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K-State Today

July 18, 2016

Math professor delivers keynote at gauge theory workshop

By Sarah Hancock

David Auckly, professor of mathematics, presented "Knotted spheres, corks and equivariant cobordism" as a keynote at the Flavours of Gauge Theory Workshop in May. The workshop was hosted at the Fields Institute at the University of Toronto, widely recognized as one of the leading mathematical research institutes in the world.

Auckly's address explained how to turn a possibly nonuniform space with an interesting symmetry into a uniformly curved space with the same symmetry and fill in the difference with an even larger space. Auckly has worked with three colleagues to solve this technical problem that has existed in geometric topology since the 1970s.

Gauge theory is a branch of geometry that inhabits the border between theoretical physics and mathematics. Theories in physics typically describe physical forces such as gravity in terms of fields that transmit force between particles. The fields themselves cannot be measured, but changes, energies and velocities can. Different configurations of the fields can result in identical quantities, a phenomenon known as gauge invariance, or gauge symmetry. Theories that explain or involve gauge invariance are known as gauge theories. Auckly investigates how gauge theories can detect subtle geometric properties of the underlying space. The underlying space is always assumed to be a manifold, a space that looks like a flat space in a small neighborhood. For example, the surface of the Earth is a two-dimensional manifold because a small patch looks just like a small patch of a plane — think about western Kansas. The powdered sugar and a powdered sugar donut has the same property — in a small neighborhood, the surface looks like part of a plane. In the same way that the earth folds back on itself to form a sphere — news flash, it is not flat — space may fold back on itself into a shape. Such a shape would be a manifold, and the gauge fields might provide clues about what the shape might be.

Auckly also was an organizer of the workshop, which hosted researchers from four different countries. One of the many highlights of the conference was Tom Mrowka's talk describing how gauge theory might lead to a new proof of the four-color theorem. The four-color theorem states that any map of the plane may be colored with just four colors so that no two countries sharing a length of border will have the same color. The only known proofs reduce the theorem to a very large number of cases that must be checked, which requires a computer. A different proof that did not require a computer would be helpful.

The conference was in honor of Daniel Ruberman on the occasion of his 61st birthday. Ruberman and Auckly have been collaborating for several years.

"It was wonderful to celebrate Danny's career and see the tremendous progress in the field. I was able to meet many young mathematicians working in the field that I hope to work with in the future," Auckly said.

Auckly met with three of his research collaborators, Hee Jung Kim, Paul Melvin and Danny Ruberman, at the workshop. They will publish a paper titled "Equivariant Hyperbolization of 3-Manifolds via Homology Cobordisms" in the near future. This research group continued to meet this summer, producing even more results in the theory of 4-manifolds.

Auckly's travel was supported by a Faculty Development Award from the Office of Research and Sponsored Programs in fall 2015 and by the mathematics department in the College of Arts & Sciences. Auckly is reinvigorating his research program after serving as associate director of the Mathematical Science Research Institute in Berkeley, California, from August 2009 through August 2012.