Patrick Naylor

I'm an NSERC postdoctoral research fellow in the Department of Mathematics at Princeton University. I recently completed my PhD at the University of Waterloo under the supervision of Doug Park.My research interests are in low dimensional and geometric topology, but I'm particularly interested in studying surfaces in 4-manifolds. My non-research interests include distance running, cycling, and drinking too much coffee.You can find a copy of my CV here.

Places I'll be soon

(Above: the model of a 120-cell currently hanging in the M3 atrium at UW)


I study geometric and low dimensional topology in dimensions 3 and 4. Questions about 4-manifolds are particularly interesting because of exotic phenomena: objects which are topologically the same but smoothly distinct. Most of my work has been constructive, i.e., about producing diffeomorphisms, or proving that certain objects are smoothly equivalent.I also study trisections (a decomposition of a smooth 4-manifold into three handlebodies), which are a bit like Heegaard splittings, and in particular may be described by diagrams of curves on surfaces.Image: an interactive 3D model of the Stevedore knot; the first nontrivial slice knot.

Publications and preprints

  • Doubling Gluck twists: a 5-dimensional approach (with David Gabai and Hannah Schwartz). In preparation.

  • Multisections of 4-manifolds (with Gabriel Islambouli). Submitted. [arXiv].

  • Negacylic weighing matrices (with Robert Craigen, Colin Desmarais, and Ted Eaton). Submitted.

  • Trisections of non-orientable 4-manifolds (with Maggie Miller). To appear in the Michigan Mathematical Journal. [arXiv].

  • Trisection diagrams and twists of 4-manifolds. Comptes Rendus Mathématique 360:845–866, 2022. [arXiv].

  • Gluck twisting roll spun knots (with Hannah Schwartz) Algebraic & Geometric Topology 22(2), 973–990, 2022. [arXiv].

  • From automorphisms of Riemann surfaces to smooth 4-manifolds (with Ahmet Beyaz, Sinem Onaran, and Doug Park). Math Res. Lett. 27(3), 629-645, 2020. [Journal].

  • Testing bi-orderability of knot groups (with Adam Clay and Colin Desmarais) Canad. Math. Bull. 59(3), 472-482, 2016. [Journal], [arXiv].


This term, I'm teaching MAT 203: Advanced Vector Calculus. All course information can be found on Canvas. This interactive plot was built using plotly, an open source graphing library for Python. The 3D-printed models for the course were built using a combination of OpenSCAD, CalcPlot3D and Blender/Cura. Interactive model files can be found on Thingiverse (more coming soon).

In Spring 2020, I taught Math 235 at the University of Waterloo. I produced a few videos that you may find useful: you can find them here.


I have been a presenter for Math Circles, an after-school mathematics enrichment program for high school students run by the Center for Education in Mathematics and Computing (CEMC). If you're interested, links to some of my worksheets are below.

I love math contests, and have been involved in designing and marking some of the national contests that the CEMC has to offer. I also help out with CEMC's Problem of the Month, a monthly problem for enthusiastic high school students!I have been a mentor for Camp Euclid , a 6 week mathematical research enrichment program for high school students run by David Gay at the University of Georgia.I also make high school visits to talk about mathematics and problem solving! If you're interested in having me visit your school, please get in touch.


909 Fine Hall

Department of Mathematics
Fine Hall, 304 Washington Rd
Princeton, NJ 08544

(Obligatory hiking photograph)

© Patrick Naylor 2022