Research
In Progress
Conflicting Claims to Hermeneutical Justice
Social Epistemology
Feminist Epistemology
In Progress
Many standard examples of hermeneutical injustice involve a gap, or lacuna, where a term describing an important social experience is missing. This paper introduces hermeneutical dilution and hermeneutical monumentalization as two categories of hermeneutical occlusions which, in contrast, do not require the absence of a term. Instead, these occlusions have to do with the inability to specify one’s particular experience using existing terms, either because the term is inaccessible or because it is not sufficiently specific. This paper is in progress; text is subject to change.
Normative Distributivity & Group Zetetic Obligation
Social Epistemology
Zetetic Epistemology
In Progress
In a recent paper, Jennifer Lackey argues for a summative account of justified group belief using a thought experiment, IGNORING EVIDENCE. In this paper, I develop the framework of normative distributivity to understand the challenge to non-summativism posed by IGNORING EVIDENCE. In light of this framework and recent research into inquisitive reasons, I argue that IGNORING EVIDENCE shows only that norms of inquiry are summative, not that norms of belief are. This paper is in progress; text is subject to change.
Grounding and Hypergraphs
Metaphysics
Combinatorics
Grounding
Unfinished
A natural model of grounding is with directed graphs: if \(\phi\) grounds \(\psi\), then there is an edge \(\phi \to \psi\). This model has been complicated by recent research into the collectivity of ground. Left-collective, and more radical bi-collective, notions of ground demand a treatment in terms of hypergraphs; however, even basic concepts for hypergraphs have numerous competing definitions. This research aims to provide definitions for these concepts tailored to the philosophical demands on their use in the study of grounding. This paper is unfinished; text is not yet available.
Published and Preprints
Pak-Stanley Labels
Combinatorics
Algorithms
Published
The Pak-Stanley algorithm offers a correspondence between superstable configurations on a graph \(G\) and the regions of the \(G\)-Shi hyperplane arrangement. In the case \(G = K_n\), this correspondence is bijective. However, it remains an open problem to characterize the failure of bijectivity for arbitrary \(G\). In this paper, we make significant progress towards resolving this open problem by solving it for trees and graphs with a single cycle, as well as indicating the next steps for further analysis.
Split Petal Projections
Combinatorics
Knot Theory
Algorithms
Published
An übercrossing diagram is a knot diagram with only one crossing; such a diagram without any nested loops is called a petal projection. Every knot has a petal projection from which the knot can be recovered using a single permutation, but these diagrams are not in the standard double-crossing form. This paper demonstrates a manipulation of the knot into a split petal projection, retaining the petal structure while in double-crossing form. It also applies this idea to the computation of the knot determinant.
Ergodicity and the Torelli Subgroup
Dynamics
Abstract Algebra
Preprint
In 2008, T. Gelander proved that if \(G\) is a compact, connected, semisimple Lie group, the orbit of almost any point in \(\operatorname{Hom}(F_n, G)\) under the natural action of \(\operatorname{Aut}(F_n)\) is dense, and furthermore that this action is ergodic. In this paper, we generalize this result to the Torelli subgroup, an important normal subgroup of \(\operatorname{Aut}(F_n)\). That is, we prove that the orbit of almost any point in \(\operatorname{Hom}(F_n, G)\) under action of \(\operatorname{Tr}(n)\) is dense and that the action is ergodic.
The Theory of Ranked-Choice Voting
Social Choice Theory
Algorithms
Preprint
Ranked-choice voting (RCV) has been proposed as a potential solution to increasing extremism in American election. However, there is a lack of mathematical research about the differences between RCV and the first-past-the-post primary system in place in most states. In this paper, we validate the hypothesis that RCV favors moderate candidates using proof-based methods relying on integration over high-dimensional polytopes and simulations on various simplified models.
Other
Tokuyama's Formula
Combinatorics
Abstract Algebra
Expository
Tokuyama’s formula connects combinatorics and representation theory by interpreting an expression involving the characters of general linear groups as a sum over combinatorial objects such as Gelfand-Tsetlin patterns, shifted tableaux, or gamma ice models. This paper reviews existing proofs of Tokuyama’s formula, presents two novel proofs avoiding complicated machinery required by previous proofs, and describes progress in extending these results toward an analogue for symplectic groups.