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Tinkertoy Models Produce New Geometric Insights

Tinkertoy Models Produce New Geometric Insights
Sam Payne. Credit: Vivian Abagiu

Equations may seem set in stone, but the shapes they describe rely on the space they occupy. This is a concept explored by tropical geometry, a relatively new field of mathematics which reduces complex shapes into simple stick drawings. Quanta Magazine featured a pair of joint papers with connections to the University of Texas at Austin that add new insights to the field.

Dave Jensen, a mathematics professor at University of Kentucky who received his PhD from UT Austin in 2010, was an author on both. One of the papers, co-authored by UT Austin mathematics professor Sam Payne, solves a conjecture known as "strong maximal rank" that was posited by former UT Austin professor Gavril Farkas.

The researchers used algebra to break down geometric shapes into lines and then form them into graphs, producing what are known as tropicalizations. These simplified drawings retain many important properties of the original shapes, but are easier to study. The findings bring mathematicians towards a systematic understanding of how equations that embed shapes, or describe them relative to the spaces they exist in, vary based on the shapes themselves. In turn, they gain further information about the nature of the shapes and their applications.

"Often to really understand [a shape] or classify it, to get a finer feel for its geometry, we're going to embed it somewhere," said Sam Payne, co-author of one of the recent papers.

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Quanta Magazine: Tinkertoy Models Produce New Geometric Insights

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Tuesday, 28 March 2023

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