Ohio State Newark faculty member part of national collaboration
Mention 3D to any grade school student and they’ll quickly share the names of 3D movies they’ve seen or about the 3D printer available at their local library. 3D is familiar. It’s how we define our world—through length, width and height.
But what about other dimensions? Possibly, infinite dimensions?
Tackling that question are three mathematical researchers from across the country. One of them is Niles Johnson, associate professor of mathematics at The Ohio State University at Newark.
Johnson and his colleagues are inventing a new algebra to measure infinite-dimensional shapes.
“Elementary algebra is 0-dimensional because it is about numbers. Our algebra is 2-dimensional and it allows us to look at patterns across multiple consecutive dimensions,” noted Johnson.
"Some physical applications treat time as a fourth dimension, and we have higher-dimensional data whenever we study something that has more than three separate properties,” explained Johnson. “For example, if you use 12 different statistics for your fantasy sports team, you are using 12 dimensional data."
The team’s most recent publication proves that the algebra they’ve developed captures all of the essential information in 2-dimensional slices of higher-dimensional shapes.
Their research fits into the broader subject of mathematics as the science of explaining shapes and numbers, and the things needed to understand those explanations. Johnson and his colleagues undertook the problem of explaining higher-dimensional spheres—think analogues of a beach ball, but in higher dimensions (four or more). These are the building blocks for more complex shapes, and their algebraic model gives new insight into the ways that different spheres can be attached together.
Working with Johnson are Nick Gurski, assistant professor of mathematics at Case Western Reserve University in Cleveland; and Angélica Osorno, assistant professor of mathematics at Reed College in Portland, Oregon.
The trio has been collaborating for four years and has published three papers from their algebraic modeling project. Two follow-up papers, "A model for the stable 2-type of the sphere" and "A 2-categorical group-completion" are nearing completion.
Academic collaboration is a critical component of the rich academic environment at Ohio State Newark, and it’s this strong spirit of cooperation and exploration that Johnson especially wants those outside the university to understand.
Collaborators give each other distinct perspectives on a problem with differing expertise and ideas for solutions, as well as building critical connections to other work, noted Johnson. The team meets periodically, but also continues their research individually, meeting once a week by video to check in and update each other on progress.
“Collaboration in math means working together for both understanding and explaining,” noted Johnson. “First we have to understand why something happens, or why it is the way it is, and then we have to learn how to explain that idea, using the other parts of mathematics that have already been explained.”
Johnson received his Ph.D. from the University of Chicago in 2009. His areas of expertise include algebraic topology, mathematics education and mathematical visualization. He holds a B.A. and M.A. in mathematics from the University of Rochester. He was awarded Ohio State Newark’s Scholarly Accomplishment Award in March 2018 and both the Teaching Excellence Award and Service Award in March 2016.
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