Homepage of Ezra Jordan Aylaian

Math PhD Student at Duke University
Email: ezra.aylaian [at] duke [dot] edu

I am a second year math PhD student at Duke University. My advisor is Adam Simon Levine. I was an undergraduate at the University of Maryland (UMD), where my advisor was Boyu Zhang. I am interested in low-dimensional geometric topology and algebraic topology.

Summer Travel 2026

Graduate Student Topology and Geometry Conference, Rutgers University, April 24-26.
Spring School on Khovanov and Floer Homotopy, Nantes Université, May 18-22.
Summer School on Homotopy Colimits, University of Regina, June 22-26.
PCMI Graduate Summer School on Knotted Surfaces in Four-Manifolds, PCMI, June 28-July 18.

Research Papers

Tile Numbers of Knot Corner Mosaics, Geombinatorics, vol. 34, no. 4, April 2025, pp. 130-43. arXiv:2311.12258 [math.GT]

Research Talks

Partitions and Foliations of Punctured Space (Topological and Geometric Structures in Low Dimensions, SLMath), July 31, 2025
Partitions and Foliations of Punctured Space (LG&TBQ2, Centre de Recherches Mathématiques), June 3, 2025
Partitions and Foliations of Punctured Space (Triangle Area Graduate Mathematics Conference, Duke), February 22, 2025
Tile Numbers of Knot Corner Mosaics (Triangle Area Graduate Mathematics Conference, NC State), November 2, 2024

Expository Talks

What Is Dehn Surgery? (Graduate Student Topology and Geometry Conference 2026, Rutgers), April 25, 2026
The Volume-Preserving Diffeomorphism Group (Topology and Hydrodynamics Seminar, Duke), February 6, 2026
Structural Stability and Anosov Flows (Topology and Hydrodynamics Seminar, Duke), January 23, 2026
Floer Homology of Double Branched Covers of Alternating Links (Knot Floer Homology Seminar, NC State), December 11, 2025
Squeezing and Non-Squeezing in Real and p-Adic Symplectic Geometry (Grapevine Conference, Duke), December 9, 2025
Manolescu and Sarkar's Knot Floer Spectra (Math 690-10 Stable Homotopy Theory, Duke), December 5, 2025
C. T. C. Wall's Amazing Theorem on Non-Smoothable 8-Manifolds (Triangle Area Graduate Mathematics Conference, NC State), November 15, 2025
Foliations on Surfaces (Fluids Reading Group, Duke), November 11, 2025
Multiply-Pointed Heegaard Diagrams for Links (Knot Floer Homology Seminar, NC State), September 25, 2025
Infinity-Categories (Special Innovative Seminar, Duke), September 13, 2025
Lozenge Tilings (Statistical Mechanics Reading Group, Duke), July 7, 2025

Teaching

Fall 2025: Lab TA for Duke's Math 111L, Laboratory Calculus I.
Spring 2025: Help Room TA for Duke's Math 112L, Laboratory Calculus II.
Fall 2024: Mentored two undergraduate students through Duke Math's Directed Reading Program on metric spaces. At the end, the students gave an expository presentation on Laidacker and Poole's metric space proof of the existence of a minimal convex worm blanket.
Fall 2022: Taught UMD's Math 299R with Sam Lidz, which explored the way the philosophy of math interacts with math. Math 299R studied platonism, formalism, intuitionism, constructivism, finitism, and structuralism, the arguments for and against each, and the mathematical systems arising from them. Math 299R was featured in a news article.

Service

Organizer of the Duke Topology and Hydrodynamics Seminar, Spring 2026
Treasurer of the Duke American Mathematical Society, Fall 2025 - Present
Board Member on the Duke University Libraries' Graduate and Professional Student Advisory Board, Fall 2024 - Spring 2026
Organizer for the Duke Math Grad-Fac Seminar, Fall 2024 - Spring 2025
Undergraduate Representative on the UMD Math Department Climate Committee, Fall 2023 - Spring 2024

Honors

Duke Sloan Affiliate
UMD Math High Honors, May 2024
UMD's Milton Abramowitz Award, May 2024 and May 2023