Alexander Zupan
University of Nebraska-Lincoln, NE·Mathematics & Statistics
Topology studies the properties of shapes that stay the same when a shape is twisted, stretched or bent, but not torn. A manifold is a space that looks flat up close.
- Fields
Mathematics & Statistics
- Topics
topology
3-manifolds
4-manifolds
- Grants
- 1 known grant, through Jun 2027
What they work on
- Prove new cases of the smooth four-dimensional Poincaré conjecture, which asks whether sphere-like four-dimensional spaces are truly standard spheres, by connecting four-dimensional handle calculus with Dehn surgery on knots and links.
- Find new relationships between the topology of certain homotopy four-spheres, spaces that resemble the four-sphere, and the Andrews–Curtis conjecture in combinatorial group theory.
- Develop connections between knot invariants in three and four dimensions, and use bridge trisections, which cut four-manifolds into three simple pieces, to build new surfaces in four-manifolds.
More about the research
The surface of a donut, for example, looks like a flat plane under high magnification. Three-dimensional spaces are now fairly well understood. Four-dimensional ones, which can be pictured as three-dimensional objects changing over time, remain full of open problems. This project looks for deep connections between dimensions three and four, and low-dimensional topology has applications in biology, chemistry, physics and quantum computing.
Why it matters
This is pure mathematics. It aims to deepen understanding of four-dimensional shapes, a setting where some of the field's most basic questions remain open, by connecting them to knots and three-dimensional spaces. Progress could clarify long-standing conjectures. Low-dimensional topology has applications in biology, chemistry, physics and quantum computing, so the ideas could eventually inform work in those fields.
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.
Joshua Greene
Boston College, MA·Mathematics & Statistics
Attack peg problems, such as whether every closed curve in the plane contains the corners of a square, using symplectic methods from the geometry of classical mechanics.
Jun Li
California State University-Northridge, CA·Mathematics & Statistics
Explore whether results on braids and Dehn twists (basic twisting symmetries) along Lagrangian spheres extend to general symplectic 4-manifolds, using surgery on Lefschetz fibrations.
The school
University of Nebraska-Lincoln: 17 labs on PI Pages, $657M a year on research, 261 research doctorates a year.
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