Publications

Published

2026

1. ASEE

   

   WIP: Using Career Panels to Spark Curiosity and Career Awareness Among Mathematics Majors

   Ann W. Clifton and Blake Farman

   In ASEE Annual Conference & Exposition, Jun 2026

   Abs DOI PDF

   This Work-in-Progress study investigates how career panels shape students’ perceptions of mathematics and the opportunities available to them. The 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences specifically recommends that “mathematical sciences major programs should offer their students an orientation to careers in mathematics,” citing benefits such as recruitment and retention of majors. Similarly, the BIG (Business, Industry, Government) Math Network urges departments to engage students in pathways beyond academia. Responding to these national calls, the Mathematics & Statistics program at ___ began hosting career panels in Fall 2025, supported by (grant). To examine student responses to these panels, we adapted a survey instrument previously validated in the field of information systems. Items were modified to reflect mathematics-specific constructs, including perceptions of the field, job availability, perceptions of professionals, usefulness of studies, confidence in learning, effectance motivation, and intention. In addition, we introduce a curiosity construct to capture new questions and interests stimulated by the panels. These constructs align with expectancy-value and career development frameworks, which highlight the roles of perceived utility, confidence, and curiosity in shaping student motivation and career exploration. The survey was administered to students attending the career panels, including both mathematics majors and students from other STEM disciplines. This paper reports preliminary data from the initial implementation of the panels and instrument. We examine patterns in students’ perceptions of mathematics-related careers and analyze open-ended responses to identify common themes in students’ expressed curiosities and questions. Quantitative Likert-scale responses are summarized descriptively to provide context for the qualitative findings. Looking forward, we will extend the instrument and undertake formal validation in the context of mathematics majors. Ultimately, this work will contribute a validated instrument for use across the mathematics community. In the immediate term, we offer a formative tool for departments seeking feedback on career-oriented programming.

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 ASEE Annual Conference & Exposition, 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 ASEE Annual Conference & Exposition, 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.

Formalization: merged into mathlib4

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

Formalization: open pull requests

1. mathlib4

   

   feat(CategoryTheory/Subobject): kernels of pullbacks of subobjects

   Blake Farman

   Sep 2026

   Adds Subobject.le_pullback_of_comp_eq_zero, Subobject.ofLE_comp_pullbackπ_eq_zero and Subobject.isLimitKernelForkPullbackπ: for a morphism f : X ⟶ Y and subobjects x of X and y of Y, if x.arrow ≫ f = 0 then x is contained in the pullback of y along f, the induced map into that pullback composed with pullbackπ f y vanishes, and if moreover x.arrow is a kernel of f then that map is a kernel of pullbackπ f y. These require only HasZeroMorphisms and HasPullbacks. Open pull request #44223.

   GitHub
2. mathlib4

   

   feat(CategoryTheory/Abelian): short exact sequences from pulling back a subobject

   Blake Farman

   Sep 2026

   Pulling a subobject B of Y back along an epimorphism f : X ⟶ Y whose kernel is A yields an extension of B by A; this records that as a short exact sequence in an abelian category. Adds shortComplexPullbackπ and shortExact_shortComplexPullbackπ, together with the special case f = cokernel.π A.arrow, which is one direction of the correspondence between subobjects of X containing A and subobjects of the quotient. Builds on Subobject.isLimitKernelForkPullbackπ from #44223. Open pull request #44229.

   GitHub
3. mathlib4

   

   feat(CategoryTheory/ObjectProperty): closure properties of orthogonals and S. E. Dickson’s theorem

   Blake Farman

   Sep 2026

   Studies how the closure properties of a property of objects P interact with the orthogonal complements P.leftOrthogonal and P.rightOrthogonal and with passage to the opposite category. In a well-powered abelian category with coproducts, if P is closed under quotients, extensions and coproducts then P.rightOrthogonal.leftOrthogonal = P, the hard direction of S. E. Dickson’s characterization of torsion classes. Open pull request #44226.

   GitHub
4. mathlib4

   

   feat(CategoryTheory/Abelian/TorsionTheory): Introduce torsion theory for abelian categories

   Blake Farman

   Mar 2026

   Introduces torsion theories on an abelian category: a pair of object properties that are each other’s left and right orthogonals. Adds Abelian.TorsionTheory, the typeclasses IsTorsionClass and IsTorsionFreeClass, the torsion theories generated and cogenerated by a property, transport along op and unop, and Dickson’s theorem isTorsionClass_iff: in a well-powered abelian category with coproducts, a class is a torsion class if and only if it is closed under quotients, extensions and coproducts. Open pull request #36744.

   GitHub