Topology presentation shows new outlooks

A Pacific University professor was invited to discuss how manifolds are constructed and explained the different aspects of surfaces. Students, faculty, and visiting professors filled the audience. The event was a part of the weekly science colloquium series.

Bill Breslin, assistant professor of Mathematics, focused his presentation on “What Are Those Surfaces Doing in My Hyperbolic 3-Manifold?”

Topics discussed included 3-manifolds, Heegard surfaces, surface bundles and fibers, hyperbolic 3-manifolds and principal curvatures.

He began by describing what 2-manifolds are and how they are constructed.

“You can’t wander off into infinity in a compact 2-manifold,” Breslin said.

Breslin then explained how to construct a surface by using the example of cutting an octagon in half.

Breslin decided to go into topology due to liking the visualization of it and the pictures. He even displayed an image of the doodles in his research journal and also those that his 6-year-old son composed.

Before going more in depth, Breslin discussed why people study 3-manifolds. One of the reasons he included was that we live in a 3-manifold.

Breslin discussed how mathematicians like cut-and-paste topology. This idea comes from the concept of cutting these objects apart and pasting them back together.

He said that by drawing
where the loops are in a surface will help determine where the 3-manifold is.

He gave an explanation of what a 3-topus was given and shown why it is a part of 3-manifolds.

“If I looked up at the top of the ceiling, I would see my feet,” Breslin said.

The main topic that Breslin covered was hyperbolic 3-manifolds. He then described principal curvatures and how they can be changed. Breslin stated that flat curvatures equal zero and that you can change the principal curvature by isotaping it.

With this information, he began to discuss his own personal results to his research.

The main question that he tried to answer was, “What can you say about the geometry of typological important surfaces?”

Breslin found that it is possible to flatten Heegaard surfaces and fibers in hyperbolic 3-manifolds to a certain extent.

During his first year teaching at the University of Michigan, Breslin decided to figure out how close to one the constant C can be. His results found that the closest it can get is to the interval [-1,1]. He received his doctorate from University of California in Davis.

“That’s what I want to do all day is draw pictures,” Breslin said. “It’s a very rich subject. A lot comes out of it.”
Ivanna Tucker/
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