   GitHub

2025

1. ASEE

   

   Fostering Growth Mindsets: Implementing Standards-Based Grading in College Algebra

   Blake Farman, Ann W. Clifton, William C. Long, and 1 more author

   In , Jun 2025

   Abs DOI PDF

   In mathematics courses, grading practices often focus on performance through a cumulative point system, which can mask students’ true understanding of the material. For students at EPSCoR institutions, many of whom are first-generation college students from rural communities, this traditional grading approach does not always foster the depth of learning or resilience required for long-term success. With the goal of encouraging a growth mindset, faculty members at these institutions sought a grading system that would better reflect students’ comprehension and support their academic growth. Standards-based grading (SBG) was implemented to meet these needs by shifting the focus away from partial credit accumulation and toward a more meaningful assessment of learning. SBG was adopted in College Algebra courses to encourage students to master specific learning objectives through an iterative process of reassessment. Unlike traditional weighted average grading (WAG), SBG emphasizes mastery of content, giving students multiple opportunities to demonstrate their understanding. This paper will present a statistical analysis between student outcomes in SBG and WAG college algebra courses. Supported by recent literature, SBG has been shown to foster student engagement, reduce math anxiety, and cultivate a growth mindset. This paper additionally explores the logistical aspects of implementing SBG, the process of gaining administrative and student buy-in, and provides resources for instructors interested in adopting SBG in STEM fields to improve student outcomes.

2. ASEE

   

   Work in Progress: First-Year Engineering Students’ Confidence in Communicating Mathematical Content

   Ann W. Clifton, Mary Fendley, Blake Farman, and 1 more author

   In , Jun 2025

   Abs DOI PDF

   This Work in Progress study explores the impact of weekly journaling assignments on engineering students’ ability to communicate mathematical concepts effectively in design projects. At _ University, first-year engineering students participate in the "Living with the Lab" course sequence, culminating in the First-Year Projects Showcase. While students excel at explaining their product’s purpose and hardware, they often struggle to articulate the underlying STEM principles, especially in mathematics. To address this gap, a targeted journaling assignment was integrated into the calculus sequence to enhance reflection on mathematical concepts and their connection to engineering applications. Using surveys, written reflections, and project presentations, this mixed-methods study evaluates the effectiveness of journaling in improving communication skills and confidence. Initial findings from the fall quarter suggest promising improvements in students’ mathematical communication skills, with ongoing data collection in winter and spring quarters. Aligned with Kern Entrepreneurial Engineering Network (KEEN) principles of Curiosity, Connections, and Collaboration, this intervention encourages students to integrate math into their hands-on work and articulate these connections. Findings aim to offer strategies to develop mathematical communication skills, supporting deeper learning and better preparation for future challenges in engineering.

2021

1. JNCG

   

   Kernels for noncommutative projective schemes

   Matthew Ballard and Blake Farman

   J. Noncommut. Geom., Nov 2021

   Abs DOI PDF

   We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along the lines of Rickard and Orlov.