Mark Brittenham
University of Nebraska at Omaha, NE·Mathematics & Statistics
How can you tell what shape a three-dimensional space has from the inside? Topologists study 3-manifolds, spaces that look like ordinary 3D space up close but can be shaped very differently overall.
- Fields
Mathematics & Statistics
- Topics
low-dimensional topology
3-manifolds
fundamental group
- Grants
- 1 known grant, through Aug 2027
What they work on
- Investigate the L-space conjecture for toroidal 3-manifolds, which links left-orderable fundamental groups, foliations and Floer homology, using the notion of slope detection.
- Study actions of 3-manifold groups on circles, lines and trees that arise from foliations and flows, including the dynamical properties of these actions.
- Determine how group actions on one-dimensional spaces relate to the existence of left-orders, rankings of group elements preserved under left multiplication, on 3-manifold groups.
More about the research
Each 3-manifold carries an algebraic object called its fundamental group, which records the loops one can draw in the space. A group is "left-orderable" if its elements can be ranked so that multiplying any two of them on the left by the same element keeps their order. Surprisingly, whether a 3-manifold's group is left-orderable seems tied to two geometric structures. One is a foliation, a way of slicing the space into 2D layers, like the bands in metamorphic rock. The other is a flow, which describes how points move through the space. This project aims to deepen understanding of these connections.
Why it matters
This is pure mathematics. It aims to explain why algebra (ordering of groups) and geometry (foliations and flows) give matching information about 3-dimensional spaces, and to make progress on the L-space conjecture. Better understanding of these links could inform low-dimensional topology, dynamics and group theory more broadly.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Adam Levine
Duke University, NC·Mathematics & Statistics
Investigate knot concordance questions, including homology slice knots, ribbon concordance and piecewise-linear concordance of knots in homology 3-spheres, using Heegaard Floer homology invariants.
Abhishek Mallick
Dartmouth College, NH·Mathematics & Statistics
Establish new structural results for instanton, monopole and Heegaard Floer theories, which are invariants built from physics-inspired equations, in settings where spaces carry symmetries such as rotations or reflections.
Michelle Chu
University of Minnesota, MN·Mathematics & Statistics·Early-career
Study virtual properties of the fundamental groups of hyperbolic 3-manifolds — properties held by finite-index subgroups, which correspond to finite-sheeted covering spaces of the manifold.
The school
University of Nebraska at Omaha: 2 labs on PI Pages, $35M a year on research, 38 research doctorates a year.
Reach out
A start for your email. Fill in the brackets.