Patrick Naylor

I'm an Assistant Professor in the Department of Mathematics & Statistics at McMaster University. Previously, I was an NSERC Postdoctoral Fellow at Princeton University, and I completed my PhD at the University of Waterloo under the supervision of Doug Park.My research interests are in low-dimensional and geometric topology. My non-research interests include running, cycling, and drinking too much coffee.With Caroline Junkins and Lee Van Brussel, I am running a summer program on 3D printing for mathematics education. You can find more information under Printing.

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

Research

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 equivalent but smoothly distinct. Most of my work has been constructive, i.e., about producing diffeomorphisms, or proving that certain objects are smoothly equivalent.Image: an interactive 3D model of the Stevedore knot, a nontrivial slice knot.

Publications and preprints

Exotic definite four-manifolds with non-cyclic fundamental group (with Robert Harris and B. Doug Park). Submitted. [arXiv].
Doubling Gluck twists: a 5-dimensional approach (with David Gabai and Hannah Schwartz). Submitted. [arXiv].
Negacylic weighing matrices (with Robert Craigen, Colin Desmarais, and Ted Eaton). Submitted.
Multisections of 4-manifolds (with Gabriel Islambouli). To appear in the Transactions of the American Mathematical Society. [arXiv].
Trisections of non-orientable 4-manifolds (with Maggie Miller). To appear in the Michigan Mathematical Journal. [DOI], [arXiv].
Trisection diagrams and twists of 4-manifolds. Comptes Rendus Mathématique 360:845–866, 2022. [DOI], [arXiv].
Gluck twisting roll spun knots (with Hannah Schwartz) Algebraic & Geometric Topology 22(2), 973–990, 2022. [DOI], [arXiv].
From automorphisms of Riemann surfaces to smooth 4-manifolds (with Ahmet Beyaz, Sinem Onaran, and B. Doug Park). Math Res. Lett. 27(3), 629-645, 2020. [DOI].
Testing bi-orderability of knot groups (with Adam Clay and Colin Desmarais) Canad. Math. Bull. 59(3), 472-482, 2016. [DOI], [arXiv].

Teaching

In Fall 2024, I'm teaching MATH 4TT3/6TT3: Topics in Topology. The course will focus mainly on knot theory and the basics of 3-dimensional topology. For more information or permission to enrol, please send me an email.Image: A Seifert surface for the figure eight knot, created using SeifertView.

In Spring 2023, I taught MAT 203: Advanced Vector Calculus at Princeton University. I produced a variety of 3D-printed models for the course using a combination of OpenSCAD, CalcPlot3D and Blender/Cura. Interactive model files can be found on Thingiverse. For more information on 3D printing for mathematics education, see the page on Printing.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.

Outreach

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.

Mathematical Games: [Games 1], [Games 2], [Games 3], [Notes]
More Mathematical Games: [More Games], [Partitions], [Dwellings], [Solutions]
Knot theory: [Lesson], [Problems], [Solutions]

I love math contests, and have been involved in designing and marking some of the national contests that the CEMC has to offer.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.

3D Printing for Mathematics Education

Are you interested in 3D printing, math, and enhancing learning through accessibility? Consider applying for a unique summer work opportunity! Caroline Junkins, Lee van Brussel and I are inviting applications from students eager to contribute to an innovative project: designing 3D Printed Mathematics Manipulatives.Through this program, you will:

Gain hands-on experience with 3D printing technology.
Explore calculus concepts through practical application.
Contribute to making math education more accessible and inclusive.

The deadline for applications is Friday, February 23 at 5pm. If you are interested, submit an application through the Office of Undergraduate Research page here.

Seminars

This term, the Geometry & Topology seminar takes place on either Mondays or Thursdays at 3:30; upcoming talks can be found here.

The learning seminar for Fall 2023 took place on each alternate Monday or Thursday at 4:30. A schedule and suggested reading from Milnor's Lectures on the h-cobordism theorem can be found below. Other useful texts are Morse Theory by Milnor and §1-4 of Morse Theory and Floer Homology by Audin and Damian. An updated schedule will be posted closer to the start of the Fall 2024 term.

Date	Speaker	Subject	Reading
Sept 25	Subhajit	Morse functions, cobordisms	§0, §1,
Oct 2	Kursat, Mahan, Subhajit	The Morse lemma, existence of Morse functions	§2
Oct 9	-- Fall Break --	--	--
Oct 16	Charles Stine	Handle/Kirby calculus	--
Oct 19	Mahan, Subhajit	Existence of Morse functions	§2
Oct 23	Jeff, Mahan	Gradient flow, product cobordisms	§3
Oct 30	Kursat, Patrick	Elementary cobordisms, handles	§3
Nov 16	Patrick	Handles and Heegaard splittings	§5
Nov 23	Jeff	Handle rearrangement	§5
Nov 30	Mahan	Handle cancellation	§6

Contact

The best way to contact me is by email, at:patrick.naylor@mcmaster.ca

319 Hamilton HallDepartment of Mathematics & Statistics
Hamilton Hall
1280 Main Street West
Hamilton, Ontario, L8S 4K1

I am currently looking for masters students to supervise. If you are interested in working with me, please contact me via e-mail. Applications from members of equity-deserving groups are especially welcome to apply.I am also looking for undergraduate summer research students, who are normally supported by an NSERC USRA. You can find more information here.

(Obligatory hiking photograph)

I am privileged to work at McMaster University, located on the traditional territories of the Mississauga and Haudenosaunee nations, and within the lands protected by the “Dish with One Spoon” wampum agreement.