Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1809.05930

Visually Representing the Landscape of Mathematical Structures

Katherine Gravel·Hayden Jananthan·Jeremy Kepner
View original source ↗Read source document ↗Export metadata ↓

Abstract

The information technology explosion has dramatically increased the application of new mathematical ideas and has led to an increasing use of mathematics across a wide range of fields that have been traditionally labeled "pure" or "theoretical". There is a critical need for tools to make these concepts readily accessible to a broader community. This paper details the creation of visual representations of mathematical structures commonly found in pure mathematics to enable both students and professionals to rapidly understand the relationships among and between mathematical structures. Ten broad mathematical structures were identified and used to make eleven maps, with 187 unique structures in total. The paper presents a method and implementation for categorizing mathematical structures and for drawing relationship between mathematical structures that provides insight for a wide audience. The most in depth map is available online for public use [1].

Explore related subjects

math.HO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Katherine Gravel, Hayden Jananthan, Jeremy Kepner. 2018-09-16. Visually Representing the Landscape of Mathematical Structures. https://arxiv.org/abs/1809.05930

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Come for the vibe, stay for the math

This article describes our experiences in mathematical outreach over the past decade. We talk about specific activities, but also general principles that we've learned along the way.

math.HO↗
arXiv2026

Graduate Mathematics in the Age of AI: Forming Mathematicians for Original, Independent, and Responsible Inquiry

Artificial intelligence can increasingly produce plausible, sophisticated mathematical material faster than a developing graduate student can understand or verify it. A sophisticated result or paper draft therefore becomes weaker evidence of the student's own mathematical development. This creates a formation gap between output and personal capacity, and a trust gap between a convincing argument and warranted acceptance. The formation gap can persist even when the student understands the output: understanding a supplied argument does not by itself establish the capacity to initiate and direct inquiry. These gaps are not the whole story. AI can also help students explore examples, compare approaches, enter unfamiliar areas, and undertake ambitious research. The task is to design an apprenticeship that realizes these possibilities while developing substantive mathematical command. The central purpose of a mathematics PhD is to form mathematicians capable of original, independent, and responsible inquiry, including inquiry conducted with AI. This document develops that objective through four connected capacities: competence, judgment, independence, and responsibility. It distinguishes a work's contribution to mathematics from the evidence it provides of a student's formation; explains how a known answer can initiate rather than end creative inquiry; and proposes changes in learning activities, assessment, doctoral originality, advising, and institutional support. Purposeful independent work and ambitious AI-assisted research are complementary parts of the model. Its recommendations include proportionate contribution statements, recognition of advising costs, and staged pilots evaluating both mathematical ability and effective human--AI collaboration. The aim is not to preserve an inherited sequence of training, but to improve mathematical formation as mathematical practice changes.

math.HO↗