The interplay between topology, group theory, and semigroup theory has yielded a wealth of information in all three mathematical fields. These connections are central to the research of our faculty working in this area.
Faculty
Mark Brittenham works in low-dimensional topology, a subject that focuses on spaces that have small dimension, such as surfaces, and knots (and their complements) in 3-space. His research involves studying surfaces in 3-dimensional manifolds, and the extent to which such surfaces can be used to reveal the structure of the manifold containing them. Depending upon the problem, the techniques involved can be combinatorial (using triangulations and `normal' surfaces), geometric (using hyperbolic geometry and foliation theory), or algebraic (using group actions and fundamental groups).
Susan Hermiller works in geometric group theory, which centers on the study of groups that can be viewed as nonpositively curved topological spaces. Her research interests include both the topological and computational/algorithmic properties of infinite groups. Her work on rewriting systems for groups also has close connections to the study of Groebner bases for commutative and noncommutative algebras..
Alex Zupan's research is in geometric and low-dimensional topology. In particular, he studies Heegaard splittings of 3-manifolds, trisections of smooth 4-manifolds, and interactions between these two theories. He is also interested in knots in dimension three and surfaces in dimensions three and four. His recent work has connections with the smooth 4-dimensional Poincaré conjecture and the slice-ribbon conjecture.
Román Aranda's research lies in the field of Low-dimensional topology. Román studies decompositions of 4-dimensional objects such as 4-manifolds and surfaces in 4-space and how they relate with techniques in dimension three. His research agenda explores 4-dimensional smooth topology through the theory of trisections of 4-manifolds.
Emeritus Professor John Meakin has research interests in semigroup theory and geometric group theory. Some of his particular interests are in the theory of inverse semigroups (essentially semigroups of partial symmetries of mathematical objects) and in the study of algorithmic problems in semigroup theory and infinite group theory. His work uses geometric and topological techniques as well as ideas from automata theory, formal language theory, and mathematical logic.
Current Graduate Students
Ellyn Collier, advised by Mark Brittenham
Audrey Goodnight, advised by Mark Brittenham and Susan Hermiller
Sam Lewis-Monkman, advised by Alex Zupan
Nick Meyer, advised by Alex Zupan
Kaitlin Tademy, advised by Mark Brittenham
Kathryn Van Etten, advised by Susan Hermiller
Petra Lynn Vanderhei, advised by Mark Brittenham and Susan Hermiller
Recent Graduates
Aurora DeBellevue (PhD 2023), advised by Mark Brittenham and Susan Hermiller
Ash DeClerk (PhD 2023), advised by Susan Hermiller
Andrew Quaisley (PhD 2023), advised by Mark Brittenham and Susan Hermiller
Ana Wright (PhD 2023), advised by Mark Brittenham and Alex Zupan
Jesse Moeller (PhD 2021), advised by Mark Brittenham and Alex Zupan
Vince Longo (PhD 2021), advised by Mark Brittenham and Alex Zupan
Robert Huben (PhD 2021), advised by Mark Brittenham and David Pitts
Marla Williams (PhD 2020), advised by Mark Brittenham and Alex Zupan
Katie Tucker (PhD 2019), advised by Mark Brittenham and Susan Hermiller
Corbin Groothius (PhD 2018), advised by John Meakin and Jaime Radcliffe
Maranda Franke (PhD 2017), advised by Mark Brittenham and Susan Hermiller
Muhammed Inam (PhD 2016), advised by Mark Brittenham and John Meakin
Nicholas Owad (PhD 2016), advised by Mark Brittenham and Susan Hermiller
Scott Dyer (PhD 2015), advised by Mark Brittenham and Susan Hermiller
Anne Kerian (PhD 2015), advised by Mark Brittenham
Anisah Nu'man (PhD 2015), advised by Mark Brittenham and Susan Hermiller
Nathan Corwin (PhD 2013), advised by Collin Bleak and Mark Brittenham
Melanie DeVries (PhD 2013), advised by Mark Brittenham and Susan Hermiller
Ashley Johnson (PhD 2013), advised by Mark Brittenham and Susan Hermiller