Groups Semigroups and Topology

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